LANGLANDS RECIPROCITY: L-FUNCTIONS, AUTOMORPHICFORMS, AND DIOPHANTINE EQUATIONS
MATTHEW EMERTON

Abstract. This chapter gives a description of the theory of reciprocity laws
in algebraic number theory and its relationship to the theory of L-functions.
It begins with a historical overview of the work of Euler, Dirichlet, Riemann,
Dedekind, Hecke, and Tate. It then proceeds through the development ofclass field theory, before turning to the theory of Artin L-functions, and thenof general motivic L-functions. It culminates in a description of Langlands’s
very general reciprocity conjecture, relating arbitrary motivic L-functions to
the automorphic L-functions that he defined. The reciprocity conjecture isclosely bound up with another fundamental conjecture of Langlands, namely
his functoriality conjecture, and we give some indication of the relationshipbetween the two. We conclude with a brief discussion of some recent results
related to reciprocity for curves of genus one and two.

The goal of this chapter is to discuss the theory of ζ- and L-functions fromthe viewpoint of number theory, and to explain some of the relationships (bothestablished and conjectured) with Langlands’s theory of automorphic L-functions
and functoriality. The key conjectural relationship, also due to Langlands, is that ofreciprocity. The word reciprocity is a storied one in number theory, beginning withthe celebrated theorem of quadratic reciprocity, discovered by Euler and Legendre,
and famously proved by Gauss (several times over, in fact, and many more times
since by others). The meaning of reciprocity, both in its precise technical usage andin its more general, intuitive sense, has evolved in a complicated way since it wascoined (by Legendre) in the context of quadratic reciprocity in the 18th century.
We hope to give the reader some sense of its modern meaning, but also to give thereader an indication of how this modern meaning evolved from the original meaningof Legendre.1 To this end, we have taken a quasi-historical approach to our subject
matter in the earlier parts of the chapter, although our historical review is by nomeans complete.
One of Langlands’s earliest statements regarding automorphic L-functions andreciprocity appeared in his Yale lectures (delivered in 1967, published as [Lan71]), in
which he wrote “Before beginning the substantial part of these lectures let me make,without committing myself, a further observation. The Euler products mentioned
above are defined by means of the Hecke operators. Thus they are defined in an
entirely different manner than those of Artin or Hasse-Weil. An assertion that an
Euler product of the latter type is equal to one of those associated to an automorphic
form is tantamount to a reciprocity law (for one equation in one variable in the case
The author was supported in part by the NSF grants DMS-1601871 and DMS-1902307.
1Although it only alludes to the general framework laid out by Langlands at its very end, thearticle [Wym72] of Wyman, titled “What is a Reciprocity Law?”, remains an excellent introduction
to the subtle notion of a reciprocity law in number theory. We strongly recommend it.
1

2

M. EMERTON

of the Artin L-series and for several equations in several variables in the case of the
Hasse-Weil L-series).” The title of Langlands’s lecture series was “Euler products”.
For this reason too it seems appropriate that our discussion should begin with thework of Euler, and trace the thread of ideas leading from that work to the work ofLanglands.
1. Introduction
We elaborate slightly on our goals in writing this chapter, before giving a brief
overview of its contents.
1.1. The Langlands program, and unification in number theory. As will
be clear to any reader, the view of history adopted in this note is not a neutral one.
Rather, in addition to explaining some of the mathematics related to the Langlands
program, I hope that my presentation of the historical development will convince
the reader of the essential unity of number theory as a discipline over time, despite
the changes in language, in technique, and in apparent focus. In fact, more than justunity, I hope to illustrate to the reader that the trend in number theory is towards
one of unification. Apparently disparate phenomena become related to one anotherover time, and the theories used to explain them become more overarching. The
Langlands program currently stands as one these overarching theories. Its apparent
abstraction, and air of inaccessibility to the novice, are manifestations of the grandscope of the ideas and problems (old and new) which it encompasses. I hope that
this note will help some non-expert readers overcome some of the difficulties ofappreciation that necessarily attend such a broad and general theory.
1.2. Problems in number theory. To counterbalance the abstract sentiments ofthe preceding paragraph, it is good to recall that number theory has always been,
and continues to be, driven by attempts to solve concrete problems. One of thebeauties of the subject is that these problems seem almost inevitably to lead totheoretical and structural considerations of the deepest nature, whether or not thenecessity of such considerations was apparent in the original question.
As a historical example, one might mention the law of quadratic reciprocity
itself, which as we already noted lies at the root of all the mathematics we will
be discussing. Actually, the point is illustrated even if one looks just at the early
history of the quadratic reciprocity law: in trying to prove this law, Legendre2 was
led to state (nowadays we might say “conjecture”) the theorem (proved later by
Dirichlet) on the infinitude of primes in arithmetic progressions. Conversely, as
we will recall in more detail below, Dirichlet’s proof of that result itself relies on
quadratic reciprocity (which had been proved in the meantime by Gauss, along
different lines to the one pursued by Legendre) — and so these two seemingly
distinct threads of research are intertwined (one example of unity in number theory).
As more recent examples, one can think of the modularity conjecture for elliptic
curves, or the Sato–Tate conjecture, originally formulated on their own terms, but
now seen as instances of the general conjectural framework of Langlands reciprocity.
2In fact Euler stated this result as a theorem, at least for progressions of the form 1+mN [Eulb];
this particular case is indeed known to admit a fairly direct proof. In his introductory discussionof the problem [Dir37], Dirichlet treats the statement as a generally observed phenomenon, anddoesn’t attribute it to any particular mathematician. He does discuss Legendre’s attempts toprove it, though (and does not mention Euler in this context).

RECIPROCITY AND L-FUNCTIONS

3

We hope that the occasional examples we have included, as well as the various
proof sketches that we have given, will enable the reader to see some of the concrete
mathematics underlying the generalities to be discussed.

1.3. Overview. The first half of our chapter discusses (what are now called)
abelian L-functions, beginning at the beginning, with the work of Euler, Riemann,and Dirichlet, and moving on to discuss the contributions of Dedekind and Hecke.
This initial discussion culminates in a brief presentation of Tate’s thesis. One important thread woven through this history is the notion of Euler product. Originallya discovery of Euler, beautiful but seemingly contingent, by the time one comes toTate’s adèlic (re)formulation, it is built into the underlying structure of the entire
theory.
Reciprocity was discovered in this abelian setting; from our perspective (i.e. in
the context of studying L-functions) it arises first when one attempts to compare
Dirichlet’s formula for the value L(1, χ), when χ is a quadratic character, with
Dedekind’s general class number formula. The point is that the former L-function
is “automorphic”, while (in the context of this particular comparison) the lattershould be thought of as “motivic”. A reciprocity law (in this particular case,quadratic reciprocity) is required for the comparison to be effected.
The consideration of abelian reciprocity laws culminates in the general statements of class field theory. We briefly recall these general statements, emphasizing
the formulations in terms of L-functions, and illustrating them with the special
cases of quadratic and cyclotomic extensions of the rational numbers. In anticipation of our dicussion of compatible families of `-adic Galois representations, we also
explain how compatible families of characters can be constructed from algebraic
Hecke characters.
We then turn to describing motivic L-functions in general. Historically, theseL-functions were first discovered somewhat implicitly, in conjunction with the investigations of abelian reciprocity: they arise (implicitly) in the factorization ofζK (s) into a product of Hecke L-functions for F (as in (32) below), when K/F
is an abelian extension. It was Artin who singled out the factorization of ζK (s)into a product of L-functions for arbitrary extension K/F (see (43) below), where
the L-functions now depend on (typically greater than one-dimensional) representations of (typically non-abelian) Galois groups, rather than on characters (i.e.
one-dimensional representations) of ray class or idèle class groups.
We discuss Artin’s L-functions, and then go on to discuss ζ-functions of finite
type Z-algebras (or, more generally, of finite type schemes over Z). We give a
brief overview of the theory in the case of varieties over finite fields, and then turnto the case of varieties over number fields, where we encounter the Hasse–Weil ζfunctions and their factorization into L-functions associated now to a compatible
family of `-adic Galois representations; these are the motivic L-functions — a vast
generalization of Artin’s L-functions.
Along the way, we also briefly describe the theory of local root numbers (or factors) in this non-abelian context. Although the problem of developing such atheory was considered by several mathematicians before him, it was Langlands who
ultimately solved it [Lanc].
We conclude our discussion of motivic L-functions by introducing the conjectural Langlands group. This is a hypothetical object, whose existence Langlands

4

M. EMERTON

proposed in [Lan79].3 It mediates the relationship between automorphic forms andmotives — that is, it mediates reciprocity. Although it is usually introduced firstin the automorphic context, we introduce it in the motivic context, by analogy
with the construction of compatible families of Galois characters from algebraic
Hecke characters. To provide motivation (and hopefully not too much confusion),our discussion of the Langlands group is intertwined with an explanation of theSato–Tate conjecture for elliptic curves; in particular, we explain its interpretation
as a Čebotarev density-type theorem for the image of the Langlands group.
We then turn to our discussion of automorphic forms and Langlands reciprocity
itself. We first briefly recall the basic notions related to automorphic forms, and also
Hecke’s work associating L-functions to modular forms. We then describe Langlands’s construction of automorphic L-functions in general — and observe that thevery definition is in terms of an Euler product! This already suggests (and already suggested to Langlands, as our quote above indicates) that these L-functions
could be related to the L-functions attached to motives. The reciprocity conjecture, which we state next, formalizes this suggestion. The reciprocity conjecture
is closely related to another conjecture of Langlands, his functoriality conjecture,
and we briefly recall this relationship as well. Indeed, if understood in a sufficiently
broad fashion, the latter conjecture subsumes the former.
Finally, we discuss some of the progress on the reciprocity conjecture, thoughwith no pretense at completeness.
1.4. A remark on terminology and notation. There is no precise rule for
distinguishing between the usage of the expressions “ζ-function” and “L-function”.
Roughly, one speaks of the ζ-function of a variety, and the L-function of a motive (soζ-functions can be broken up into products of L-functions, analogously to the waythat varieties can be decomposed into motives). In the automorphic context, one
speaks primarily of L-functions (although Tate uses the notation ζ in his thesis).
The reader unfamiliar with the nuances of contemporary usage shouldn’t worry
overly much about trying to discern any distinction between the two choices ofterminology and notation. Of course, the attribution of proper names to ζ- and Lfunctions is an even more fraught enterprise. In the discussion that follows, I have
tried to observe both tradition and contemporary practice as best I know how.
2. The early history of ζ- and L-functions — Euler, Riemann, andDirichlet
The problems that Langlands is concerned with, in his theory of automorphic
L-functions and his reciprocity conjecture, have their origins in the very beginningsof modern number theory, and I hope that it will be useful to the reader to review
some of the history of this “early modern” number theory, if only to convince themof the essential continuity between the contemporary Langlands program and thenumber theory of previous eras.
2.1. Euler. The story of ζ- and L-functions begins with the study of the series
(1)

∞
X
1

nk
n=1

=1+

1
1
+ k + ··· .
2k33Actually, we follow Kottwitz [Kot84] and Arthur [Art02] in modifying Langlands’ construction.

RECIPROCITY AND L-FUNCTIONSZ ∞A comparison with the improper integral
1

while it diverges if k = 1; indeed,
N
X
1

(2)

n=1

n

5

1
shows that (1) converges if k > 1,xk

∼ log N.

The investigation of this series for various values of k seems to date back to the very
beginning of the study of mathematical analysis in the 17th century. For example,
the Basel Problem, posed in 1650, asked for the value of the sum
1 + 1/4 + 1/9 + · · ·(i.e. the value of (1) when k = 2). Eighty-odd years later, this problem was solved
π2. His method of proof
by Euler [Eulc], who showed that the value was equal to6
was to factor the function sin z as
∞
∞
Y
Y
z 
z2 
z 
1+
=z
1− 2 2sin z = z
1−
πn
πn
π n
n=1
n=1
(via a consideration of its zeroes, and via the analogy with polynomials); expandingout the product and equating the resulting power series with the usual Taylor series
for sin z yields the formulaπ26(equate the coefficients of z 3 on either side of the equation) and, with some effort,
analogous formulas for all higher even values of k. (As is well known, one obtains
rational multiplies of π k .)In fact, in the same paper, Euler considered other analogous factorizations derived in the same manner, beginning with the factorization
∞
∞
Y
Y
z
z
1 − sin z =
1−
)2
(1 −
)2 ,
(4n
+
1)π/2
(4n
−
1)π/2
n=0
n=1
1 + 1/4 + 1/9 + · · · =(3)which yields for example the formula
π
1 1 1
(4)
1 − + − + ··· =3 5 7
4(by identifying the coefficients of z on either side of the equation), due originally to
Leibniz, as well as another (Euler’s first) proof of (3) (by identifying the coefficientsof z 2 ).
An even more consequential factorization, more directly related to the series (1),was discovered by Euler soon afterwards. Namely, Euler [Eulf] observed that thefundamental theorem of arithmetic (which states that each natural number admits
a unique factorization into a product of prime powers) leads to the identity
(5)

∞
X
1
n=1

nk

=

Y
p prime

1+

Y

1
1
1
+ 2k + 3k + · · · =kpppp prime

1−

1 −1
.
pk

As an immediate consequence, by considering the case k = 1, Euler deduced that
there must be infinitely many primes, since a minimal necessary condition for the
product over p to diverge (as it must when we set k = 1, since the sum over n does)is that it involve an infinite number of factors. More quantitatively, one may take6

M. EMERTON

logarithms of both sides of this formula when k = 1; taking into account (2), one
deduces thatX
1
(6)
∼ log log N.
pp prime ≤N

Euler obtains an analogous factorization of Leibniz’s formula (4), to the effect
that
Y
Y
π
1 −1
1 −1
= .
1+
(7)
1−
p
p
4
p prime ≡1 mod 4

Taking logarithms, one finds that
X
1
(8)
p
p prime ≡1 mod 4
p≤N

p prime ≡−1 mod 4−

1p
p prime ≡−1 mod 4
Xp≤N

converges as N → ∞. Comparing this with (6), one finds that each of the sums
appearing in (8), considered individually, diverges as N → ∞, and thus concludes
that there are infinitely many primes congruent to each of ±1 mod 4.4
In his paper [Eula], Euler attempted to evaluate (1) at odd values of k. He was
unsuccessful at this, but instead obtained generalizations of the formula (4), to theeffect that
11
(9)
1 − k + k − · · · = a rational multiple of π k
35when k is an odd natural number. In the subsequent paper [Eule], Euler studied theseries (1), as well as the series appearing in (9), at negative integral values of k. Of
course these latter series diverge, but Euler assigned them values via a regularizaP∞tntion technique (“Abelian summation”): he considered the series n=1 (−1)n−1 k
n
P∞
t2n+1
n
and n=0 (−1)
as functions of t (functions which can be explicitly de(2n + 1)ktermined using formulas for geometric series and their derivatives, if k ≤ 0), andthen let t → 1.5 Euler then found explicit formulas relating the resulting valuesof (1), as well as the series appearing in (9), at k and at 1 − k, and he conjectured
that these formulas hold for arbitrary values of k. (And he rigorously verified his
conjecture at k = 1/2 for the series (1), and verified it numerically at additional
values of k.)
2.2. Riemann. Riemann [Rie] took up the investigation of the series (1) where
Euler left off. He introduced the notation
∞
X
1
1
1
(10)
ζ(s) == 1 + s + s + ··· ,
s
n
2
3
n=1
4 Much of Euler’s results on what are now called Euler products are also discussed in Chapter
15 of his text [Euld]. The study of the particular difference appearing in (8) is the subject of his
paper [Eulb]. This method of argument was vastly generalized by Dirichlet, as we recall below.
5Note that 1 − 1/2k + 1/3k − 1/4k + · · · = 1 − 21−k  1 + 1/2k + 1/3k + 1/4k + · · · ); thus
one may regard this series as being obtained from the series (1) via a modification of the factor at
p = 2 in the product formula (5). For a modern explication of Euler’s method, and an indicationof its relationship to more contemporary ideas in number theory, the reader can consult e.g. thefirst part of the paper [Kat75] of Katz.

RECIPROCITY AND L-FUNCTIONS

7

and considered ζ(s) as a function of a complex variable, initially for <s > 1 (where
the series (10) converges absolutely), but then for all s, after effecting a meromorphic continuation. Indeed, he showed that ζ(s) is holomorphic on C \ {1}, and has
a simple pole at s = 1, with residue 1.6 He also established the famous functional
equation for ζ(s), namely that1−sss1 − s
(11)
π− 2 Γ
ζ(s) = π − 2 Γ
ζ(1 − s).
22This recovered the results of Euler for integral values of s already mentioned, andamounts to Euler’s conjectural generalization of those results (for arbitrary complex
values of s!).
Riemann gave two proofs of the analytic continuation and functional equationof ζ(s). His second proof established a connection between ζ(s) and the Jacobi theta
function, beginning an intimate relationship between the theories of L-functions andof automorphic forms which continues to this day.
Jacobi’s theta function is defined by the formula
∞
∞
X
X
2
2
ϑ(τ ) =eπin τ = 1 + 2eπin τ = 1 + 2q + 2q 4 + 2q 9 + · · · ,
n=−∞

n=1

πiτ

where q = e
and =τ > 0 (equivalently, |q| < 1). An application of the Pois2
son summation formula to the Gaussian e−x (which is essentially its own Fouriertransform) yields the functional equation
r
τ
−1 
=
ϑ(τ ).
(12)ϑτ
i
Combining this with the integral formulaZ i∞s
ϑ(τ ) − 1 τ s/2 dτ
− 2s
(13)
π Γ
ζ(s) =22iτ0yields both the analytic continuation and functional equation.7
We can rewrite the formula (5) asY(14)
ζ(s) =(1 − p−s )−1 (<s > 1);
p primethis is now referred to as the Euler product formula for ζ(s). Riemann famously
employed (14) to study the distribution of the prime numbers (building on Euler’sapplication of it to reprove the infinitude of the set of primes). His ideas were further
developed by Hadamard and de la Vallée Poussin to prove the Prime Number
Theorem (that the number of primes ≤ x is asymptotic to x/ log x; note that thiscan be regarded as a strengthening of Euler’s asymptotic (6), since it immediately
implies that result). A fundamental conjecture of Riemann is that the zeroes ofζ(s) (other than those at the negative even integers, which were already discovered
6If one considers the function η(s) = (1 − 21−s )ζ(s) = 1 − 1/2s + 1/3s − 1/4s + · · · , then η(s)is holomorphic on the entire complex plane — the zero at s = 1 of the factor 1 − 21−s cancels
out the simple pole of ζ(s). The determination of the residue at s = 1 of ζ(s) is then seen to beequivalent to another formula of classical analysis, namely that 1 − 1/2 + 1/3 − 1/4 + · · · = log 2.
7The functional equation (12) lets us rewrite the integral appearing in (13) in the form
R
1−s 
s
ϑ(τ )−1
1+ ii∞( τi ) 2 + ( τi ) 2 dτ
, from which the meromorphic continuation and funcs(s−1)
2τtional equation both immediately follow.

8

M. EMERTON

by Euler) all lie on the line <s = 12 — this is the famous Riemann Hypothesis. The
crux of both Hadamard and de la Vallée Poussin’s proofs of the Prime Number
Theorem is a weaker result in the direction of the Riemann Hypothesis, namelythat ζ(s) is zero-free on the line <s = 1.
We remark that the two properties of ζ(s) that we have discussed — (i) its analytic continuation and functional equation (which are mediated via its relationshipto the automorphic form ϑ(τ )), and (ii) its Euler product — are completely unrelated in Riemann’s treatment of them. This apparently technical point is in fact ofa more than merely technical nature; it is crucial, and we will return to it below.
2.3. Dirichlet. Dirichlet introduced what are now called Dirichlet L-functions in
his celebrated investigation of primes in arithmetic progressions [Dir37].
Following contemporary notation,8 if χ : (Z/N Z)× → C× is a character (i.e. ahomomorphism between the indicated multiplicative groups), then we writeYX
χ(p) −1
χ(n)
=
1− s
,
(15)
L(s, χ) =snpp primen≥1
n coprime to N

p-N

where the validity of the Euler product follows from the multiplicative propertyof χ. Dirichlet cites Chapter 15 of Euler’s text [Euld] as an inspiration for his
consideration of such Euler products.
If χ is not primitive, i.e. if χ can be written as a composite (Z/N Z)× →ψ

(Z/M Z)× −→ C× for some proper divisor M of N and some character ψ (morecolloquially, if the period of χ is a proper divisor of N ), then L(s, χ) and L(s, ψ)coincide, up to a finite number of factors in their Euler product related to primes
that divide N but not M . Thus we may, and typically do, restrict attention toprimitive L-functions, i.e. those L(s, χ) for which χ is primitive.
Note that if N = 1, so that χ is trivial, then L(s, χ) = ζ(s), which has asimple pole at s = 1. On the other hand, it is easily seen that L(s, χ) extends
holomorphically over the entire complex plane if χ is non-trivial.9 Dirichlet thenproves that L(1, χ) 6= 0 if χ is non-trivial. Passing to logarithms and taking into
account the Euler product, he obtains the result thatX χ(p)(16)
p
p primeconverges if χ is a non-trivial character. Combining this with (2) (the corresponding
statement for the trivial χ), an application of finite Fourier theory shows thatP
1
diverges if (a, N ) = 1, proving in particular that there are infinitelyp≡a mod N
pmany primes in the arithmetic progression a + mN.
Dirichlet proves that L(1, χ) 6= 0 (for non-trivial χ) by first considering the sumof the expressions (16), namelyPX
X χ(p)Xχ χ(p)(17)=.
pp
χ a char.
mod N

p primep prime

8This notation differs from that of Dirichlet, who wrote simply L, or L with a subscript if hewished to indicate a particular character.
9We mention, though, that Dirichlet only considers s as a real variable.

RECIPROCITY AND L-FUNCTIONS

9

The orthogonality relations for characters imply that this series has non-negative
terms, and hence if it diverges, it must diverge to +∞. On the other hand, each
expression (16) is either finite (if L(1, χ) is finite and non-zero), diverges to +∞(which happens precisely when χ is trivial), or diverges to −∞ (if L(1, χ) = 0).
Making an analysis of the rates of convergence, one finds that at most one L(1, χ)
can possibly equal zero (since the sum of at least two divergences to −∞ would
overwhelm the divergence to +∞ that comes from the contribution of the trivial
character, contradicting the positivity already noted). Since L(1, χ) = L(1, χ), thisshows that L(1, χ) 6= 0 if χ is a genuinely complex character (i.e. if χ 6= χ).
If χ is a real- (i.e. ±1-) valued character (one also refers to such characters as
quadratic), then Dirichlet’s proof that L(1, χ) 6= 0 is more involved, and is numbertheoretic in nature — if χ is a primitive quadratic character mod N , then he shows[Dir]√ that L(1, χ) is a certain positive multiple of the class number of the field
Q( ±N ) (here the sign is chosen equal to χ(−1)).10 Since this class number is
non-zero, so is L(1, χ).
As a special case, when N = 4 and χ is the unique non-trivial character, herecovers Leibniz’s formula (4); from Dirichlet’s point-of-view, this formula can beinterpreted as showing that Q(i) has class number 1.

3. Dedekind ζ-functions and Hecke L-functions
Over the course of the 19th century, algebraic number theory emerged as a topic
of central importance, and many preceding preoccupations of number theorists(such as the theory of Diophantine equations, and the theory of quadratic forms)
became to a large extent absorbed into the general apparatus of algebraic number
theory. This is in particular true of the theory of L-functions.
Two of the principal contributors to these developments, both the development ofalgebraic number theory in general and the positioning of the theory of L-functions
at the centre of it, were Kummer and Dedekind. Lack of space prohibits us fromdiscussing Kummer’s seminal contributions in any detail (though we refer the reader
to the review [Maz77] for a wonderful overview), and we instead limit ourselves torecalling Dedekind’s definition of the ζ-function of a number field, and the statement of his class number formula. This formula in a sense generalizes Dirichlet’s
formula for the special value L(1, χ) when χ is a quadratic character, but there is afundamental point that has to be understood to make precise the sense in which thisis true; indeed, as we already remarked in the introduction, comparing these two
formulas is a first example of reciprocity, and we discuss it more fully in Section 4.1
below.
In this section, after briefly recalling Dedekind’s contributions, we move onto thekey (for our purposes) further developments of the analytic theory of L-functions in
the first half of the 20th century — first those of Hecke (whose results generalized
those of both Riemann and Dirichlet from the field Q to a general number field),
and then those of Tate, as famously developed in his thesis.
10Dirichlet actually employs the older language of quadratic forms, due to Legendre and Gauss.
The transition to the language of algebraic number theory comes about in the work of Kummer
and Dedekind; see e.g. this MathOverflow answer by Lemmermeyer [hl]. See also (24) below for
the explicit form of Dirichlet’s result.

10

M. EMERTON

3.1. Dedekind. If K is a number field (i.e. a finite extension of Q), with ring ofalgebraic integers OK , then we may imitate the definition of the Riemann zetafunction, and define a ζ-function attached to K, via the formulaζK (s) =

X
a6=0

1
;
N (a)shere the sum is over all non-zero ideals a ⊆ OK , and N (a) denotes the norm of a, i.e.
the order of the (finite) quotient ring OK /a. We call this the Dedekind ζ-functionof K. If we take into account the unique factorization of ideals into products ofprime ideals, along with the multiplicativity of the norm, we find that ζK (s) admits
an Euler product(18)ζK (s) =

Y

1−

p

1
)−1 ;
N (p)shere the product runs over non-zero prime ideals (equivalently, maximal ideals)of OK .
We remark that the series and Euler product converge, and hence these formulas
make sense, when <s > 1, and indeed Dedekind regarded ζK (s) as a function on
the halfline s > 1.11
Following Dirichlet’s arguments for evaluating L(1, χ) for quadratic characters χ,
Dedekind proved the famous class number formula(19)

lim (s − 1)ζK (s) =s→1

2r1 (2π)r2 Rh
p
.
w |D|

where r1 and r2 denote the number of real and complex places of K respectively,
h and R respectively denote the class number and regulator of K, w denotes thenumber of roots of unity contained in K, and |D| is the absolute value of thediscriminant of K.
Subsequently, Landau [Lan03] proved that ζK (s) has an analytic continuation to
the left of <s = 1, holomorphic except for a simple pole at s = 1 (whose residue isthen computed by (19)). He was then able to follow the method of Hadamard–dela Vallée-Pousin to establish the Prime Ideal Theorem (that the number of prime
ideals of norm ≤ X is asymptotic to X/ log X), which is the number field analogueof the prime number theorem.12
It is natural to consider number field analogues of Dirichlet’s theorem on primes
in arithmetic progression as well. Already Dedekind suggested considering (what
11Dedekind developed the theory of ζ (s) — which he denotes Ω(s), in line with his notational
K
choice of Ω for the number field that we are denoting K — in §184 of his Supplement XI to
Dirichlet’s lectures on number theory [Ded].
12 We recall one particular consequence of this theorem. Suppose that K/F is a Galois extension of number fields. As is well-known, 100% (in the sense of density) of prime ideals of K
are split completely over F . (A non-split prime q of K lying overP some prime p of F has norm atleast N (p)2 , and so the contribution of these primes to lims→1 q N (q)−s is actually convergent.)
Since there are [K : F ] primes of K lying over a given prime of F that splits in K, we then deduce— by comparing the prime ideal theorems for F and for K — that the primes in F that split
completely in K have density 1/[K : F ] among all the primes in F . This proves the Čebotarev
density theorem (see Subsection 5.3 below) for the trivial conjugacy class in Gal(K/F ).

RECIPROCITY AND L-FUNCTIONS

11

we would now call) L-functions
L(s, χ) =X χ(a)
a6=0

N (a)s,where χ is a character of the class group of K [Ded, §184]. One might hope to use
such L-functions to prove a result analogous to Dirichlet’s theorem on primes inarithmetic progression, to the effect that each ideal class contains infinitely many
prime ideals. This idea was taken up by Hecke.
3.2. Hecke. Hecke [Hec17, Hec20] proved the analytic continuation of and functional equation for Dedekind ζ-functions, and for a more general class of L-functions
X χ(a) s
L(s, χ) =,N (a)a(a,m)=1

where χ is (what is now called) a Hecke character, or equivalently a Grössencharacter,or (again equivalently, after an appropriate reformulation) an idèle class character,of conductor m.
If m is a non-zero ideal in OK , and if S denotes the finite set of prime ideals
dividing m, then a Hecke character of conductor13 m is a homomorphism of multiplicative groupsχ : {fractional ideals coprime to S} → C×for which there is some continuous character of multiplicative groups14
θ : (R ⊗Q K)× → C×such that for any α ∈ K × with α ≡ 1 mod m, we have χ (α) = θ(α) (where wethink of α as lying in (R ⊗Q K)× via the natural embedding of K × into the formergroup).

If we take θ to be the trivial character (so that we simply insist that χ (α) = 1
if α ≡ 1 mod m), then we obtain the notion of a ray class character of conductor m.
These are precisely the Hecke characters of finite order.
The multiplicative property of a Hecke character χ ensures that Hecke’s Lfunctions again admit an Euler productYχ(p) −1
L(s, χ) =1−.
N (p)sp6∈S

3.2.1. Example. If K = Q, then conductors are either of the form N or N ∞(where N > 0 is a natural number, and ∞ denotes the embedding Q ,→ R).
Ray class characters of conductor N are precisely the even primitive charactersχ : (Z/N Z)× → C× , while those of conductor N ∞ are the odd primitive charactersχ : (Z/N Z)× → C× . (We say a character is even or odd according to whether
13In fact, the notion of conductor should be broadened in the usual way: a conductor m should
consist of a non-zero ideal (the finite, or non-archimedean part of the conductor) together withany subset of the set of real embeddings of K (the infinite, or archimedean, part of the conductor).
If σ : K ,→ R is a real embedding, and if α ∈ K × , then the condition α ≡ 1 mod σ is interpreted
as meaning that σ(α) > 0 (i.e. that σ(α) and 1 lie in the same connected component of R× ).
14Note that R ⊗ K is just a more functorial way of writing Rr1 × Cr2 , and so we are simplyQ
considering characters (R× )r1 × (C× )r2 → C× . The canonical embedding of K into R ⊗Q K (andhence of K × into (R ⊗Q K)× ) is just a functorial way of describing the embedding of K into
Rr1 × Cr2 ∼= Rr1 +2r2 that is traditionally considered in algebraic number theory.

12

M. EMERTON

χ(−1) = 1 or −1.) Thus in this case, Hecke’s ray class L-functions reduce toDirichlet’s L-functions.
The notion of Hecke character in this case (i.e. when K = Q) is scarcely moregeneral: any Hecke character has the form (a) 7→ χ(a)|a|s0 for some mod N character χ and some s0 ∈ C, in which case the corresponding L-function is simply
L(s − s0 , χ) — i.e. a Dirichlet L-function with the variable shifted. In particular,
the only unitary Hecke characters (i.e. those that land in the group of complex numbers of modulus 1) are products of the finite order characters that already appear
in Dirichlet’s theory with a unitary character a 7→ |a|iy of R× .
3.2.2. Example. If K = Q(i), then the ring of integers is Z[i], which is well-known
to have trivial class group. The unique prime ideal above 2 is (1 + i); all otherprime ideals (which we might call odd) have a unique generator π satisfying thecongruence π ≡ 1 mod (1 + i)3 . The character ψ : (π) → π (where the generator ischosen to satisfy the preceding congruence) is an infinite order Hecke character ofconductor (1 + i)3 . (The corresponding character θ is simply the identity characterz 7→ z on C× .) The character ψ is not unitary, but it is easy enough to modify itπ.
so as to make it so, e.g. by defining χ (π) = p
N (π)(More generally, for any number field K, we always have the one-parameterfamily of Hecke characters of trivial conductor a 7→ |N (a)|s . If ψ is any Hecke
character, then |ψ| belongs to this one-parameter family, and χ = ψ|ψ|−1/2 isunitary.)
Hecke proved the analytic properties of his L-functions by combining Riemann’s
arguments using theta functions with Dirichlet’s and Dedekind’s arguments (appearing in their proofs of the class number formula) involving sums over idealclasses. As in Riemann’s case, the analytic aspects of his theory apparently bearno relation to the Euler product structure of the L-series. Rather than writing
more about Hecke’s arguments, though, we now turn to Tate’s reworking of them,since this marked a decisive shift in the analytic theory of L-functions.
3.3. Adèles and idèles. In his thesis [Tat67], Tate gives a reformulation of thetheory of Hecke L-functions in terms of adèles and idèles. We begin by briefly
recalling the definitions of these latter objects.
b denote the profinite completion of Z, so
As usual, we let ZY
b = lim Z/N Z =
Z
Zp ,
←−
N

p

the second equality holding by virtue of the Chinese Remainder Theorem. By its
b is a profinite (and, in particular, a compact) commutative ring. We
construction, Zb ⊗Z Q, which
may extend scalars to Q to obtain a locally compact Q-algebra 15 Z
we also denote by A∞
,
and
refer
to
as
the
ring
of
finiteadèlesofQ.
We may alsoQY0∞describe AQ as the restricted product
Qp of the various p-adic fields Qp : itpQconsists of those tuples (xp ) ∈ p Qp for which xp is an integer for all but finitely
15Writing Q = S

S
1
1 b
∞
b
N >0 N Z, we find that AQ =N >0 N Z is simply a union of copies of Z, inS
1a manner analogous to that in which Qp = Zp ⊗Z Q = n≥0 pn Zp is a union of copies of Zp .

RECIPROCITY AND L-FUNCTIONS

13

many p. We then define the full ring of adèles of Q to be the product
AQ = R × A∞
Q ;it is again a locally
Y compact Q-algebra, and also admits a description as the re0
stricted productQv , where v now ranges over all places of Q, including thev

infinite (archimedean) place.
If K is any number field, then we define∞bA∞
K = AQ ⊗ Q K = Z ⊗ Z K =

Y

0

Kp ,

p

the final description here being the restricted product of the various non-archimedean
completions Kp of K (in the same sense as used above, i.e. consisting of those elements of the product for which all but finitely many of their components areintegers). We define
Y
r1
r2
∞
0
AK = AQ ⊗Q K = (R ⊗Q K) × A∞Kv ,K = R × C × AK =v

where now the restricted product is over all places of K. We refer to these two
K-algebras as the ring of finite adèles over K and the ring of adèles over K, respectively. They are both locally compact K-algebras. (As a Q-vector space, i.e.
forgetting the K-algebra structure, AK is just a product of [K : Q] copies of AQ .)If G is any affine algebraic group over K, then we may consider its group G(AK )of AK -valued points; the hypothesized Zariski closed embedding of G into some ddimensional affine space over K induces an embedding of G(AK ) as a closed subsetof AdK , giving G(AK ) the structure of a locally compact group. For now, we apply
this to the group
Gm = GL1 = {(x, y) ∈ A2 | xy = 1}.
As a group, GL1 (AK ) is simply the group of units A×
K ; the algebraic group viewpoint
is important for endowing it with the correct topology. We refer to the abelian group
A×
K , endowed with the locally compact topology just discussed (so we identify it
A2K ) as the group of idèles of K.
with the closed subset {(x, y) ∈ A2K | xy = 1} of Y0
We may also write A×Kv× consisting of elements in
K as the restricted productv

the product which are integral units at all but finitely many finite places.
More important for our purposes is not the idèle group of K itself, but theidèle class group of K. Before defining this, we note that the natural (diagonal)
embedding Q ,→ AQ has discrete image.16 Extending scalars we obtain the discrete
embedding K ,→ AK , and consequently a discrete embedding K × ,→ A×
K . The idèleclass group of K is defined to be the quotient17 K × \ A×.
We
arequotientingaK
locally compact abelian group by a discrete subgroup; the result is again a locallycompact abelian group.
16This embedding is an analogue of the more classical discrete embedding Z ,→ R, and one of
the technical jobs that the adèles perform is to mimic this latter embedding, while neverthelessallowing one to work with structures over Q — such as Q-vector spaces, or groups like GLn (Q),rather than having to contend with more general abelian groups, or groups like GLn (Z), whose
technical properties are typically not as pleasant.
17It is traditional at this point, if disorienting for the uninitiated, to write the group by which
we are taking the quotient on the left.

14

M. EMERTON

In fact, the group K × \ A×
K is very close to being compact. Namely, the productformula for global fields shows that the surjective homomorphismY0| | : A×Kv× → R×
>0
K =
v

Q

defined by (xv ) 7→ v |xv |v (where | |v is an appropriately normalized absolutevalue on Kv× ) contains K × in its kernel. If we let A1K denote this kernel, thenK × \ A1K is compact,18 and we have the short exact sequence×
1 → K × \ A1K → K × \ A×
K → R>0 → 1.

The first key reinterpretation of Hecke’s theory in terms of these objects is thata Hecke character is the same thing as a continuous character×χ : K × \ A×
K →C .
××
Indeed, composing χ with the inclusion Kv× → A×
K → K \ AK yields a continuous××
character χv : Kv → C , and we may factor χ as
Y
(20)
χ=
χv
v

The continuity of χ ensures that for all v outside a finite set S (which we now take toinclude the infinite places) the character χv is unramified, in that it factors through×— i.e. χv (x) depends only on the v-adic valuation of x.
the quotient Z = Kv× /OKv
We now define a character19 as an idèle class character via the discussion ofχ̃ : {fractional ideals coprime to S} → C×
via
a=

Y
p6∈S

pnp 7→Yχp (πp−1 )np ,p6∈Swhere πp is some choice of uniformizer in Kp (it doesn’t matter which). If we let θ
denoteYχv : (R ⊗Q K)× → C× ,v infinite

then the continuity of χ will ensure that χ̃ satisfies the requirements to be a Hecke
character with respect to the character θ and some conductor m involving just theplaces in S. Note in particular that although the definition of χ̃ only involves theχv for v 6∈ S, we may recover χ, and in particular the remaining χv for v ∈ S,
from χ̃, using the fact that χ is continuous and is defined on the quotient K × \ A×K.
The finite order idèle class characters correspond to Hecke’s ray class characters.
These necessarily factor through a finite quotient of K × \ A×
K . In fact, the group××
of connected components π0 (K × \ A×
)
of
K
\
A
is
naturallya profinite group,KK
and may be described as the inverse limit, over all conductors m, of the ray classgroups of conductor m. The connected component of the identity in K × \ A×
K is the×
product of a copy of R×withtheconnected
componentoftheidentityinK\A1K ,>018This is a reformulation of two key theorems of elementary algebraic number theory, namely
the finiteness of the class group and Dirichlet’s unit theorem.
19 The exponent −1 on π is chosen to ensure a compatibly between various conventions andpconstructions; e.g. if we identify a character χ : (Z/N Z)× → C× with an idèle class character
via the discussion of Example 3.3.1, then the character χ̃ coincides with the ray class character
associated to χ via the discussion of Example 3.2.1.

RECIPROCITY AND L-FUNCTIONS

15

this latter group being a connected compact group. It is trivial in the simplest caseof K = Q, but is otherwise non-trivial.
3.3.1. Example. One easily computes that×b×
Q × \ A×
Q = R>0 × Z .

This recovers the description of Hecke characters for Q given in (3.2.1) above.
3.3.2. Example. It is also easy to compute that××
3d
Q(i)× \ A×
Q(i) = C × {x ∈ Z[i] | x ≡ 1 mod (1 + i) }.

(More precisely, there is an obvious map from the right-hand side to the left, which is
an isomorphism.) In these terms, the Hecke character ψ of (3.2.2) may be described
simply as the projection onto the C× factor.
√
√
2)
has
class
number1,
that
1+
2 is a funda3.3.3. Example. Recalling
that
Q(
√
√
mental unit of Q( 2), and that N (1 + 2) = −1, one finds that
√
√ ×
√ 2
\
×
√ = h(1 +Q( 2)× \ A×2) i \ R×.
>0 × R>0 × Z[ 2]Q( 2)In particular, the connected component of the identity is equal to√
lim h(1 + 2)2n i \ R×
× R×
,
>0
>0
←−
n

which is the product of R×>0 with a solenoid (the inverse limit of an inverse systemof circles with respect to non-trivial covering maps).
3.4. Tate. In his thesis [Tat67], Tate reproved Hecke’s results on L-functions ofHecke characters, using the formalism of the idèles and idèle class characters. As
in Riemann’s and Hecke’s arguments, he uses an integral transform to obtain theanalytic continuation and functional equation. However, the integral is now takenover the group of idèles A×
K (from this point of view, the integral in Riemann’s
formula (13) should be viewed as taking place over the purely archimedean group
R×>0 ), and the application of Poisson summation that underlies the proof of thefunctional equation is now performed with respect to the discrete group K ⊂ AK .
The fact that the integral is taken over A×
K has a crucial consequence: the Euler
product for the Hecke L-functions appears naturally, as a part of the same sequenceof arguments that proves the analytic continuation and functional equation!
In a little more detail:20 Tate considers a unitary idèle class character χ, andthen considers integrals of the formZsζ(f, χ| | )) =f (x)χ(x)|x|s d× x,
A×K

where f is a function on AK which decays suitably at infinity (i.e. belongs to an
appropriately defined Schwarz space of functions on AK ), and d× x is a suitably
20We sketch the contents of Tate’s thesis here, since it played, and continues to play, such a

key role in the development of the the theory of automorphic L-functions. For more details, we
encourage the interested reader to study the thesis itself, as well as Weil’s important Seminaire
Bourbaki [Wei95], which gives a distribution-theoretic interpretation of Tate’s arguments; suitable
extensions of the ideas in Weil’s lecture are basic to the contemporary analysis of integral formulasin the theory of automorphic forms. There are also many excellent commentaries on Tate’s thesis
available; among these, we particularly recommend the article [Kud03] of Kudla.

16

M. EMERTON

normalized Haar measure on the idèle group A×
K , obtained as a product of Haar
measures d× xv on each of the multiplicative groups Kv× .
If the function f factors as a product f = ⊗v fv of (appropriately Schwarz)functions fv on each Kv , then Tate’s ζ-integral admits a factorization
Z(21) ζ(f, χ| |s ) =f (x)χ(x)|x|s d× x
A×
K

=

YZ
v

Kv×

fv χv (xv )|xv |sv d× xv =Yζv (fv , χv | |sv ),v

where the local ζ-integrals are defined according to the indicated formula. Directly
evaluating these local ζ-integrals, and then (via the factorization (21)) analyzing
the global ζ-integrals, Tate finds that the local ζ-integrals converge for <s > 0 —this directly corresponds to the fact that the geometric series
∞
X
1 −1
1
= 1− snsppn=0
defining the Euler factors in (5) converge when <s > 0 — and that the global
ζ-integrals converge for <s > 1 — directly corresponding to the fact the Euler
product (5) itself converges21 precisely when <s > 1.
Tate establishes two crucial results. The first, local, result, concerns the ratio(22)ζv (fv , χv | |sv ),ζv (fˆv , χ | |1−s )vwhere fˆv denotes the additive Fourier transform of fv (with respect to some chosen
additive character ψv ); note that both integrals are defined when 0 < <s < 1, andso this ratio makes sense (provided that the denominator does not vanish). What
Tate shows is that the ratio (22) is in fact independent of the choice of the function
fv , and analytically continues to a meromorphic function of s. This is proved byan elementary manipulation of integrals.
The second, global, result, states that ζ(f, χ| |s ) has a meromorphic continuation
to the entire complex plane, and satisfies a functional equation
ζ(f, χ| |s ) = ζ(fˆ, χ| |1−s );here fˆ denotes the additive Fourier transform of f , computed with respect to an
additive character of AK which is chosen to be trivial on K. The proof of thisresult follows Riemann’s proof of (11) via the integral transform of a theta series;
in particular, as already remarked, Poissson summation (but now with respect tothe discrete subgroup K of AK ) plays a key role.
By choosing f = ⊗v fv to be (essentially) Fourier self-dual (so at an infinite
place, fv will be an appropriate Gaussian — and so we connect back to Riemann’s
proof via Jacobi’s ϑ — while at a finite place, fv will be the characteristic functionof OK ), we find that ζ(f, χ| |s ) is essentially Hecke’s L-function L(s, χ). In fact,
we essentially obtain the completed L-function Λ(s, χ) — i.e. with the appropriate
21It is important that χ be unitary for these bounds on the region of convergence to becorrect. As we already saw in (3.2.1), and as is clear from the formulas defining the ζ-integrals,
if we multiply χ by some power ||s0 so as to make it possibly non-unitary, then this amounts
to shifting the variable s in the ζ-integral by s0 , and correspondingly translating the regions ofconvergence by <s0 .

RECIPROCITY AND L-FUNCTIONS

17

Γ factors, powers of π, etc., included — and (via (21)) we obtain it directly in
its Euler product form! We say “essentially”, because at the primes dividing the
conductor of χ the local ζ-integrals are a little more finicky (they involve Gauss
sum-like expressions); and we also have factors at the infinite places. When we put
everything together, we obtain Hecke’s functional equation22
Λ(1 − s, χ) = (χ)Λ(s, χ),
where the quantity (χ) which mediates the functional equation is itself written as
an “Euler product”
Y(23)
(χ) =v (χ, ψv );v∈Shere S denotes the set of ramified primes for χ, together with the infinite places,
and the local factors v arise from an explicit evaluation of the ratios of (22) at theplaces v ∈ S.23 (As the notation indicates, the local  factors depend on the additive
character ψv used to compute the local additive Fourier transforms — although inan easily understood way — whereas the global  factor does not. Indeed, since
the functional equation is what it is, regardless of how we normalize the Fouriertransforms used to establish it, the global  factor cannot depend on the choice of ψ;
more concretely, one sees that this is so via an application of the global productformula.)If we take χ to be the trivial character, Tate’s ζ-integral essentially computes the
Dedekind ζ-function ζK (s). In this optic, the class number formula (19) emerges
quite directly from a computation of the volume of the compact group K × \ A1K .
3.5. Applications to density. Given the analytic continuation of his L-series,
Hecke was able to generalize Dirichlet’s arguments (and indeed the subsequent
improvements on them by Landau) to prove the infinitude (and indeed, the appropriate density) of prime ideals in a given ray class (by working with L(s, χ) forχ running over all characters of the appropriate ray class group), and even moregeneral density theorems (by using L(s, χ) for infinite order χ), such as results on
the density of Gaussian primes lying in a given angular sector of the complex plane.
(One uses Fourier theory on C× to approximate the indicator function of the sector
in terms of characters θ on C× , and then uses a version of Dirichlet’s argument,
but applied to Hecke characters χ that extend the relevant θ.)
22 In the case of the Riemann, or more generally Dedekind, ζ-function, which in Hecke’s optic isthe case of the trivial character, one has the simplest possible functional equation: if ξK (s) denotes
the appropriately completed form of ζK (s), then ξK (1 − s) = ξK (s). For quadratic (equivalently,
real-valued unitary) characters, the functional equation again takes a simple form, namely

Λ(1 − s, χ) = ±Λ(s, χ)(a factor of ±1 is the only possibility which is consistent with the involutive nature of the symmetrys 7→ 1−s). But for genuinely complex characters, the functional equation relates Λ(s, χ) and Λ(1−s, χ); since χ and χ are now distinct, the constant in the functional equation is less constrained;a priori it is merely a complex number of absolute value 1.
23It is more correct to include the variable s in the global and local  factors as well; the factor
s/2|D|N (m)which appears in the classical completed L-function is then incorporated into the 
factors, and written as the product of its various local contributions. The  factors in our formula
are obtained by evaluating at the point of symmetry, i.e. at s = 1/2. When we evaluate the 
factors at a specific value of s in this way, it might be better to refer to them as local and global
root numbers.

18

M. EMERTON

3.6. Summary. If one unpacks Tate’s proof of Hecke’s functional equation in sufficient detail, it does not differ so much in its technical details from Hecke’s original
proof. (As one example, the step in Hecke’s argument, going back to Dirichlet’s
original analysis of his L-functions, in which one decomposes the L-function intoa sum of separate series, each indexed by ideals lying in a given ideal class, is related to the proof of the compactness of A1K /K × , in which (as we noted above) the
finiteness of the class group plays a key role.) But the difference in presentationis enormous! Tate’s integral formulas lead directly to the presentation of the Lfunctions as an Euler product, his local calculations explicate the structure of the
global root number by writing it as a product of local factors, and the class number
formula falls out as another manifestation of the basic structure of A1K /K × .
We close this discussion by a recapitulation of the role of the adèles in thisargument, and, more generally, in contemporary number theory. Like many mathematical innovations, the use of the adèles at first seems to be a merely technicaldevice, which one becomes used to over time; and which one might eventually regard as natural and even intuitive, just through habit alone. There is certainly an
aspect of this with regard to the use of the adèles, and I believe that it reflects theattitude of many, perhaps even most, students of number theory upon encountering
them for the first time. And the adèles do have many advantages which appear
purely technical at first; e.g. as we already indicated (in footnote 16 above), onesuch advantage is that we get to replace the ring Z with the field Q. It seems, then,worthwhile to reflect on what fundamental number theoretic ideas and intuitions
might be being captured in the concept of the adèles, beyond their purely technicaladvantages. We conclude with a short series of remarks in this spirit:
The product structure of the adèles, a structural product in the sense of modern
algebra, reflects in a subtle but important way the literal, arithmetic product thatappears in the statement of the fundamental theorem of arithmetic (that any natural number factors uniquely as a product of prime powers). The restricted natureof the product reflects the boundedness of the denominator of any rational number.
And even though Q is a field, and so in some sense has no intrinsic arithmetic (unlike the ring Z), the primes are still present and making their influence felt, once weconsider the embedding Q ,→ A. One might summarize these points by saying thatthe embedding Q ,→ A “externalizes” the traditional arithmetic concepts of prime
number, unique factorization, and boundedness of denominators, and reinterpretsthem in a framework which more comfortably accommodates the mores of modern
algebra, which require one to manipulate rings and fields themselves, rather thantheir individual elements.24

24One might regard the passage from elements of sets to sets themselves, in one’s view of what

is an “atomic” mathematical object, as a first step in the process of categorification, in which thepassage from sets or other mathematical objects to the categories of which they are constituents,
then to higher categories, and so on, make up the subsequent steps. The transition from Hecke’s
viewpoint to that of Tate’s thesis can then be regarded as just one step in the ongoing evolution of
number theory in general and the Langlands program in particular. Nevertheless, however muchthe ideas and problems apparently transform, the connection to the concrete mathematics, notjust of Hecke, but of Euler, Riemann, and Dirichlet, persists, since the inner nature of the number
theory remains the same.

RECIPROCITY AND L-FUNCTIONS

19

4. Class field theory
We now begin our discussion of reciprocity, by treating the so-called abelian case,which is encapsulated in the statements of class field theory. The history of classfield theory is such a vast topic that we are unable to do it any justice here.25 It
has its origins in a multitude of topics from 19th century algebraic number theory,
such as Gauss’s theory of quadratic forms, the theory of quadratic and higher
reciprocity laws (studied first by Gauss, then by Eisenstein, Jacobi, and Kummer,
among others), much of the theory of cyclotomic fields that Kummer developed in
his study of higher reciprocity, the theory of complex multiplication elliptic curves
as studied by Kronecker (among others); and also the relationship between algebraic
number theory and L-functions brought to light by Dirichlet’s proof of his theorem
on primes in arithmetic progressions. This last topic will be the starting point forour own discussion of the theory.
4.1. Comparing Dirichlet and Dedekind. As is well-known, and as we described above, the crux of Dirichlet’s proof of his theorem on primes in arithmetic
progression is the non-vanishing of the special values L(1, χ) when χ is a non-trivial
character mod N , and the difficult case of this result occurs when χ is a primitive
quadratic (i.e. real-valued, i.e. ±1-valued) character.
There is a bijection between non-trivial primitive quadratic charactersp and quadratic extensions K of Q, given by associating to χ the field K = Q χ(−1)N .
What Dirichlet shows is that
 2Rh

p
if χ(−1) = 1



 |D|
(24)
L(1, χ) =


2πh


if χ(−1) = −1,
 p
w |D|
where h, etc. are the usual invariants attached to K, as in our discussion of (19).
Indeed, a moment’s consideration shows that the right hand expression in thisformula is the value on the right hand side of the class number formula (19) (forthe particular quadratic field K at hand). Clearly, there is a relationship between
the two formulas!
Let us consider ζK (s). To reduce the number of symbols on the page, andremembering that χ(−1) is simply a sign, we write ±N in place of χ(−1)N (theparticular√ choice of sign of course being dictated by the value χ(−1)). The field
K = Q( ±N ) is Galois over Q, with Galois group of order 2; we identify its Galois
group with {±1}. For each unramified prime in K, i.e. for each p - N , there isa Frobenius element σ(p) ∈ {±1} which determines the splitting behaviour of pin K: it equals +1 (respectively −1) if p splits completely (respectively remains
inert) in K. If p is odd, then this splitting behaviour depends just on the questionof whether or not ±N is a square modp, and so we find that ±N 
σ(p) =p(the right-hand side denoting the Legendre symbol). We can now compute ζK (s)via its Euler product (18); if p is ramified, i.e. if p|N , then there is one prime of K
25For the reader interested in the history, we refer the reader to the survey of Hasse [Has67].

20

M. EMERTON

above p, having norm p; if p is unramified and split there are two such; and if p isunramified and inert then there is one prime above p of norm p2 . The corresponding
factors in (18) multiply out to either
1 −1
1 −2
1 −1
1 −1
1 −1
1− s
, (1 − s, or 1 − 2s
= 1− s
1+ s
,
p
p
p
p
p
depending on which case we are in. The latter two cases admit the uniform description
σ(p) −1
1 −1
1− s
,
1− s
p
p
and so we obtain the formula
Y
Y
1 −1
σ(p) −1(25) ζK (s) =
1− s
×
1− s
p
p
p primep primep-N= ζQ (s) ×Yp primep-N

1−σ(p) −1,
ps

where we have used (5) to rewrite the first of the two products over primes. If werecall that ζQ (s) has a simple pole with residue 1 at s = 1 (the specialization of (19)to the case of the number field Q), then we may rewrite the class number formulafor K in the form
 2Rh

p
if χ(−1) = 1



|D|

Y

σ(p) −1
1− s
(26)
lim
=
s→1

p

p prime2πhp-N
if χ(−1) = −1.
 p
w |D|
Most readers who have made it through to this point no doubt know where weare heading, and those few who don’t can surely guess: the key to Dirichlet’s
formula (24) is to identify the Euler product appearing in (26) with L(s, χ). But,
taking into account the Euler product formula of (15), this amounts to verifying
an identity of Euler productsYYχ(p) −1
σ(p) −1
(27)
1− s
=
1− s
,
p
p
p primep primep-N

p-N

which in turn amounts to verifying an identity of individual Euler factors, whichitself comes down to establishing the formula26
±N 
(28)
χ(p) = σ(p) =;pbut if we recall that χ is a primitive quadratic character modulo N , then thisformula reduces to the law of quadratic reciprocity (together with its supplementarylaws, if N is even). We sketch a proof of this law in Subsection 4.3 below.
This is the crux of reciprocity: a Diophantine quantity, such as σ(p), which we
may think of as being defined by the condition
1 + σ(p) = the number of solutions to the equation x2 = ±N mod p,
26For odd primes; we ignore the details related to the special case p = 2 when N is odd.

RECIPROCITY AND L-FUNCTIONS

21

is related to an “automorphic” quantity, in this case the character χ. (“Automorphic” because it can be interpreted as an idèle class character, i.e. as a characterof GL1 (Q) \ GL1 (AQ ).)And so we obtain an interpretation of the notion of reciprocity which goes wellbeyond the evident “switching p and N ” meaning which is at the origin of its use
in the term “quadratic reciprocity”: reciprocity is an identification of one kindof L-function, the automorphic kind with which we already have some familiarity,with a quite different kind of L-function, defined by Euler products such as the one
appearing in (26) above, which are defined not by mod N characters or the like,but in terms of Frobenius elements of Galois groups!
In the subsequent sections we turn to a detailed discussion of these new kindsof L-functions (one might reasonably call them “Diophantine”, but in fact thetraditional label is “motivic”); but before doing that, we explain how the preceding
example can be generalized; this is the content of class field theory.
4.2. Class field theory (I). If we take into account the identification of Euler
products (27), then we may rewrite the factorization (25) in the simple form(29)

ζK (s) = ζQ (s)L(s, χ).

As we have seen, it is this factorization that yields Dirichlet’s class number formula (24).27 Now the Dedekind ζ-function ζK (s) may be thought of as either automorphic or motivic, from the perspective of the field K (for any number field K): itis the Hecke L-function for the trivial character on K × \ A×
K (and so automorphic)
but it admits the Euler product
Y
1 −1
,
1−
N (p)sp
and the coefficient 1 that we have emphasized (in order to make the analogy with thecoefficients σ(p) considered above) counts the number of solutions to the (extremely
trivial!) equation x = 0 in the residue field at p. (Just to be clear, this equationhas 1 (!) solution in any field.) This makes ζK (s) motivic.
However, from the perspective of Q, the L-function ζK (s) is inexorably motivic— as we already saw in (25), its Euler factors (when we index them by rational
primes p, rather than by the primes p of K) relate to counting the number of solutions to a quadratic equation mod p. What (29) shows is that ζK (s) is neverthelessalso automorphic from the perspective of Q: it can be written as a product of Hecke
L-functions for characters of Q \ A×
Q.
It is natural to ask: for which other extensions K of Q does ζK (s) have thisproperty? More generally, we can ask: for which extensions K/F of number fields
27 We note that Dirichlet’s theorem L(1, χ) 6= 0 follows immediately from this factorization,
since we know that each of the two ζ factors have a simple pole at s = 1. It is also interesting tocompare the functional equation for L(s, χ) that one obtains from this factorization of ζK (s) withthe functional equation that one obtains directly from Hecke’s theory. Indeed, since each of ζQ (s)and ζK (s) have a sign of +1 in their functional equations, we find that also Λ(s, χ) = Λ(1 − s, χ)for the quadratic character χ; i.e. the global root number for L(s, χ) is also equal to +1. If one
compares this with the formula (23) and computes the product of the local root numbers explicitly,
one recovers Gauss’s celebrated theorem on the sign of the quadratic Gauss sum (another exampleof unity and unification!). Indeed, the theory of  factors is a modern descendent of that classicalline of enquiry by Gauss. See Subsection 5.2 below for a discussion of one aspect of this theory.

22

M. EMERTON

can ζK (s) be written as a product of Hecke L-functions with respect to idèle classcharacters of F ? Class field theory answers these questions.
4.2.1. Theorem (Main theorem of class field theory, in terms of L-functions). If
K/F is an extension of number fields, then ζK (s) may be written as a product ofHecke L-functions with respect to idèle class characters of F if and only if K/F
is Galois with abelian Galois group. Furthermore, the L-functions that appear in
the factorization are ray class L-functions, and any ray class L-function over F
appears in such a factorization (for an appropriately chosen abelian extension K
of F ).
We elaborate on the statement of this theorem. Suppose to begin with that K/F
is an abelian Galois extension, with Galois group G. We begin by generalizing
the factorization (25). If χ : G → C× is any character, let Hχ ⊆ G denote its
kernel, and let Eχ denote the subfield of K fixed by Hχ (so, by Galois theory, χ
induces an embedding Gal(Eχ /F ) ,→ C× ). Let Sχ denote the set of primes in
F which ramify in Eχ . Given a prime p 6∈ Sχ , we have a Frobenius element28
Frobp ∈ Gal(Eχ /F ), which is taken, via χ, to an element χ(Frobp ) ∈ C× . It is now
an exercise, generalizing the computations leading to (25), to show thatYY
χ(Frobp ) −1.
(30)ζK (s) =1−
N (p)sχ p a prime of F
p6∈SχFor example, the factor corresponding to the trivial character is precisely ζF (s).
∼If K/F is quadratic, then there is a unique non-trivial character χ : G −→ {±1},so (30) has just two factors. If we further assume that F = Q, then (30) reducesto (25). (The quantity denoted there by σ(p) coincides with the quantity χ(Frobp )of the present discussion.)
The first statement in Theorem 4.2.1 is then proved by establishing a generalization of the equality (27) — one proves that each of the various Euler products
appearing in (30), labelled by the various characters χ of the Galois group, is in facta Hecke L-function for some ray class character of F . In fact, one proves a morestructural statement — one shows that, in an appropriate sense, the character χ isa ray class character, in such a way that χ(Frobp ) just becomes χ(p). The Euler
products in (30) will then literally be Euler products for ray class L-functions, andthe factorization of ζK (s) as a product of such L-functions will be established.
4.3. Cyclotomic fields. We begin by explaining a special case of the preceding
discussion. Let N > 0 be a natural number, and set K = Q(ζN ). Then theirreducibility of the N th cyclotomic polynomial implies that there is a canonical∼isomorphism Gal(K/Q) −→ (Z/N Z)× , given as follows: an element a ∈ (Z/N Z)×a
is identified with the Galois automorphism which maps ζN to ζN. If p - N , so that
K is unramified above p, then one sees that Frobp corresponds to the the residue
28Typically, the Frobenius elements Frob ∈ Gal(E/F ), for a Galois extension of numberqfields, are associated to unramified primes q of the extension field E. Any two q lying over agiven unramified prime p of F are conjugate by some element of Gal(E/F ), implying that thecorresponding Frobenius elements in Gal(E/F ) are conjugate (in the sense of group theory); we
write Frobp to denote this conjugacy class (or, sometimes, some choice of element in it). If
Gal(E/F ) is abelian — which is the case for E = Eχ — then conjugacy classes in Gal(E/F ) aresingletons, so that Frobp is indeed a well-defined element.

RECIPROCITY AND L-FUNCTIONS

23

class of p itself. The characters of Gal(K/Q) are thus identified with the charactersof (Z/N Z)× .
A character χ has conductor M (i.e. is a primitive character mod M , for somedivisor M of N ) if and only if Eχ is contained in Q(ζM ), but in no smaller cyclotomic
subfield of K. One then sees that Eχ is ramified at precisely the primes dividing M ,and so, identifying the Galois character χ with the corresponding character of(Z/M Z)× , we find thatYχ(p) −1
χ(Frobp ) −1 Y=
1− s= L(s, χ).
1−
ps
p
p6∈Sχ

p-M

Combining this identification of Euler products with (30), we find thatY
ζK (s) =L(s, χ),χ a char. mod N

establishing the claim of Theorem 4.2.1 for the cyclotomic extensions K = Q(ζN )of Q. Additionally, since Dirichlet’s L-functions are precisely the ray class Lfunctions for Q, we have established the second (“Furthermore, . . . ”) statement ofTheorem 4.2.1 in the case F = Q.
Theorem 4.2.1 in general, for F = Q, then comes down to the following celebrated
theorem of Kronecker and Weber.
4.3.1. Theorem. Any abelian Galois extension K/Q is contained in Q(ζN ) forsome N .
Let us sketch a proof of this result in the case of quadratic extensions K/Q.
Firstly, we note that if χ : Gal Q(ζN )/Q = (Z/N Z)× → {±1} is a primitive
quadratic character, then Eχ is a quadratic subextension of Q(ζN ) ramified precisely
at the primes dividing N . Furthermore, it is real or imaginary depending on whether
χ(−1) = 1 or −1 (since −1 ∈ (Z/N Z)× corresponds to complex conjugation in
Gal Q(ζN )/Q ). From the explicit classification of quadratic extensions of Q, thisinformation uniquely determines Eχ , and indeed shows thatp(31)Eχ = Q χ(−1)N .
As palready noted,
as χ runs over all primitive quadratic characters, the fieldsQ χ(−1)N run over all quadratic extensions of Q. Thus we find that, indeed, ev29
ery quadratic extension of Q is contained in some Q(ζN ). Furthermore,×from (31),and the fact that Frobp ∈ Gal Q(ζN )/Q corresponds to p ∈ (Z/N Z) , we findpthat Frobp ∈ Gal Q χ(−1)N /Q = {±1} is equal to χ(p). This establishes thelaw of quadratic reciprocity (28), and illustrates how the apparently more abstract
analysis of the Galois theory of abelian extensions can be used to deduce concrete
reciprocity laws.
4.3.2. Remark. Certainly Q admits other abelian extensions than quadratic ones.
For example, since there are infinitely many primes p ≡ 1 mod 3, there are infinitely
many extensions Q(ζp ) which contain a degree 3 subextension. One might wonder,
then, why there is not a more classical reciprocity law which relates to (say) the
29A more traditional proof of this result uses quadratic Gauss sums to express

explicitly as an element of Q(ζN ).

p
χ(−1)N

24

M. EMERTON

degree 3 case of Theorem 4.3.1 in the same way that the quadratic case is related
to quadratic reciprocity.
The reason is that there is no simple family of Diophantine equations that gives
rise to a corresponding family of degree 3 abelian extensions of Q in the same waythat the family of equations X 2 = D gives rise to the family of quadratic extensions.
Indeed, while we can write down the family of cubic equations x3 + ax + b = 0, sucha cubic will typically have Galois group S3 rather than C3 . In order to have abelian
Galois group, we need to impose the additional condition that −4a3 − 27b2 be asquare. Thus parameterizing such cubics amounts to finding rational points on thesurface 4a3 + 27b2 + c2 = 0 — which is not a particularly simple parameter space.
In practice, the simplest way to describe abelian extensions of Q of degree > 2 isas subfields of appropriate cyclotomic fields.
4.4. Class field theory (II). In its more Galois-theoretic formulation, class field
theory provides an analogue of Theorem 4.3.1 for general number fields. Moreprecisely, if F is any number field, and m is any conductor in F , then one introduces
the notion of a ray class field Km of F . This is a finite Galois extension of F
satisfying the following properties:(1) Gal(Km /F ) is isomorphic to the ray class group of K of conductor m;(2) Km is unramified30 at places of F not dividing m. Further, if p - m, thenthe Frobenius element Frobp ∈ Gal(Km /F ) is identified with the class of pin the ray class group (for an appropriate choice of the isomorphism in (1),which is then pinned down uniquely by this requirement).
4.4.1. Example. The cyclotomic field Q(ζN ) is the ray class field of Q of conductor N ∞.
One then proves the following theorem.
4.4.2. Theorem (Main theorem of class field theory, in terms of abelian extensions).
For any choice of conductor m in F , a corresponding ray class field Km exists (andis unique up to isomorphism). Further, if K/F is an abelian Galois extension, then
K is contained in some ray class field of F .
If we take into account (4.4.1), then this result generalizes Theorem 4.3.1 in an
evident sense. We also see (applying (30)) thatYζKm (s) =L(s, χ),χwhere the product is taken over all ray class characters of conductor dividing m.
More generally, if K/F is a subextension of Km , and if we write H = Gal(Km /K) ⊆Gal(Km /F ), so that H is a subgroup of the ray class group of conductor m, thenY(32)
ζK (s) =L(s, χ).
χ which are trivial on H

Thus Theorem 4.2.1 follows from Theorem 4.4.2.
30Here one has to extend the notion of ramification to the infinite places: if a real place of F
lifts to a complex place of Km , we say it is ramified; otherwise we say it is unramified. Complex
places are always unramified.

RECIPROCITY AND L-FUNCTIONS

25

As the conductor m varies, the ray class fields Km form a tower of field extensions(if m|m0 then Km ⊆ Km0 ), and so we may form their union. Theorem 4.4.2 showsthat this is precisely the maximal abelian Galois extension F ab of F . Thus
Gal(F ab /F )=lim Gal(Km /F )
←
−
m

=

lim (ray class group of conductor m).
←−m

Recalling that this last inverse limit is precisely the group of connected componentsof the idèle class group, we obtain the following reformulation of the main theoremof class field theory.
4.4.3. Theorem (Main theorem of class field theory, in terms of idèles). There is∼a canonical isomorphism Gal(F ab /F ) −→ π0 (F × \ A×F ).
4.4.4. Attributions. Class field theory was first proved in general by Takagi [Tak20],
although the characterization of ray class fields that he used was different. Class
fields were first introduced by Weber, who defined them in terms of the splitting
behaviour of primes in Km /F (but could not prove their existence in general).31
In his work Takagi begins with a different definition, related to Weber’s as wellas to norms;32 we refer the reader to Hasse’s excellent historical survey [Has67]
for more details on Takagi’s point-of-view (and on all of the history that we aresummarizing here). In our formulation they are characterized by the descriptionof Frobenius elements in the Galois group (the splitting behaviour of primes thenbeing a consequence of this description).
The emphasis on Frobenius elements is due to Artin [Art27], who reformulated
the theory in terms of his famous reciprocity law. Although none of our various
formulations of the main theorems of class field theory state Artin’s reciprocity law
explicitly, it follows from condition (2) in the definition of the ray class field Km ,together with the statement that any abelian extension of a given ground field F
is contained in a ray class field.
The idelic reformulation is due to Chevalley [Che40].
4.5. Algebraic Hecke characters. Since the ray class characters of a number
field K are precisely the finite order idèle class characters, they may also be regarded
as the characters of π0 (K × \ A×
K ). Theorem 4.4.3 then gives a very direct sense to
the notion that characters of Galois groups are the same as ray class characters.
What is less obvious from Theorem 4.4.3 is that certain infinite order idèle classcharacters are also related to characters of Galois groups, in a manner which we
now explain.
Firstly, among all the charactersθ : (R ⊗Q K)× = (R× )r1 × (C× )r2 → C× ,31Suppose that K is an extension of F in which the primes that split completely are precisely
the primes that are trivial in a given ray class group. Then the result described in footnote 12,
together with Hecke’s density result discussed in Subsection 3.5, shows that [K : F ] necessarilyequals the order of the ray class group. This argument sketch may give the reader a sense of how
one can even begin to relate the splitting behaviour in a field to other of its properties.
32
The role of norms from an extension K down to F is fundamental in the study of class field
theory, although since it is not so relevant for our purposes, we have omitted any discussion of it
here.

26

M. EMERTON

let us single out those that are algebraic, in the sense that they are given by aformula (using obvious notation)npq(x1 , . . . , xr1 , z1 , . . . , zr2 ) 7→ xn1 1 . . . xr1r1 z1p1 z q11 · · · zr2r2 z rr22 ,for some integers n1 , . . . , nr2 , p1 , q1 , . . . , pr2 , qr2 . We then say that an idèle classcharacter χ is algebraic33 if the associated character θ is algebraic in the above
sense.
∼
Now let ` be a prime,34 and choose an isomorphism35 ı : C −→ Q` .
Let ψ be an algebraic Hecke character, with associated algebraic character θ :(R× ⊗Q K)× → C× . Note that (C ⊗Q K)× = (C× )r1 × (C× × C× )r2 , and so θ
extends to a character(C ⊗Q K)× → C×
via the formula (again using obvious notation)npq(x1 , . . . , xr1 , z1 , w1 , . . . , zr2 , wr2 ) 7→ xn1 1 . . . xr1r1 z1p1 w1q1 · · · zr2r2 wr2r2 .
We continue to denote this extension via θ.
The isomorphism ı allows us to regard θ as a character×(Q` ⊗Q K)× → Q` .
××
Composing this with
Q the embedding (Q` ⊗Q K) ,→ (Q` ⊗Q K) , and rememberingthat Q` ⊗Q K = λ|` Kλ , we obtain a characterYθ` :Kλ× → Q` .
λ|`Furthermore, this character is again continuous.36
×
Now define a character A×
K → Q` via the formula(33)
x 7→ ı ψ(x)θ(x)−1 θ` (x);here θ(x) is computed by projecting x to its archimedean components, while θ` (x)is computed by projecting x to its components at primes λ|`. This a continuous
character, which by its definition and the construction of θ` is trivial on K × . Furthermore, since, in the formula (33), we have shifted the “infinite order” aspects
contributed by θ from the archimedean primes to the `-adic primes of K, we seethat (33) is also trivial on the connected part of A×
K (as indeed it must be, since
33In the original terminology of Weil [Wei56b], such a character is said to be of type (A ).
034The use of `, rather than p, in this context is traditional.
35
Such an isomorphism exists, by the axiom of choice. Its precise nature is not terribly
important, and making a choice of ı is largely an expedient. Its main purpose is to allow us tomatch the collection of embeddings K ,→ C with the collection of embeddings ı : K ,→ Q` , andto do this it suffices to identify the algebraic closures of Q inside C with the algebraic closure of
Q inside Q` . Such a choice of identification is provided by ı, but is a considerably tamer piece ofdata than ı itself.
36It is here that we use the assumption that θ is algebraic. Indeed, the isomorphism ı iscertainly not continuous, since C and Q` have utterly different topological natures. But thecharacter θ is given simply by raising to various integer powers, and such homomorphism arecontinuous on any topological abelian group.

RECIPROCITY AND L-FUNCTIONS

27
×

it is continuous and takes values in the totally disconnected group Q` ). Thus (33)
defines a continuous character37×ψe` : π0 (K × \ A× ) → Q ,`Kwhich by Theorem 4.4.3 we may equally well regard as a continuous character(34)×ψe` : Gal(Q/F ) → Q` .

Thus the algebraic Hecke character ψ gives rise to a family38 of `-adic characters (34). If ψ has conductor m, then one sees from the construction that ψe` isunramified at primes p - m`, and for such a prime p, one computes39(35)
ψe` (Frobp ) = ψp (π −1 ),pwhere ψp denotes the local factor of ψ at p, and πp denotes some uniformizer
at p (the indicated expression being independent of this choice). This value isindependent of ` (it depends just on the original character ψ), and gives meaning
to the idea that the ψe` are a compatible family of `-adic Galois characters.
4.5.1. Remark. If ψ is in fact a ray class character, then θ is trivial, the character×ψ takes values in Q , and ψe` coincides with ψ (just using ı to relocate Q frombeing a subfield of C to being a subfield of Q` ). So in this case the family ψe` ise reinterpreted as a finite order Galois character
really just the single character ψ,
via Theorem 4.4.3.
If, however, θ is non-trivial, then the ψe` are truly distinct characters as ` varies(for example, ψe` will be infinitely ramified above `, but not above any other rational
prime), and the sense in which they are related is a subtle one, mediated by thecompatibility condition (35).
The notion of compatible families of `-adic Galois representations was introduced
by Taniyama [Tan57], and further developed by Serre [Ser68]. It has evolved intoa fundamental notion in number theory, and as we will see in what follows, itis the notion that underlies the definition of motivic L-functions. The preceding
construction allows us to produce compatible families of `-adic characters whose
associated L-function is just the Hecke L-function L(ψ, s); thus these are compatible
families for which reciprocity is known to hold.
×
4.5.2. Example. The absolute value character ψ = | | : Q× \ A×
Q → R>0 is algebraic,
and the associated family of `-adic characters is just the family of `-adic cycloχ` (σ)
tomic characters χ` : Gal(Q/Q) → Z×for any Galois` , defined by σ(ζ) = ζ
automorphism σ and any `-power root of unity ζ.

4.6. Brauer groups and Weil groups. Another aspect of class field theory that
we have so far not mentioned is the role of the theory of Brauer groups (over both
local and global fields). The reader can consult [Has67] for some indication of thehistorical importance of Brauer groups in the development of the subject. Fromthe modern cohomological perspective, the construction of the fundamental class in
37We use the notation ψe` just to avoid conflicting with our already-established convention, inthe particular case when K = Q, of writing ψ` for the local component of ψ at `.
38“Family”, because we may vary the prime `.
39Whether or not one includes the power −1 in the following formula depends on the normalization of the isomorphism in Theorem 4.4.3; it is adapted to our (admittedly implicit) choice of
normalization. See also footnote 19.

28

M. EMERTON
H 2 Gal(K/F ), K × \A×K is one of the pivotal steps in the entire development of thetheory. We may interpret this class as an extension of Gal(K/F ) by K × \A×K ; we callthis extension the Weil group WK/F . (See [Wei51] for Weil’s original construction,and [Tat79] for a presentation along more modern lines.)ab
Just as K × \A×K relates to Gal(K /K), the group WK/F also relates to a Galoistheoretic object. Namely, if we let Gal(Q/K)c denote the closure of the commutator
subgroup of Gal(Q/K), so that Gal(K ab /K) = Gal(Q/K)/ Gal(Q/K)c , then we
have a canonical morphism of short exact sequences of groups(36)
1/ K × \ A×

/ WK/F/ Gal(K/F )

/1

1


/ Gal(K ab /K)/ Gal(Q/F )/ Gal(Q/K)c/ Gal(K/F )/1

K

where the left vertical arrow is induced by (the inverse to) the isomorphism ofTheorem 4.4.3.
If ψ denotes an algebraic Hecke character on K × \ A×
K , then we may induce ψto obtain a representation ρ : WK/F → GLn (C), if n = [K : F ]. Correspondingly,
we may induce each of the `-adic characters ψe` of Gal(K ab /K) to obtain a representation ρe` : Gal(Q/F ) → GLn (Q` ) (which is trivial on Gal(Q/K)c ). The ρe` givean example of a compatible family of n-dimensional `-adic Galois representations.
We finish this discussion by recalling how to construct the absolute Weil groupof F . First, we recall that if E ⊆ K with E again Galois over F , then there isa natural surjection WK/F → WE/F , inducing the natural surjection on Galois
groups, and the norm map on idèle class groups. In particular, taking E = F , andnoting that by construction WF/F = F × \ A×
F , we obtain a surjection WK/F →F × \ A×that
identifies
thetargetwith
theabelianizationof its source. We thenF1find that the kernel WK/F
of
||

×
ab
WK/F → WK/F
= F × \ A×
F −→ R>0sits in a short exact sequence
1
1 → K × \ A1K → WK/F
→ Gal(K/F ) → 1,1
and since K × \ A1K is compact, so is WK/F
. Thus WK/F is an extension of R×>0 by
a compact group.
Now, just as we may take an inverse limit to form the profinite group Gal(Q/F ) =limK Gal(K/F ), we may form an inverse limit WF = limK WK/F . This is a locally←−←−1
compact group (in fact, it is an extension of R×limK WK/F, and the latter>0 by ←−
group is compact, being an inverse limit of compact groups), which admits a continuous morphism WF → Gal(Q/F ). The kernel of this morphism is connected; it
is an inverse limit, under the norm maps, of the connected parts of all the various
×
K × \A×K , and so is an extension of R by a very complicated solenoid. (See (3.3.3).)

RECIPROCITY AND L-FUNCTIONS

29

5. Motivic L-functions
Our discussion up to this point essentially completes the story of reciprocity inthe abelian case (in summary form, to be sure!). As far as chronology is concerned,
it takes us up to more-or-less the middle of the 20th century.
We must now turn to the discussion of non-abelian reciprocity, which mightalso be called non-abelian class field theory. The search for such a theory beganalready in the first half of the 20th century, but even the conjectural statements
and frameworks remained entirely elusive outside of certain special cases, and it was
only with Langlands’ letter to Weil [Lana], and his elaborations [Lan70] and [Lan71]
on this letter, that a general framework for non-abelian reciprocity was proposed.
There is no doubt that this is indeed the correct framework within which to consider
the problem, although much remains to be proved.
To begin with, we will describe non-abelian L-functions on the motivic side,
since it is these L-functions which were the first to be discovered in the non-abeliancontext. Indeed, while these L-functions exist in abundance, the absolutely fundamental difficulty in making even a conjectural statement of a general non-abelian
reciprocity law, prior to Langlands’ letter to Weil, was the seeming lack of any
kind of L-functions on the automorphic side that remotely resembled the creatures
littering the zoo of non-abelian motivic L-functions.
5.1. Artin. Suppose that K/F is a finite Galois extension of number fields, andwrite G = Gal(K/F ). Artin [Art31] was the first to observe that even when G
is non-abelian, the Dedekind ζ-function ζK (s) admits a factorization analogous
to (30).
Let ρ : G → GLn (C) be a representation. If p is a prime of F unramified in K,then the conjugacy class in G of Frobp is well-defined, and so ρ(Frobp ) is well-defined
as a matrix up to conjugation. In particular, its characteristic polynomial is welldefined. Actually, we are more interested in its reciprocal characteristic polynomial
det Idn×n − T ρ(Frobp ) , which is a degree n polynomial in T with constant term
1. We use this latter polynomial to construct an Euler factor
−1(37)
det Idn×n − N (p)−s ρ(Frobp );this expression is now the reciprocal of a degree n polynomial in the quantity
N (p)−s . In fact, if p is unramified in the splitting field of ρ (i.e. the fixed field of thekernel of ρ) then this same definition makes sense. Note that if ρ is 1-dimensional,
i.e. just a character χ, then we recover the definition of the Euler factors appearing
in the Euler product labelled by χ that appears in the factorization (30).
If p is ramified in the splitting field of ρ, then the definition of the Euler factor p
is a little more involved. It will be of degree < n, but (since n > 1 if ρ is not a character) will typically not be trivial (whereas in the abelian contexts we’ve consideredup till now, the Euler factors at ramified primes have always been trivial.) To define
it, we let V denote the vector space Cn on which G acts via ρ. We also make achoice Ip ⊂ Dp ⊂ G of inertia and decomposition group at p (again well-defined up
to conjugation). If Frobp ∈ Dp denotes a choice of element lifting the Frobenius
element of Dp /Ip , then the reciprocal characteristic polynomial of ρ(Frobp ) acting
on the invariant subspace V Ip is well-defined independent of choices, and we form
the Euler factor
−1(38)
det IdV Ip − N (p)−s ρ(Frobp )|V Ip.

30

M. EMERTONOf course, if p is unramified in K (or, more generally, in the splitting field of ρ),then V Ip = V = Cn , and so the two Euler factors (38) and (37) coincide. In
general, the quantity in (38) is the reciprocal of a polynomial in N (p)−s of degree
equal to dim V Ip .
We now define the Artin L-function of ρ as the product over all p of these Euler
factors, i.e.
Y
−1
(39)
L(s, ρ) =det IdV Ip − N (p)−s ρ(Frobp )|V Ip.
pThese L-functions are perhaps the first to appear in the literature that are truly
defined as an Euler product, rather than as a Dirichlet series which is subsequently
factored into a product over primes. The usual comparisons with ζ(s) show thatthe product defining L(s, ρ) converges to a holomorphic function on the half-plane<s > 1; and this all one can say about them to begin with.
In the case of a character χ : G → C× , class field theory allows us to identify
χ with a ray class character, so that L(s, χ) (as defined above) coincides with the
Hecke L-function L(s, χ) (as the formula (35) shows). In particular, in this case,L(s, χ) has an analytic continuation and functional equation.
One aspect of defining L-functions for general representations of G, rather thanjust 1-dimensional representations, is that the set of all representations of G has
more structure: for example, we can take direct sums of representations, or induce
representations from subgroups. (Note that both of these processes tend to output
representations of dimension > 1, and so were not visible in the context of the
L-functions of characters that we’ve studied up till now.)
One rather easily verifies the following properties of Artin L-functions:(40)

L(s, ρ1 ⊕ ρ2 ) = L(s, ρ1 )L(s, ρ2 ),and, if H ⊂ G is a subgroup and θ is a representation of H, then(41)

L(s, IndG
H θ) = L(s, θ).

This second formula looks a bit paradoxical at first, since the dimensions of thetwo representations IndG
H θ and θ itself are different (assuming H is a proper subgroup of G) — the dimension of the former is [G : H] times the dimension of thelatter, and so the Euler factors in the Euler product (at least all the unramified
primes) are of different degrees! But the Euler products are taken over different
collections of primes! The representation IndG
H θ is a representation of G, and so
L(s, IndG)
isdefinedasan
Euler
product
overprimes of F . On the other hand,Hθ is a representation of H = Gal(K/E), if E denotes the fixed field of H, and so
L(s, θ) is defined as an Euler product over the primes of E. Now [E : F ] = [G : H],
and so degree d Euler factors indexed by primes in E, if we write them in terms
of primes of F , will indeed become of degree [G : H]d. Thus we see that (41) isrelated to, indeed formalizes and generalizes, the computations we made to derive
the factorization (25).
Indeed, let us apply these formulas in the case when H = {1} ⊂ G, with θ being
the (necessarily) trivial character of H. Then the fixed field of H is just K itself,
and the Artin L-function of the trivial character equals the Hecke L-function of
the trivial character equals ζK (s). On the other hand, the induction of the trivial
character from {1} to G is just the regular representation C[G] of G, and we have

RECIPROCITY AND L-FUNCTIONS

31

the well-known decomposition
C[G] =

Mρ⊕ dim ρ ,ρwhere ρ runs over (a set of isomorphism class representatives of) all the irreducible
representations of G. Thus, applying first (41) and then (40), we find thatY(42)
ζK (s) =L(s, ρ)dim ρ ,ρwhere the product is taken over the all the irreducible representations ρ of G.
If G is abelian, then the irreducible representations of G are exactly the various
characters of G, and so (42) reduces to (30), which we now recognize as a factorization of ζK (s) into a product of Artin L-functions. Class field theory then allows
us to reinterpret this as expressing ζK (s) (for the abelian extension K of F ) asa product of Hecke L-functions for F . The factorization (29), essentially due toDirichlet, gives one of the simplest illustrations of this phenomenon, and alreadyhas far-reaching consequences (being the crux of Dirichlet’s proof of this theorem,
in so far as it leads to the non-vanishing of L(1, χ)).
What are the number-theoretic consequences of the more general factorizations (42)? Is there a non-abelian class field theory which gives us an alternativeinterpretation of the L-functions L(s, ρ) that appear in it? The answers to thesequestions (some known, most still conjectural) are provided by Langlands’s notionsof reciprocity and functoriality, to be discussed in the following section.
For now, we state the following celebrated conjecture of Artin.
5.1.1. Conjecture (Artin). If ρ is an irreducible representation of G distinct fromthe trivial character, then L(ρ, s) admits a holomorphic continuation to the complex plane and satisfies a functional equation relating (a suitably completed formof) L(ρ, 1 − s) and (a suitably completed form of) L(ρ∨ , s). (Here ρ∨ denotes thecontragredient to ρ.)
5.1.2. Example. We explain one concrete consequence of Artin’s conjecture. If
E/F is an arbitrary finite extension, let K denote the Galois closure of E, andwrite H = Gal(K/E) ⊆ Gal(K/F ) = G. If θ denotes the trivial character of H,
then IndG
H θ is a transitive permutation representation, and so contains exactly one
copy of the trivial character of G as direct summand. Thus we find thatY(43)
ζE (s) = L(s, θ) = L(s, IndG
L(ρ, s),H θ) = ζF (s)ρwhere ρ ranges over the collection of non-trivial irreducible direct summands ofIndG
H θ. Thus, if Artin’s conjecture is true, we find that ζE (s)/ζF (s) is an entire
function, or (more expressively), that ζF (s) divides ζE (s).
One approach to studying Artin’s conjecture, initiated by Artin himself andfurther pursued by Brauer, is to attempt to express representations of G in termsof inductions of characters (i.e. 1-dimensional representations) of subgroups of G.
This becomes a group-theoretic problem, via the study of which Brauer was led toprove [Bra47a] the following result.
5.1.3. Theorem (Brauer). If ρ is a non-trivial irreducible representation of G =Gal(K/F ), then L(ρ, s) is a product of integral powers of Hecke L-functions (attached to Hecke characters of various fields intermediate between F and K). In

32

M. EMERTON

particular, L(ρ, s) has a meromorphic continuation to the entire complex plane,and satisfies a functional equation of the expected form.
Artin had already proved the corresponding statement with rational rather thaninteger powers, thus obtaining an a priori multi-valued meromorphic continuation.
The integral powers appearing in Brauer’s theorem can be negative in general, andso it does not suffice to prove the holomorphic continuation that Artin conjectured.
For particular (not necessarily irreducible) ρ, variants are possible; for example, insome contexts one can obtain powers that are positive rational numbers. Combinedwith Brauer’s general theorem, this suffices to prove that L(s, ρ) is entire. E.g. inthis way Brauer proved [Bra47b] the divisibility discussed in (5.1.2) when E/F isalready Galois.
5.2. Epsilon factors. There is a subtle but important point that arises with regard
to the functional equation for L(s, ρ) that comes out of Brauer’s theorem. It indeed
has the “expected” form (as formulated by Artin — it is a natural generalization of
the form of the functional equation in the abelian case, involving Γ-factors for thearchimedean places, and a power of the discriminant of K and also of the conductorof ρ; the notion of conductor having been formulated by Artin with exactly thispurpose in mind). But the root number in the equation comes as a single quantity;
it does not appear as a product of local factors as abelian root numbers do (via (23)).
It is natural to hope that such a factorization of the non-abelian root numbers
should exist, especially if one anticipates that Artin L-functions, and Artin’s conjecture, are to be explained by a non-abelian class field theory. Versions of thisproblem were studied by Hasse [Has54] and by Dwork [Dwo56], among others; andDwork succeeded in constructing a local root number “up to a sign”, in an appropriate sense.
The difficulties in studying this problem are two-fold. Firstly, there is the theproblem of characterizing the local  factors. One imagines that they should satisfy
analogues of the properties (40) and (41); given this, one can follow Brauer’s proofof Theorem 5.1.3 and show that these properties allow one to uniquely compute thelocal root numbers, assuming they exist. The second, and seemingly more serious,
difficulty is that the product appearing in Brauer’s theorem is not at all unique,and so correspondingly, the local root numbers whose computation we have justdescribed are not at all obviously well-defined. In fact, if one adapts (41) in themost naive manner, it turns out the local root numbers can only be well-defined up
to a sign, and this explains the limitation on Dwork’s result.
The problem was subsequently taken up by Langlands, who realized that the key
to finding the correct construction of the local root numbers was to adapt (41) ina more subtle manner.40 With the correct characterization of the local  factors in
hand, Langlands proceeded, in his manuscript [Lanc], to show that they are welldefined. Again, this required dealing with the non-uniqueness in Brauer’s theorem;
the necessary calculations are famously elaborate, and make [Lanc] very long. By
employing Langlands’s characterization, Deligne soon after found a more indirect,
but shorter, local-global argument that establishes the well-definedness [Del73].
40We don’t go into the details here, but the key point is that local  factors are literally
induction invariant only for the induction of virtual representations of degree 0.

RECIPROCITY AND L-FUNCTIONS

33

5.3. Čebotarev. We recall an important consequence of Brauer’s Theorem 5.1.3.
Namely, an appropriate generalization of Hadamard and de la Vallée Poussin’s workshows that Hecke L-functions are zero-free on the closed half-plane <s ≥ 1. Thus
the ratio of Hecke L-functions appearing in Brauer’s theorem is holomorphic in
the same half-plane. This is enough to deduce an analogue of the prime number
theorem, which in this context proves that as p ranges over the primes in F (excluding the finitely many primes that ramify in K), the Frobenius elements Frobp
are equidistributed in Gal(K/F ). More precisely, since Frobp is only well-defined
up to conjugacy, the asymptotic proportion of p for which Frobp lies in a given
conjugacy class is proportional to the size of the class; in particular, there are an
infinitude of p for which Frobp lies in a given conjugacy class. Note that if K/F
is abelian, and thus contained in a ray class field, this reduces to Hecke’s theorem
regarding primes lying in a given ray class. (And the proof becomes Hecke’s proof.)
This result was first proved by Čebotarev [Tsc26], using a different argument,
building on earlier partial results of Frobenius (see e.g. the result mentioned infootnote 12 above for the simplest such precursor). Čebotarev’s argument was theinspiration for Artin’s proof of his reciprocity law.
5.4. Motivic ζ-functions. If we recall the definition of the Dedekind ζ-function
X 1, where a runs overof a number field K as a Dirichlet series ζK (s) =N (a)sa
all non-zero ideals of OK , an obvious generalization suggests itself. Namely, if
A is any ring (commutative, and with a unit), one could attempt to imitate thisdefinition so as to obtain a ζ-function attached to A, by taking a sum over all
cofinite ideals in A (i.e. ideals a for which A/a is finite; the point being that, inthe number field case, N (a) = |OK /a|). However, there is a serious defect withthis definition: ignoring for now the question of convergence of such a series (which
would be related to bounding the number of ideals a in A with |A/a| = n, as n
grows), we see that, since there is no unique factorization of ideals into prime ideals
in a general ring,41 there is no reason in general for a ζ-function, so defined, tohave an Euler product; and hence there is no reason (at least for our purposes) toimagine that this proposed definition is useful!
Indeed, we immediately abandon the momentary proposal of the preceding paragraph. Instead, we will insist that ζ-function of A (for suitable A) have an Euler
product, and we will enforce this by definition! That is, we will imitate not thedescription of ζK (s) as a series, but its description as an Euler product. Any cofinite prime ideal in A is certainly maximal, and the natural condition to place on A
(which will also lets us control the counting problem related to convergence) is thatit be finitely generated as a Z-algebra. Indeed, the Nullstellensatz then implies thatany maximal ideal of A is cofinite, and so we define
Y
1 −1
(44)
ζA (s) =1−.
|A/p|sp maximal in A

41Essentially the only interesting and general context in which we have unique factorization ofcofinite ideals into cofinite prime ideals is when A is a product of finitely many Dedekind domains
which are of finite type over Z. Working on each Dedekind domain factor separately, and soassuming that A is a Dedekind domain, we then find that either A = OK [1/a] for some number
field K and some a ∈ K, or that Spec A is a smooth curve over a finite field; we will discuss thislatter case in the following subsection.

34

M. EMERTON

Any cofinite ideal of A has a positive residue characteristic, and so (assuming somehalf-plane of convergence) we may rearrange the product in (44) to write
Y
Y
1 −1
(45)
ζA (s) =
1−
.
s
|A/p|
p max. in A
p primeof res. char. p

Each Euler factor in the product labelled by the prime number p is of the form
1 −1
, if A/p is a finite field of order pf . The difference between this general
1 − fspsetting and the case when A = OK is that there might be infinitely many suchfactors for each prime p!
There is an important geometric perspective on this construction: maximal ideals
in A are the same as closed points in Spec A. In this optic, we can generalize thepreceding construction to any finite type scheme X over Spec Z. If X is such ascheme, write κ(x) for the residue field of a point x ∈ X; then we can define42
Y
1 −1
.
(46)
ζX (s) =1−|κ(x)|sx∈X closed

Since the underlying topological space of X, and the residue fields at its closed
points, are insensitive to any nilpotent elements in the structure sheaf of X, we
may certainly assume that X is reduced when we analyze ζX (s). (In the affinecase when X = Spec A, this amounts to quotienting out A by its nilradical, and`so
supposing that A contains no non-zero nilpotents.) Also, if we write X = Y Z
as the disjoint union of two subschemes, then clearly(47)

ζX (s) = ζY (s)ζZ (s).

Since any reduced scheme of finite type over Spec Z may be stratified into a disjoint
union of integral affine locally closed subschemes, in analyzing some of the coarser
properties of ζX (s), e.g. its convergence properties, we may assume that X = Spec A
with A an integral domain.
Momentarily suppose, then, that A is an integral domain of finite type over Z.
Either A is torsion free, in which case A contains maximal ideals of residue characteristic p for all but finitely many primes p, or A is of characteristic p for somefixed prime p. In the second case, a rough count of closed points on Spec A (e.g.
via a Noether normalization to compare Spec A with an affine space of the same
dimension) shows that ζA (s) converges if <s > dim A. In the first case, we findthat Spec A/p is either empty (if p is invertible in A, which happens only for a finite
number of p) or of dimension equal to dim A − 1. Inputting the estimates from ouranalysis of the second (i.e. positive characteristic) case into the product over (allbut finitely many) p, we again find that ζA (s) converges if <s > dim A. Working
backwards through the stratification argument, we then find, for any scheme X offinite type over Z, that ζX (s) converges if <s > dim X.
42The author first learnt this definition from the note [Ser65] of Serre. The author’s impressionis that this definition developed via a complex and somewhat organic process over several decades,
beginning with the function field analogue of Dedekind ζ-functions of quadratic field in Artin’s
thesis [Art24], progressing through the work of several others, especially Hasse and Weil, until by
the 1960s, this emerged as the evidently natural definition. We strongly recommend [Ser65] tothe reader, as well as Tate’s article [Tat65], which immediately follows it in the same volume; ourdiscussion here owes much to both papers.

RECIPROCITY AND L-FUNCTIONS

35

Note that the preceding argument provides an important geometric interpretation of the first product (the product over p) in (45). If we consider the more generalcase of a finite type Z-scheme X, then we may break up the product in (46) into a
product over primes p of a product over all closed points x of residue characteristic p.
Since these latter points coincide with the closed points in the base-change X/Fp ,we find thatY Y
1 −1 Y=
ζX/Fp (s).
(48)
ζX (s) =1−|κ(x)|spp x∈X/FpclosedThis suggests that in order to analyze ζX (s) for general X, we should first consider
the case when X lies over a particular finite field, say Fp .
5.4.1. Remark. How do these considerations connect to number theory? Well, a
finite type Z-algebra A is just a quotient
A = Z[x1 , . . . , xn ]/(f1 , . . . , fr )for some variables x1 , . . . , xn , and some polynomials fi (with integer coefficients) inthese variables. Maximal ideals of A, or equivalently, closed points of X = Spec A,
correspond (more or less — we will see the precise relationship soon) to solutions
to the equations(49)

f1 = · · · = fn = 0

over some finite field Fq . Maximal ideals of fixed residue characteristic p thencorrespond to such solutions with q taken to run through the powers of some fixed p.
So ζX/Fp (s) encodes information about solutions to the system of simultaneous
Diophantine equations (49) modulo the given prime p, while ζX (s) then packages
this information together as the prime p varies.
5.5. ζ-functions for varieties over finite fields. We very briefly recall the theory of ζ-functions of varieties over finite fields. This is an enormous topic in its own
right, which is the subject of Weil’s celebrated series of conjectures [Wei49] — now
theorems of Dwork, Grothendieck (and his collaborators), and Deligne.
Suppose that X is of finite type over Fp . We first recall the precise relationshipbetween ζX (s) and counting solutions to Diophantine equations. If X is cut out bysome equations, then giving an Fpn -valued solution to the equations cutting out X
is the same as giving a morphism Spec Fpn → X, i.e. an Fpn -valued point of X — asusual, we denote this set of solutions, or equivalently, morphisms, by X(Fpn ). Sucha morphism has a closed image, say x ∈ X, and so determines, and is determinedby, an embedding k(x) ,→ Fpn . Such an embedding exists if and only if k(x) = Fpm
for some m | n, and the number of such embeddings is then equal to m (the orderof the automorphism group of Fpm ). From these remarks, we compute that(50)

log ζX (s) =∞X|X(Fpn )|
n=1

npns

.

5.5.1. Example. If X = Ad/Fp (affine d-space over Fp ), then (50) shows that
ζAd (s) =/Fp

1.
(1 − pd−s )36

M. EMERTON

If X = Pd/Fp (projective d-space of Fp ), then writing Pd/F1 as a union of Ad/Fp anda copy of Pd−1
/Fp at infinity in the usual way, taking into account (47), and arguing
inductively, we find that1
ζPd (s) =.
/Fp(1 − p−s )(1 − p1−s ) · · · (1 − pd−s )As we noted in footnote 41 above, if A is the ring of regular functions on asmooth affine curve over Fp , then A is a Dedekind domain, and in this case theEuler product defining ζA can be expanded as a sum over non-zero ideals in A, andso it is in this case that ζA is most similar to a Dedekind ζ-function. In fact, it ismore natural to consider ζX (s) for X a complete curve over Fp (this is analogous toforming the completed ζ-function by adding in the Γ-factors, etc., in the number
field case). In the number field case, the completed ζ-function conjecturally has
all its zeroes along the line <s = 1/2 (the Γ-factors absorb the trivial zeroes), andArtin [Art24] proposed the analogous conjecture for a complete curve X.
The case of elliptic curves was proved by Hasse [Has36] and the general case wasproved by Weil.43 In fact, Hasse (in the elliptic curve case), and Weil (for generalcurves), proved a much more precise statement, which we illustrate by recalling theelliptic curve case.
5.5.2. Example. If E is an elliptic curve over Fp , then there are two algebraic√
numbers α and β both of absolute value p, satisfying αβ = p, such that|E(Fpn )| = 1 + pn − αn − β n ,for any n. Thus
(1 − αp−s )(1 − βp−s ).
(1 − p−s )(1 − p1−s )Note that the Riemann Hypothesis for ζE (s) is encapsulated in the statement about
the absolute values of α and β.
ζE (s) =By examining the precise shape of ζX (s) in a range of examples for which it
could be computed (such as each of the previous examples, the case of generalcurves, and the case of diagonal hypersurfaces — this last case being discussed ina very readable form in [Wei49]), Weil was led to make his eponymous conjectures
on ζX (s) for a general smooth projective variety X over Fp . These conjectures
suggested the existence of a robust cohomology theory of such varieties, compatible
in a suitable sense with the usual (that is, singular) cohomology of (the spaces
consisting of the complex points of) varieties over C. Such a cohomology theory
was then constructed, in the form of Grothendieck’s `-adic cohomology, and, using
it as a tool, Weil’s conjectures were proved.44 Rather than recapitulating Weil’s
well-known conjectures here,45 we proceed straight to the upshot, by recalling thedescription of ζX (s) that comes out of the `-adic theory.
43The precise history of and citations for Weil’s multiple proofs are involved, and rather thangiving them here, we refer the reader to Milne’s recent excellent survey [Mil16] for a thoroughdiscussion and references.
44The rationality part of the conjectures was in fact proved first by Dwork [Dwo60], using p-adic
techniques. It was reproved by Grothendieck [AGV72] using `-adic cohomology, as was the functional equation of ζX (s). Finally, the general Riemann Hypothesis was proved by Deligne [Del74].
45
We again refer the reader to [Mil16] for more details. Appendix C of Hartshorne’s textbook[Har77] also gives a short readable summary.

RECIPROCITY AND L-FUNCTIONS

37

If X is a variety over a separably closed field k, then we can define its `-adic
cohomology groups H i (X, Q` ) for any prime ` different from the characteristicof k (i.e. if we are in positive characteristic p, we require ` 6= p). When k = C,
we recover singular cohomology with Q` coefficients. If X is defined over a notnecessarily separably closed field k, and we let X denote the base-change of X
to some separably closed field Ω containing k, then we may form the cohomology
groups H i (X, Q` ). These are in fact independent of the choice of Ω, and so it is noloss of generality to take46 Ω = k, a separable closure of k. The action of Gal(k/k)
on X then induces, by an appropriate functoriality, a continuous action of Gal(k/k)
on H i (X, Q` ).
Now if k = Fp , then Frobp generates Gal(Fp /Fp ), and so if X is a variety over Q,
we obtain an action of Frobp on each H i (X, Q` ). Now the Fpn points of X are
precisely the Frobnp -fixed points of X, and so if X is projective, we may computethe fixed points of Frobnp via the `-adic form of the Lefschetz fixed point formula.47
The upshot is that48(51)

ζX (s) =2dY(−1)i−1i;det (Id − p−s Frob−1p ) acting on H (X, Q` )i=1

here d denotes the dimension of X (or, equivalently, of X, so that the cohomology
lies in degrees between 0 and 2d). Each factor in this product is a polynomial inp−s (raised to a power ±1), of degree equal to the dimension of the corresponding
cohomology group. Thus this expression exhibits, for general X, the structure ofζX (s) that can be observed in the simple examples described above. In particular,from this formula one immediately obtains the meromorphic continuation of ζX (s).
If X is smooth as well as projective, then the cohomology of X satisfies Poincaré
duality, and this gives rise to a functional equation for ζX (s). Finally, the RiemannHypothesis in this context (i.e. for X smooth and projective over Fp ) states thaton H i are algebraic numbers all of whose conjugates
the eigenvalues of Frob−1
pi/2
have absolute value p . As Serre explained in his letter [Ser60] to Weil, thisalso admits a cohomological “explanation”, by analogy with the Hodge theory ofsmooth projective complex varieties (the particular focus of the analogy being thedefiniteness of the polarization pairings on primitive cohomology that comes out
46The point of allowing more general Ω is that if, e.g., X is a variety over Q, then we find thatthe cohomology of X base-changed to Q, which inherits a Gal(Q/Q)-action, coincides with thecohomology of X base-changed to C, which can then be interpreted as singular cohomology.
47If X is not necessarily projective, then this still works, provided that we use compactly
supported `-adic cohomology. We also remark that Weil proposed this “Lefschetz formula” interpretation of his conjectures in his ICM address [Wei56a].
48
The reason that Frob−1
p , rather than Frobp , appears in the formula is slightly technical:
in order to apply the Lefschetz formula, we have to work with a morphism of varieties, not a
Galois automorphism. Thus we work with the Frobenius endomorphism FX of X, defined byraising the coordinates of the a point to the pth power. If we compose this endomorphism with
the Galois automorphism Frobp of X, we obtain the absolute Frobenius of X (defined simply by
the map f 7→ f p on the structure sheaf OX ). This latter morphism induces the identity on theunderlying topological space of X, and consequently acts via the identity on the cohomology of X.
Thus the traces of powers Frob−1
on cohomology coincide with the traces of powers of FX , andpso it is Frob−1that appears in the Lefschetz formula for ζX (s). Motivated by this relationshippbetween Frob−1
and FX , the Galois element Frob−1
is referred to as the geometric Frobeniusppautomorphism, in contrast to the arithmetic Frobenius Frobp itself.

38

M. EMERTON

of the Hodge–Riemann bilinear relations). Grothendieck proposed an approach toproving the Riemann Hypothesis in the present context via a series of so-called“standard conjectures” [Kle94], which would allow one to make the same kind of
definiteness arguments as in the context of complex varieties. (Such arguments werealready employed by Hasse and Weil in their proofs of the Riemann Hypothesis in
the case when X is a curve.) Unfortunately, these conjectures remain unproved ingeneral, and Deligne [Del74] found a different approach to proving the RiemannHypothesis.49
We close this discussion by pointing out that if X is smooth and projective, thenthe various factorsi(52)
det (Id − p−s Frob−1p ) acting on H (X, Q` )in (51) are mutually coprime as i varies, since the Riemann Hypothesis ensuresthat they have no roots in common. This implies that the factorization of ζX (s)given by the right-hand side of (51) is intrinsic to ζX (s), and can be constructedindependently of the choice of ` 6= p. In particular, the various determinants (52)have integer coefficients (as polynomials in p−s ), and are independent of the choiceof `!
5.6. Hasse–Weil ζ- and L-functions. Let us return now to the case of a finite
type scheme over Z, which we now assume to be flat over Z, so that it does notlie in any particular residue characteristic p; the reader should have in mind an
integral model of a variety over Q or some other number field. If we combine (48)with (51), we obtain the formula (in which d denotes the relative dimension of X
over Z)(53)

ζX (s) =2dYY(−1)i−1i,det (Id − p−s Frob−1p ) acting on H (X/Fp , Q` )p i=1

which expresses ζX (s) as an Euler product, where the Euler factors are now determined by the action of (geometric) Frobenius on the `-adic cohomology of thevarious base-changes X/Fp . But this looks very similar50 to Artin’s definition (39)of his L-functions! It is also slightly inconsistent: for any particular p, we are free
to choose any ` 6= p to compute the Euler factor at p; but since ` is a prime, andso appears as one of the indices in the product over p, there is no single choice of `which works for every factor in the product (53)!
To further develop the observations just made, we now change notation. Namely,
we now let X be a Q-scheme, assumed to be smooth and projective, and we let
X be a model of X over Z, i.e. a projective flat Z-scheme whose fibre over Spec Q
equals X. (We can obtain X just by suitably clearing denominators in some systemof equations defining X; and X will now be the Z-scheme that was previously
denoted by X.) There is a finite set S of primes (the set of primes of bad reduction)such that X/Fp is smooth if p 6∈ S.
Now write X to denote the base-change of X to Q. If we choose a prime `, thenwe may form the `-adic cohomology H i (X, Q` ), with its natural Gal(Q/Q)-action;
49There is still a certain “definiteness”, or “positivity”, that goes into Deligne’s proof, but it

is more elementary — it is, grosso modo, just the fact that the point counts that are encoded by
ζX (s) are necessarily non-negative.
50The use of geometric Frobenius rather than arithmetic Frobenius can be thought of as simply
a certain choice of normalization.

RECIPROCITY AND L-FUNCTIONS

39

we are in characteristic zero, so no primes ` are excluded. General principles of`-adic cohomology (“local acyclicity”) imply that if p 6∈ S, and if ` 6= p, then
∼ H i (X/F , Q` )(54)
H i (X, Q` ) =p(where, as above, X/Fp denotes the base-change of X/Fp , the reduction mod p of X ,to Fp .) To make this identification canonical, we should choose a place of Q above p;
or equivalently, fix an embedding Q ,→ Qp , up to the action of Gal(Qp /Qp ) on
the target; or, again equivalently, fix a choice of decomposition group Dp at p in
Gal(Q/Q); or, yet again equivalently, fix a choice of surjection from the integral
closure of Z in Q onto Fp . With this choice made, the isomorphism (54) is functorial,and so is equivariant for the action of the chosen decomposition group Dp , whichacts on the left-hand side via restricting the Gal(Q/Q)-action to Dp , and acts on
the right hand side via the surjection Dp → Dp /Ip = Gal(Fp /Fp ), and the actionof Gal(Fp /Fp ) on the right-hand side.
In particular, we find that the Gal(Q/Q)-action on H i (X, Q` ) is unramified at
p 6∈ S ∪ {`}, and (for such p) we have(55)

(−1)i−1idet (Id − p−s Frob−1p ) acting on H (X, Q` )i= det (Id − p−s Frob−1p ) acting on H (X/Fp , Q` )(−1)i−1.

Finally, the “independence of `” noted above shows that the determinant (55) is
independent of `, as long as we assume that p 6∈ S and ` 6= p.
The conclusion is that (for each i) we have a family of `-adic representationsof Gal(Q/Q), namely the representations H i (X, Q` ), each of which is unramified
outside S∪{`} (where S is fixed independently of `), and for which the characteristic
polynomial of Frobenius at p, and hence the Euler factor (55), is independent of the
choice of ` 6= p, for all p 6∈ S. We say that the H i (X, Q` ) form a compatible family,
and we define the associated (partial) Hasse–Weil ζ-function via Artin’s formula51
(56)

S
ζX
(s) =

2d
YY

(−1)i−1idet (Id − p−s Frob−1,p ) acting on H (X, Q` )p6∈S i=0

where it is understood that to compute the factor at a given prime p, we choose any` 6= p. The superscript S indicates that we are forming an imprimitive ζ-function,in which we have omitted Euler factors at the finitely many bad primes in S. (Wewill return to this point below.)S
The formula defining ζX(s) then suggests that we should follow Artin in defining an L-function for each of the individual compatible families of representations H i (X, Q` ), and indeed we can do just that, setting
 Y
−1
i
LS s, H i (X) =det (Id − p−s Frob−1p ) acting on H (X, Q` )p6∈Swith the same convention as above, namely that in computing the pth factor wechoose any ` 6= p. We then have
S
ζX
(s) =

2d
Y

(−1)iLS s, H i (X).

i=051Up to working with geometric rather than arithmetic Frobenius.

40

M. EMERTONThe Riemann Hypothesis gives an improvementon the half-plane of convergence
for the individual L-functions LS s, H i (X) ; namely, this L-function converges for<s > 1 + 2i .
How should we eliminate the imprimitivity of the L-functions LS s, H i (X) , orS
the ζ-function ζX(s)? I.e., what Euler factors should we include for the missing
primes p ∈ S? One would like to use Artin’s recipe (forming the reciprocal characteristic polynomial of geometric Frobenius on the Ip -invariants). Unfortunately,
the problem now arises as to whether or not this is independent of `. This problem
S
is related to the following one: our definition of ζX(s), and of the LS s, H i (X) ,no longer uses the integral model X ! For example, it could be that for some primes
in S, the action of Gal(Q/Q) on H i (X, Q` ) happens to be unramified for somechoice of `; does this imply the same for other choices of `? Even if it does, arethe resulting characteristic polynomials of Frobenius independent of `? One way
this could happen is if X admitted a different integral model X 0 which did havegood reduction at S. Then the answers to these questions would be “yes”. But ingeneral the answers to these questions aren’t known (although the answers are still
expected to be “yes”).
The preceding question is closely related to the problem of finding “good” (e.g.
semistable) integral models of smooth projective schemes over Q, or over Qp (since
the ramification behaviour at p can be investigated locally at p). And this is related
to problems such as resolution of singularities in positive characteristic, or in mixed
characteristic (i.e. over Zp ) — problems which also remain open. For example,
the isomorphism (54) typically won’t hold at primes of bad reduction, and so thequestion arises as to what the true relationship between the two sides is in that case;the theory of nearby and vanishing cycles is aimed at answering this question. But
even for varieties over Fp , if they are singular then we don’t know independence of` for their `-adic cohomology in general, and without knowing this it doesn’t seemreasonable to imagine we could prove an independence of ` result at a prime of bad
reduction for a general smooth projective variety over Q.
All that being said, one does expect that the compatible families of `-adic representations H i (X, Q` ) are independent of ` 6= p locally at every prime p in a suitablesense, and in particular, that if we apply Artin’s formula, we obtain a well-defined
L-function
 Y

i
Ip −1
L s, H i (X) =det (Id − p−s Frob−1,p ) acting on H (X, Q` )p

as well as a ζ-function
ζX (s) =2dY(−1)i
L s, H i (X).

i=0

Independence of ` is known in many interesting cases, including the case when X
is a curve, or an abelian variety. We refer to Serre’s lecture [Ser70] for a moredetailed discussion, as well as for a discussion of how to complete these functions
at the archimedean place by adding an appropriate Γ-factor.
Once we have added the appropriate factors, we conjecture that ζX (s) will havean analytic continuation (this doesn’t actually depend on the manner in which
ζX (s) is completed at the p ∈ S or at ∞, since the Euler factors that we add

RECIPROCITY AND L-FUNCTIONS

41

are themselves meromorphic functions) and a functional equation (and having afunctional equation is very much dependent on completing in the correct way!).
More precisely, the functional equation will relate ζX (s) to ζX (d + 1 − s); in
terms of the individual L-factors, it will relate L s, H i (X) to L 1 + d − s, H 2d−i );if one takes into account the cup product pairing of H i and H 2d−i into H 2d ,and the Galois-action
on H 2d , this in fact becomes a functional equation relatingi
L s, H (X) to L 1 − s, H i (X)∨ , just as in the formalism of Artin L-functions.
And just as is the case for Artin’s L-functions, we anticipate that the global root
number will be a product of local root numbers, which can be determined using
the formalism of Langlands–Deligne discussed above.

5.7. Motives. When we write L s, H i (X) , what “is” H i (X)? For now, it is justa symbol standing for the compatible family of `-adic Gal(Q/Q)-representations{H i X, Q` )}.
Grothendieck’s theory of motives is intended to supply a more substantive answer
to this question.
Grothendieck’s proposal is based on the fact that although `-adic cohomology
is a functor on the category of varieties (e.g. over Q, or Fp ), it is functorial withrespect to a larger class of morphisms then just the morphisms of varieties. Namely,
because cohomology is valued in linear objects (vector spaces), multi-valued functions between varieties can induce single valued functions between their cohomology
groups (by simply adding up the multiple values!). Now multi-valued functions between varieties are actually very natural objects to consider — they are just thecorrespondences. (The reader might think of the Hecke correspondences on modular curves, or if f : C → C 0 is a surjective morphism of curves of degree > 1, one can∼think of the “reflected” graph of f in C × C 0 −→ C 0 × C as providing a multivalued−10morphism f : C → C.) So cohomology extends to a functor on the category ofvarieties, with suitable correspondences as morphisms. But we can also do spectral
theory of operators on cohomology groups — decomposing them into eigenspaces.
So we should also allow ourselves to decompose varieties into “eigenspaces” for theaction of correspondences. Roughly speaking, this gives us the category of motives. But many technicalities arise — for example, two coresspondences that canbe algebraically deformed, one into the other, will induce the same morphism oncohomology, and so we might want to identify them. Very quickly, in trying to makea precise definition and investigate its properties, one gets into thorny questions ofequivalence relations on cycles. Grothendieck’s standard conjectures [Kle94] areintended in part to resolve some of these questions, and thus to define a category— the category of so-called Grothendieck motives. The objects H i (X) are thensupposed to be motives. (How would we cut them out as eigenspaces of correspondences? We saw the answer above, at least over Fp , in the course of proving
independence of `: the characteristic polynomials of Frobenius on the different H iare coprime polynomials, and so we can use the Frobenius endomorphism of X to
cut X up into the various H i (X).)
Let us return momentarily to the context of Artin L-functions. If K is a number
field, then Spec K is a zero-dimensional smooth Q-scheme. If F ⊂ K is a subfield
for which K/F is Galois, then Gal(K/F ) acts as automorphism of Spec K, andelements in the group ring of Gal(K/F ) induces correspondences on Spec K. “Cutting up” Spec K using these correspondences amounts to decomposing Spec K in

42

M. EMERTON

the category of motives. Thus the Galois representations appearing in the theory ofArtin L-functions are examples of motives (0-dimensional motives). Note that there
was no ` in that story; this is because H 0 (the only degree of cohomology involvedin the 0-dimensional context) can be perfectly well defined with Q-coefficients, so“independence of `” is elementary for H 0 .
Note, though, that not all Gal(K/F )-representations will necessarily be definedover Q. In the theory of Artin L-functions we used representations on C-vector
spaces, but since Gal(K/F ) is a finite group, we could just as well have considered
representations on Q-vector spaces. For higher degree cohomology, when we haveto allow ` to vary, we should take cohomology with Q` -coefficients, so as to allow
finer motivic decompositions than can be obtained with Q` -coefficients. (One way
to express this passage from Q` to Q` -coefficients is to say that we are allowing
motives “with coefficients”.)
Whatever a motive M is, then, it should have cohomology (however we cut up
our original scheme X, we can correspondingly cut up its cohomology into the appropriate eigenspaces), and this cohomology will have a Gal(Q/Q)-action (as longas we are using correspondences defined over Q), or at least a Gal(Q/F )-action (ifwe use correspondences defined over some extension F of Q, as we did above when
we decomposed Spec K into its various irreducible Artin representations). We anticipate that, as ` varies, the cohomology of M will give compatible families of Galois
representations.52 Each motive will then have an associated L-function L(s, M ),which we again anticipate will admit an analytic continuation and functional equation (the latter relating L(s, M ) to L(1 − s, M ∨ ) for a suitably understood dual53
motive M ∨ , with a root number determined by the Langlands–Deligne theory).
We have written “whatever a motive is” above, because Grothendieck’s standard
conjectures remain unproved, and so the theory of motives is not complete. There
are other standard conjectures relating cohomology and cycles (not in the senseof being on Grothendieck’s list, but in the usual meaning of “standard”), such as
the Hodge conjecture and the Tate conjecture [Tat65], which would also have to
be solved to give anything like a complete theory of motives. (For example, one
conjectures54 that if X is smooth and projective over Q, then the Gal(Q/Q)-action
on each H i (X, Q` ) is semisimple, and that furthermore each simple constituent
is cut out by a suitable correspondence, i.e. is motivic. This would make theanalogy between our general context and the original context of Artin L-functions
particularly tight and compelling; but this semisimplicity conjecture is proved invery few cases.)

52Meaning that there is a number field E independent of ` such that, for ` 6= p, the character-istic polynomial of Frobp has coefficients in E (thought of as lying inside Q` ), and is independentof `.
53To obtain duals, there is an additional step that we should perform in the construction of
the category of motives, namely “invert the Lefschetz motive”, i.e. the motive H 2 (P1 ). If we dothis, then in fact H 2d (X) becomes an invertible motive for any smooth projective geometricallyconnected X of dimension d, and so the cup product pairings on cohomology can be used toconstruct dual motives.
54This conjecture is usually included under the general umbrella of “the Tate conjecture”, although in [Tat65], Tate attributes it to Grothendieck and Serre. For a discussion of this conjecture
and its relation to other conjectures regarding the Galois action on `-adic cohomology, one canconsult [Tat94].

RECIPROCITY AND L-FUNCTIONS

43

5.8. Compatible families of Galois representations. Number theorists often
circumvent the difficulties of rigorously defining/constructing motives by working
directly with compatible families of `-adic representations. Since this notion was
introduced by Taniyama [Tan57], it has evolved into a fundamental concept, which
provides a practical and flexible replacement for the notion of motive. Being inherently linear, Galois representations are easier to manipulate than varieties!
One technical difficulty in working with compatible families, such as the families {H i (X, Q` )} considered above, is ensuring compatibility at all primes p (i.e.
including the bad ones). Although, as already noted, arithmetic geometry is not
currently advanced enough to prove this directly in all cases of interest, in practice,there are many compatible families for which one does have control at all primes,
and hence for which one can define complete L-functions unambiguously. As wealready mentioned, the families {H 1 (X, Q` } for a smooth projective curve X overa number field give examples of such families.
In Subsection 4.6 we gave examples of compatible families of `-adic representations {eρ` } of Gal(Q/F ) (for some number field F ), constructed from algebraic
Hecke characters ψ on some extension K of F . These particular compatible families are in fact motivic. Indeed, the compatible families of characters {ψe` } from
Subsection 4.5 and 4.6 are associated to motives over K that can be constructedfrom abelian varieties with complex multiplication. (This goes back to Taniyama[Tan57] and [Ser68].) The group-theoretic induction that constructs {eρ` } from {ψe` }then corresponds in the world of varieties and motives to restriction of scalars fromK to F . Thus the L-functions L(s, {eρ` }) are motivic L-functions. In this case, theinduction formalism for L-functions (41), together with (35), shows that
(57)

L(s, {eρ` }) = L(s, ψ),and so the L-functions L(s, {eρ` }) are also automorphic55 (in this particular instancethey are Hecke L-functions). In particular, these L-functions do admit analytic
continuations and satisfy functional equations.
5.9. The Sato–Tate conjecture, unitary normalization, and the Langlands
group. We begin by considering a specific, historically important, class of examples: elliptic curves E over Q.
Suppose that E has good reduction outside the finite set S of primes; we abusenotation by writing simply E/Fp for the reduction modulo p 6∈ S. For each p 6∈ S,
we see from (5.5.2) that
ζE/Fp (s) =(1 − αp p−s )(1 − βp p−s ),(1 − p−s )(1 − p1−s )√where |αp | = |βp | = p, and αp βp = p. The numerator arises from H 1 , and thefactors in the denominator from H 0 and H 2 . We deduce that
L s, H 0 (E) = ζ(s) and L s, H 2 (E) = ζ(s − 1)(strictly speaking we haven’t computed the missing Euler factors at p ∈ S, but
completing to the Riemann ζ-function and its translate in this way are the only
55Automorphic over K, since we obtain Hecke L-functions for K. The theory of automorphic
induction, briefly recalled in Subsection 7.3 below, shows that they are also automorphic over F .

44

M. EMERTON

possibilities compatible with a functional equation), while
Y
 Y
1
1
,
=
LS s, H 1 (E) =−s−s(1 − αp p )(1 − βp p )1 − ap p−s + p1−2s
p6∈S

p6∈S

where
ap = αp + βp = 1 + p − |E/Fp (Fp )|.
√p implies (indeed, is equivalent to) the condition√ √ap ∈ [−2 p, 2 p];

The condition |αp | = |βp | =(58)

and both conditions are equivalent to the characteristic polynomial of Frob−1
on
p
56
1
2
H (E, Q` ) — which equals X − ap X + p — having non-positive discriminant.
Our next immediate goal is to discuss the Sato–Tate conjecture, which is aconjecture describing the distribution of the quantities ap as p varies. But we also
intend to explain how it relates to a more far-reaching idea, that of the Langlands
group.
To begin with, a consideration of (58) suggests that we rescale the ap , andconsider (for each p 6∈ S) the quantityap
Ap = √ ∈ [−2, 2];p
the quantities Ap now lie in the same interval, as p varies, and so it makes sense
to discuss their distribution. It turns out to be convenient to write Ap = 2 cos θp ,√where θp ∈ [0, π]. (Then, up to switching the labelling, we have αp = peiθp and√ −iθp.)βp = peIf E admits CM by an imaginary quadratic field K, then it is well-known (andeasily seen) that ap = 0 if p is inert in K (which happens half the time, by Dirichlet),so that θp = π/2 for all such p; while for the p that are split in K, one finds that
θp is distributed uniformly throughout [0, π] .
If E does not admit CM, then the Sato–Tate conjecture57 states that θp is dis2
tributed over [0, π] according to the measure sin2 θdθ.
π
Before we try to explain the meaning of these results, we begin by describing
a more structural interpretation of the above rescaling by p−1/2 . If we let {χ` }denote the compatible family of cyclotomic characters (see example (4.5.2)), then−1
χ(Frob−1. Suppose for a moment that we could form a compatible familyp ) = p
1/2

1/2

−1/2
{χ` }, with the property that χ` (Frob−1
for each p 6= `. Then wep ) = p1
1/2
could form the twisted compatible family H (E) ⊗ χ , and for a given p 6∈ S
(working with ` 6= p) we would find that the characteristic polynomial of Frob−1pon this twist is(59)

X 2 − Ap X + 1.

56The `-adic H 1 of E is dual to the `-adic Tate module of E, and so this polynomial is also
the characteristic polynomial of Frobp on the `-adic Tate module of E; which is how it is often
described in discussions of elliptic curves.
57 Now a theorem for elliptic curves over any totally real of CM number field. Over totallyreal fields, the proof is due to a large number of people, including Clozel–Harris–Taylor [CHT08],
Harris–Shepherd-Barron–Taylor [HSBT10], Taylor [Tay08], Shin [Shi11], Clozel–Harris–Labesse–
Ngô [CHLN11], and Barnet-Lamb–Geraghty–Harris–Taylor [BLGHT11]. The recent extension[ACC+ 18] to the case of CM fields is discussed in Subsection 7.3 below.

RECIPROCITY AND L-FUNCTIONS

45

Now unfortunately, the characters χ` don’t admit square roots. (For example,
complex conjugation is an element c ∈ Gal(Q/Q) of order 2 for which χ` (c) = −1.)On the other hand, the compatible family {χ` } arises from the absolute value
character | | on Q× \ A×
Q , and this character evidently admits a square root, namely
1/2|| .
This prompts the question: is there a process via which we can “unravel” thevarious two-dimensional `-adic representations H 1 (E, Q` ), so as to obtain a single
two-dimensional representation, over the complex numbers, of some group — in
the way that we can “undo” the passage from | | to {χ` }, at the cost of passing
×
×from Gal(Qab /Q) = π0 (Q× \ A×
Q ) to the group Q \ AQ itself? If so, then we
could twist this unravelled representation by | |1/2 , and obtain a representation
which (we might hope) would give a conceptual sense to the rescaled characteristic
polynomials (59).
5.9.1. Example. Consider the elliptic curve E cut out by the Weierstrass equationy 2 = x3 − x.
This is an elliptic curve with complex multiplication by Z[i], defined over Q(i), via
the formula[i](x, y) = (−x, iy).
This action makes each H 1 (E, Q` ) free of rank one over Q` ⊗Q Q(i), so that the
Gal Q/Q(i) -action on H 1 (E, Q` ) (which commutes with the Q` ⊗Q Q(i)-action) is
abelian. In fact, this action is given by the character ψe` , where ψe` is constructedfrom the (algebraic!) Hecke character ψ which is inverse to the character of examples (3.2.2) and (3.3.2). It is then easy to see that the Gal(Q/Q)-action on
H 1 (E, Q` ) is given by the two-dimensional representation ρe` obtained by inducing ψe` .
In this case, an “unravelling” of the desired type is possible: we have the Weilgroup representation ρ : WQ(i)/Q → GL2 (C) obtained by inducing ψ, as in Subsection 4.6, which we can then twist to form
W

Q(i)/Q(ψ ⊗ | |1/2 ).
ρ ⊗ | |1/2 = IndQ(i)
× \A×
Q(i)

1/2

Thus ρ ⊗ | |is also obtained by the induction construction of Subsection 4.6, but
now applied to ψ ⊗| |1/2 , which is the unitarization of ψ (see (3.2.2)). Thus ρ⊗| |1/2
is a unitary representation of WQ(i)/Q , and we see that the polynomials (59) are
the characteristic polynomials of geometric Frobenius elements58 in WQ(i)/Q . Thus
they are the characteristic polynomials of unitary 2 × 2-matrices — which explains
why their roots have the form e±iθp !
The result about the distribution of the Ap is then seen to be a result about the
distribution of the images ρ(Frob−1p ), and so is a certain kind of generalization ofČebotarev density.
On the other hand, if E is not a CM elliptic curve, then Serre showed in [Ser68]
that the action of Gal(Q/Q) on H 1 (E, Q` ) is irreducible, and remains so after
restriction to any open subgroup Gal(Q/K) of Gal(Q/Q). Thus ρ` is not induced
58We don’t recall the details of the construction of these elements right now; it is part of thelocal-global compatibility theory for Weil groups, which is discussed in the following subsection.

46

M. EMERTON

from a character, and so we cannot describe the compatible family {ρ` } in termsof a Weil group representation ρ.
However, we have the following conjecture. (We remind the reader that we
introduced the absolute Weil group WF of F in Subsection 4.6 above.)
5.9.2. Conjecture (Existence of the Langlands group). For any number field F ,there is a locally compact topological group LF , the Langlands group of F , admitting
a surjection LF → WF with compact kernel and for which the induced morphism
ab
×
Lab
\ A×
F → WF = F
F is an isomorphism, and such that compatible families ofn-dimensional `-adic Galois representations {ρ` } arising from motives arise (in asuitable sense) from certain (“algebraic”) complex representations LF → GLn (C).
This conjecture is somewhat informal as stated, but we will make it more precise
in the following subsection. For now, we just remark that the notion of “algebraic”,
for representations of LF , should be analogous to the corresponding notion for Hecke
characters, and that the mechanism which produces the compatible family {ρ` }from ρ should be (at least formally) analogous to the mechanism which produces
the compatible family {ψe` } from an algebraic Hecke character ψ.
If we admit this conjecture, then we find, for any elliptic curve, that the familyof representations H 1 (E, Q` ) arises from a representation LQ → GL2 (C). If we
let ρ denote the twist of this latter representation by | |1/2 (this twist now makes
sense!), then ρ in fact factors through SU (2) ⊂ GL2 (C). Indeed its determinantis trivial, as each characteristic polynomial (59) has constant term 1, and since
LF is compact outside of its R×>0 part coming from WF , we see that the image iscompact with trivial determinant, and so can be factored through SU (2) after anappropriate choice of basis. Now the image is a compact subgroup, and there aren’t
many possibilities for it: either a copy of U (1); the normalizer of a copy of U (1)in SU (2) (which is a non-split extension of a group of order 2 by U (1), with the
non-trivial element in the group of order 2 acting on U (1) via inversion); or SU (2)itself. The first possibility can’t occur, since the representations of Gal(Q/Q) on
H 1 (E, Q` ) aren’t abelian, and so one of the latter two possibilities must occur.
In the second case, restricting to an index two open subgroup of LQ , i.e. to
LK for some quadratic extension of K, yields an abelian representation. This isevidently the CM case. Otherwise, we must be in the third case. And now we see
the meaning of the results on the distributions of the θp : what they are saying isthat ρ(Frob−1
p ) is equidistributed in the image of ρ, as p ranges over all primes 6∈ S.
Just as in the case of the Čebotarev’s theorem, we must remember that Frob−1
p is
only defined up to conjugacy, so we have to understand this equidistribution in theframework of conjugacy classes.
The normalizer of U (1) has two connected components. The non-identity component is a single conjugacy class. The other component is just U (1) itself, and the
conjugacy classes are just the pairs {z, z} for z ∈ U (1), which can be parameterized
by the elements z = eiθ , with θ ∈ [0, π]. Thus, when E has CM, we see that half theρ(Frob−1p ) land in the non-identity component — these are the half of the primes
for which ap = Ap = 0. And the other half of the primes satisfy Frob−1
p ∈ U (1),and θp is equidistributed in [0, π].
The conjugacy classes in SU (2) are again parameterized by their pairs of eigenvalues {z, z}, and hence by z = eiθ with θ ∈ [0, π]. But the uniform (i.e. Haar)
measure on SU (2) doesn’t induce the uniform measure on [0, π]. Rather, it induces

RECIPROCITY AND L-FUNCTIONS

47

the Sato–Tate measure π2 sin2 θdθ. And the Sato–Tate conjecture affirms that if E
does not have CM, then the θp are indeed equidistributed according to this measure.
How does one prove this equidistribution? It is a density theorem, and the proof
proceeds like the proof of all such theorems, via an appropriate Fourier analysis
facilitated by a result on the non-vanishing of appropriate L-functions. Consider
first the example of (5.9.1). For primes p that are split in Q(i), say p = ππ,with π ≡ 1 mod (1 + i)3 , the corresponding conjugacy class in U (1) is simply√√(π/ p, π/ p) (see the definition of ψ in Example (3.2.2)), and so the claimed
equidistribution amounts to the equidistribution of Gaussian primes in angularsectors that was already mentioned in Subsection 3.5.
What about the non-CM case? Then we have to perform a Fourier analysis of
the conjugacy classes in SU (2). To expand the indicator function of a conjugacy
class in terms of traces of representations, we have to use not just the character ofthe standard two-dimensional representation of SU (2), but the characters of all of
its irreducible representations. These are obtained by taking the symmetric powers
Symn ρ of the two-dimensional representation ρ. Now Symn ρ is a obtained as adirect summand of ρ⊗n by writing it as the image of a certain symmetrization
projector, and since ρ arises from H 1 (E) by a twist, the Künneth theorem assures
us that ρ⊗n , and hence also Symn ρ, will be cut out of (some twist of) H n (E n ) bysome correspondence. Thus Symn ρ is motivic. So what we need to know, to prove
the Sato–Tate conjecture, is that each of the motivic L-functions L(s, Symn ρ) has
an analytic continuation up to <s ≥ 1, with no zeroes or poles in this region.
Now, although the representation ρ is only hypothetical, its L-function is not. Indeed, since hypothetically ρ = H 1 (E) ⊗ | |1/2 , we see that (again, hypothetically)59
Symn ρ = Symn H 1 (E) ⊗ | |n/2 . ThusnL(s, Symn ρ) = L s, Symn H 1 (E) ⊗ | |n/2 = L s + , Symn H 1 (E) ,2and this equation gives the left-hand L-function a meaning, even if ρ is only hypothetical. Furthermore, the Riemann Hypothesis ensures that the Euler productdefining L(s, Symn H 1 (E) converges if <s > 1 + n2 , and so we see that the Euler
product defining L(s, Symn ρ) converges if <s > 1 — so that L-functions of theunitary Langlands group representation Symn ρ behave formally just like the Lfunctions of unitary Weil group representations, or unitary Hecke characters: theEuler products converge if <s > 1 — and our problem is to analytically continue
them to the line <s = 1.
For Artin L-functions, this was achieved by Brauer’s Theorem 5.1.3. We will
briefly recall in Subsection 7.3 below that Brauer’s theorem plays a role in achieving
the analogous result here as well.
We remark that in general, for any smooth projective Q-scheme X (or, moregenerally, any motive), we may hypothetically form H i (X) ⊗ | |i/2 as a representation of the Langlands group LQ , and the Riemann Hypothesis will ensure that
this representation, if it existed, would be unitary. The Riemann Hypothesis also
ensures thati
L s, H i (X) ⊗ | |i/2 = L s + , H i (X))
2
59Now H 1 (E) just stands for the compatible family {H 1 (E, Q )}, and we can simply form`
its symmetric power to obtain a compatible family Symn H 1 (E), given by the representations{Symn H 1 (E, Q` )}.

48

M. EMERTON

converges if s > 1. The left-hand L-function is sometimes said to be unitarily
normalized. This can be thought of60 as referring to the hypothetical unitary representation H i (X) ⊗ | |i/2 .
5.10. The conjectural Langlands group — an elaboration. We first recall
the definition of the local Weil groups. They come in two flavours, archimedean andnon-archimedean. The archimedean local Weil groups are the easiest to describe:
WC = C× , while WR is the unique non-split extension of Gal(C/R) by C× (with
Gal(C/R) acting on C× via complex conjugation).61
In analogy with the short exact sequence (36) (and thinking of WC as WC/C —since C has no non-trivial finite extensions — and of WR as WC/R — since C is thealgebraic closure of R), we may place these Weil groups in short exact sequences
(60)

1

/ C×

/ WC

/ Gal(C/C)

1

1

/ C×

/ WR

/ Gal(C/R)

/1.

and
(61)

Of course, WCab = WC/C = WC = C× , while WRab = WR/R = R× , the identification with R× being uniquely determined by the requirement that it induces themorphism z 7→ zz when pulled back to C× ⊂ WR .
If Fv is a non-archimedean local field (which we denote in this fashion in anticipation of the fact that it will arise by completing a global field F at some
non-archimedean place v), with residue field Fq , then there there is a short exact
sequence of groups
1 → Iv → Gal(F v /Fv ) → Gal(Fq /Fq ) → 1,
where Iv denotes the inertia subgroup of Gal(F v /Fv ). The Galois theory of finite
b with topological genfields shows that the group Gal(Fq /Fq ) is isomorphic to Z,
erator the Frobenius element Frobv = Frobq (the automorphism of Fq defined by
x 7→ xq ). Inside this Galois group we may consider the cyclic subgroup (just a copyof Z) that is literally generated by Frobv . Then the Weil group WFv is the preimageof this cyclic group in Gal(F v /Fv ), so that WFv sits in a short exact sequence
1 → Iv → WFv → hFrobv i → 1.
We recall that if Kv is a finite extension of Fv , then Gal(F v /Kv )c (the closure of
the commutator subgroup of Gal(F v /Kv )) is contained in Iv , and so we may set
WKv /Fv = WFv / Gal(F v /Kv )c . Local class field theory then yields a morphism ofexact sequences analogous to (36)
(62)
1

/ Kv×

/ WK /F/ Gal(Kv /Fv )

/1

1


/ Gal(Kvab /Kv )/ Gal(F v /Fv )/ Gal(F v /Kv )c/ Gal(Kv /Fv )

/1

v

v

60Although it might be historically more accurate to say that this name is derived from its
meaning on the automorphic side of reciprocity, where the corresponding process amounts totwisting so as to obtain a unitary central character.
61If we regard R× as a subgroup of scalar matrices in GL (C), then C× = R× U (1) embeds
2>0>0
into R×
SU
(2),
and
W
can
also
be
described
asthenormalizerof C× in R×R>0>0 SU (2).

RECIPROCITY AND L-FUNCTIONS

49

and of course WFv = limK WKv /Fv .
←− v
If Fv is a completion of the number field F at a place v, then there is a morphism62
WFv → WF compatible with the short exact sequences (36) and whichever choiceof (60), (61), or (62) is appropriate to v.
We may also define local versions of the Langlands group. If v is archimedean,
then LFv = WFv . If v is non-archimedean, then LFv = WFv × SU (2). (From theviewpoint of Conjecture 5.9.2, relating the hypothetical global Langlands group tocompatible families of `-adic Galois representations, the role of the SU (2)-factor isto account for the potentially infinite-order action of tame inertia on `-adic representations.) In either case, we have a surjection LFv → WFv .
We now elaborate on Conjecture 5.9.2. We conjecture the existence of a locallycompact group LF , the Langlands group of F , equipped with a surjection LF → WF
whose kernel is compact and which induces an isomorphism
×
ab
×
Lab
F → WF = WF/F = F \ AF ,and also equipped with morphisms LFv → LF for each place v that are compatible
with the maps to and between local and global Weil groups. Granting this, if
ρ : LF → GLn (C) is a continuous representation, then we may pull-back ρ to amorphism ρv : WFv = LFv → GLn (C) for each archimedean place v of F . We saythat ρ is algebraic if each of these pull-backs ρv becomes algebraic when restricted
to the copy of C× ⊂ WFv ; i.e. if the induced morphism C× → GLn (C) is a productof n algebraic characters of C× , i.e. characters of the form z 7→ z p z q for somep, q ∈ Z.
We now conjecture that algebraic representations of LF correspond to motivic
compatible families of representations of Gal(Q/F ). To make this precise, we follow
Langlands [Lan79] in using the language of Tannakian categories.63 We can consider
the category of algebraic representations of LF ; this will be a Tannakian category,
and we can form its Tannakian Galois group, a pro-algebraic group over C, which wedenote by Lalg
F . Assuming that Grothendieck’s category of motives exists, we may
also form its Tannakian group Gmot
F ; this is a pro-algebraic group over Q, or overQ if we allow our motives to have coefficients. The existence of compatible families
of Galois representations can be expressed in terms of the existence of morphisms
Gal(Q/F ) → Gmot
F (Q` ) (one for each ` and each choice of embedding Q → Q` )having certain properties. A more precise formulation of Conjecture 5.9.2, then, isthat there is an isomorphism of pro-algebraic groups over C
∼

mot
Lalg
F −→ (GF )/C .

This isomorphism should again satisfy various compatibilities, whose details weomit. (For example, it should be compatible with the construction of Subsection 4.5.)
This more precise form of Conjecture 5.9.2 is the top layer of a rather tall stack ofconjectures, since its formulation depends on the existence of a category of motives
satisfying a range of desirable, but so far unproved, properties. Nevertheless, in his
work (e.g. [Lan79] and [LR87]) Langlands has consistently shown the value of substantively engaging with this circle of ideas, and of incorporating them into one’s
62To determine this unambiguously, we should extend the embedding F ,→ F to an embeddingv
Q = F ,→ F v .
63For abelian representations, this viewpoint was already introduced in [Ser68].

50

M. EMERTON

thinking. For this reason, we believe it is worthwhile, and in keeping with Langlands’s own point-of-view, to place Conjecture 5.9.2 at the centre of our discussionof reciprocity.
6. Automorphic L-functions and reciprocity
6.1. Automorphic representations. Following the definition in Hecke [Hec27](or Serre [Ser77] for a more contemporary reference), we may regard a modular
form of weight k and level 1 as a function
f : {Λ | Λ ⊂ C a lattice } → C
which satisfies the condition f (λΛ) = λ−k f (Λ) for any λ ∈ C× , which is holomorphic as a function of those τ for which =τ > 0, if we associate to τ the lattice
Z + Zτ, and which satisfies a moderate growth condition as τ → i∞.
Now choosing a based lattice in C ∼= R2 is the same as choosing a basis of R2 , i.e.
an element of GL2 (R). Changing basis amounts to applying an element of GL2 (Z).
So modular forms of weight k and level 1 are functions on the quotient(63)

GL2 (Z) \ GL2 (R)satisfying certain additional conditions. We now follow the adèlic philosophy of
replacing Z ⊆ R by Q ⊆ AQ , and observe that we can rewrite (63) asb
GL2 (Q) \ GL2 (AQ )/ GL2 (Z),so that modular forms of weight k and level 1 are functions on the quotient
GL2 (Q) \ GL2 (AQ ) satisfying certain conditions. Being invariant under the rightb is one of those conditions, and it is natural to omittranslation action of GL2 (Z)it, thus incorporating modular forms of higher level into our picture. Relaxing
the archimedean conditions (holomorphicity and the precise weight k condition) is
also a sensible step; this incorporates Maass forms and non-holomorphic Eisensteinseries into our picture.
Thus we are led to define the space of automorphic forms A GL2 (Q) \ GL2 (A) ;it is the space of functions f : GL2 (Q) \ GL2 (A) → C which are smooth in thearchimedean variables and of moderate growth as these variables tend to ∞; invariant under some open subgroup of GL2 (A∞Q ); and which are generalized eigenvectors
for the centre of the enveloping algebra of the Lie algebra gl2 of GL2 (R) (so essentially we are asking them to be eigenvectors for the Casimir operator, which
can be interpreted as the hyperbolic Laplacian in the classical picture). The space
A GL2 (Q) \ GL2 (A) is a complete locally convex topological vector space,64 whichbecomes an admissible representation of GL2 (A) under the action by right translation. An automorphic representation of GL2 (A) is then, by definition, an irreducible
subquotient (in the topological sense, i.e. the quotient of one closed invariant subspace by another that it contains, which is irreducible in the topological sense ofcontaining no proper closed invariant subspace) π of A GL2 (Q) \ GL2 (A) .
These notions extend directly to any connected reductive affine algebraicgroup
G over a number field F — we may define the space A G(F ) \ G(AF ) of automorphic forms, which is an admissible representation of G(AF ) via right translation,and then define an automorphic representation π of G(AF ) to be a topologically
64We work with topological vector spaces so as to avoid having to take O(2)-finite vectors;
the equivalence between this viewpoint and the O(2)-finite vector viewpoint follows from HarishChandra’s theory of admissible representations of real reductive groups.

RECIPROCITY AND L-FUNCTIONS

51

irreducible subquotient of this space. Since G(AF ) is the restricted direct productof the groups G(Fv ) (as v runs over all places of F ), we may factor the irreducible
representation π as a restricted tensor product (suitably defined)
O
0
πv ,
(64)
π=
where each πv is an irreducible admissible representation of G(Fv ) [Fla79]. (Thisis the analogue in the non-abelian setting of the factorization (20) of an idèle classcharacter.)The space A G(F ) \ G(AF ) contains a subspace A◦ G(F ) \ G(AF ) of cuspforms, whose elements are characterized by the vanishing of certain integrals thatcompute their “constantterms” at the boundary of G(F ) \ G(AF ). The spaceA◦ G(F ) \ G(AF ) has a positive definite G(AF )-equivariant inner product (theL2 -inner product, also called the Petersson inner product), and decomposes as adirect sum of irreducible representations. These are the cuspidal automorphic representations.
We refer the reader to [BJ79] for more details regarding these definitions.
Returning to the case of GL2 over Q and classical modular forms: if f is a classicalcuspform, then we have seen that we can interpret f as an automorphic form, i.e.
as an element f ∈ A0 GL2 (Q) \ GL2 (A) , and we may then consider the GL2 (AQ )subrepresentation that it generates. It is a theorem, essentially due to Atkin andLehner[AL70], that f (thought of as an automorphic form) generates an irreducible
subrepresentation (i.e. an automorphic representation) if and only if f (thoughtof as a modular form of some weight and level) is an eigenform for the Hecke
operator Tp for all but finitely many primes p. Thus automorphic representations
can be regarded as a representation-theoretic generalization of the notion of Hecke
eigenform, and the list of the factors πv occurring in the the decomposition (64) asa refinement of the classical data consisting of the list of Hecke eigenvalues of f .
In particular, returning to the general case, if πv is a constituent of an automorphic form which is unramified, then it gives rise to a Frobenius–Hecke conjugacy
b × Frob−1
class cv ∈ G
v (as explained by Langlands in [Lan70], building non-triviallyon Satake’s work [Sat63]; see also Shahidi’s chapter in this volume [Sha21], and thediscussion of Section 6.3 below); this class is an analogue of the classical pth Hecke
polynomial X 2 − ap X + ε(p)pk−1 associated to a modular Hecke eigenform. (Up toa possible issue of normalization, this polynomial is the characteristic polynomialof the Frobenius–Hecke conjugacy class that Langlands associates to the factor πpof the automorphic representation π generated by f .)6.2. Hecke. In his paper [Hec36], Hecke generalized Riemann’s second proof of
the analytic continuation and functional equation of ζ(s) to show that the Mellintransform Λ(f, s) of a cusp form f of level one is the product of a Dirichlet seriesL(f, s) with an appropriate Γ-factor, which admits an analytic continuation to an
entire function and satisfies a functional equation. He also proved the converse:
any entire function which satisfies a functional equation of the appropriate form,
and which satisfies an appropriate growth condition, arises from a modular form oflevel one.
In the papers [Hec37], he introduced the operators which now bear his name, andproved that f is a Hecke eigenform if and only if L(f, s) admits an Euler product— in which case the Euler factors are of the form (1 − ap p−s + pk−2s )−1 , where ap
is the pth Hecke eigenvalue, and k is the weight of f . Although he extended the

52

M. EMERTON

theory of Hecke operators to higher level modular forms (and the theory, at leastfrom a classical — i.e. non-adèlic — viewpoint was essentially completed by Atkinand Lehner [AL70]), the converse theorem for levels > 1 is not as obvious. It was
eventually proved by Weil [Wei67]. Jacquet and Langlands [JL70] proved anotherform of the converse theorem, this time in the context of automorphic representations of GL2 (AF ) for arbitrary number fields F . A key feature of their converse
theorem is that the hypothesized functional equations must have the correct factors. This provides one of the motivations for developing the theory discussed
in Subsection 5.2.
We close this discussion by emphasizing that, just as in Riemann’s work, Hecke’s
description of his L-functions associated to automorphic forms via an integral formula, and his analysis of their Euler product factorization, are completely independent of one another.
6.3. Langlands. We sketch Langlands’s definition of automorphic L-functions, referring to Shahidi’s chapter in this volume [Sha21] for more details.
Recall that if G is a connected reductive affine algebraic group over the numberb which is a group over C withfield F , then we may form first the dual group G,b depends only on G over Q),
dual root datum to that of G (the construction of Gb so as to form
and then construct an action of the Galois group Gal(K/F ) on G,
the L-groupL
b o Gal(K/F )G=G
b and hence the L-group of G, only depends onof G. The action of Gal(K/F ) on G,
the inner class of G over F .
Note that the L-group isn’t entirely well-defined, since we are free to enlarge K,
so it is better to speak of an L-group of G. This is actually an advantage in thetheory. We are not even restricted to taking the semi-direct product with a Galois
group. Any group that surjects onto Gal(K/F ) (and hence can be made to act onb through the original Gal(K/F )-action) will do. Thus we can form the L-groupG
b o WK/F , or L G = G
b o WF . If we grant ourselvesin its “Weil form”, as L G = G
b o LF .
the existence of the Langlands group, we could even form L G = G
In general, suppose that Γ is a group that surjects onto Gal(K/F ), and that web o Γ. Suppose now that H is any group equipped with a homomorform L G = G
phism to Γ (and so also with a homomorphism to Gal(K/F ), by composition). Thenb o Γ, i.e. a homomorphism which induces
giving an L-homomorphism H → L G = G
the given map upon projection to Γ, is the same as giving an L-homomorphism
b o Gal(K/F ). (This just follows from description of Gb o Γ as the
H → LG = Gb o Gal(K/F ) ×Gal(K/F ) Γ.) Thus the notion of L-homomorphism
fibre product (Gis suitably compatible with the flexibility in the formation of the L-group of G.
If p is a prime F , and Kp is an extension of Fp that splits G, then we may performb o Gal(Kp /Fp ), anda similar local construction, obtaining a semi-direct product G
similar considerations apply. In particular, if G is unramified at p (i.e. quasi-split
and split over an unramified extension), then we may choose Kp /Fp to be unramib
fied, so that Gal(Kp /Fp ) = hFrobp i. Then G-conjugacy
classes of L-homomorphisms
L
ϕ : LFp → G (for any choice of L-group; and remember that LFp = WFp ×SU (2) isb
the Langlands group of Fp ) correspond to G-conjugacy
class of L-homomorphismsb
LFp → G o hFrobp i. We define one of the latter (and hence also one of the former)

RECIPROCITY AND L-FUNCTIONS

53

homomorphisms to be unramified if it is trivial on Iv × SU (2); giving an unramib × Frob−1bfied ϕ then amounts to giving an element c ∈ G
p , up to G-conjugacy, theFrobenius–Hecke conjugacy class associated to the unramified L-homomorphism ϕ.
The class c and the unramified L-homomorphism ϕ determine one another.
We now consider a representation ρ : L G → GLn , required to be algebraic whenb and continuous on the Galois/Weil factor in L G. Thus ρ will be
restricted to G,
unramified at all but finitely many primes p of F . In other words, there is a finite
b o WF for p 6∈ S,
set of primes S such that for p 6∈ S, when we restrict ρ to Gpb
the action of WFp on G factors through the quotient WFp /Ip = hFrobp i, and theb o WF factors through Gb o hFrobp i.
restriction of ρ to Gp
Suppose now that π is an automorphic representationof G(AF ). Then we mayO0form the restricted tensor factorization π =
π
.
The
group
G is unramified atv v
all but finitely many primes p, and for all but finitely many of these primes, thelocal factor πp will be unramified. As recalled in Subsection 6.1 above, for eachsuch prime p, we obtain a corresponding Frobenius–Hecke conjugacy class, i.e. an
b o Frob−1belement cp ∈ G
p , well-defined up to G-conjugacy.
Now enlarge the finite set of places S so that it includes all the bad primes for ρ,
all the bad primes for π (i.e. the primes for which cp is not defined), and also all theinfinite places. Langlands then defines the (imprimitive) automorphic L-function
associated to π and a representation ρ : L G → GLn as an Euler product, via theformulaY1(65)
LS (s, π, ρ) =−s ρ(c )
det
Id
−
N
(p)
n×n
p
p6∈S
Note the similarity to Artin’s definition (39) of his L-functions!
The problem of completing the L-function (65) by adding Euler factors at theprimes in S is a difficult one. We first recast the Frobenius–Hecke classes cp asunramified L-homomorphisms ϕp : LFp → L G, in the manner described above.
We would then like to use the remaining πp to produce L-homomorphisms ϕv :LFv → L G which are no longer necessarily unramified. The local Langlands conjecture for G states that this is possible. It is proved in a number of cases (we canvary G, and also, having fixed G, impose various ramification conditions on the π),
and in particular is proved in the archimedean case for arbitrary G by Langlands
himself [Lan89], and is proved completely for GLn by Harris–Taylor and Henniart[HT01, Hen00]; but it remains open in general.
Langlands proves that the Euler product (65) converges in some right half-plane.
There is one particular case in which we can say more. Let G = GLn , so thatb = GLn also, and take ρ to be the standard representation, i.e. simply the idenGtity map of GLn to itself; the resulting L-functions are referred to as the standard L-functions, and we denote them L(s, π, standard). The work of Godement–
Jacquet[GJ72] establishes the analytic continuation and functional equation forthese L-functions by methods generalizing those of Tate’s thesis. In general, Langlands conjectures that, after completing the Euler product (65) by adding factors
at the missing places, the resulting function will admit an analytic continuation
to the entire plane, and satisfy an appropriate functional equation. But in fact,
he conjectures more: as we recall below, his functoriality conjecture states that anarbitrary automorphic L-function L(s, π, ρ) should in fact be a standard L-function

54

M. EMERTON(for some other π, to be sure), whose analytic continuation and functional equationis then assured by [GJ72].
6.4. The reciprocity conjecture. If π is an automorphic representation of G(AF ),then we obtain the unramified L-homomorphisms ϕp : WFp → L G, for p 6∈ S, and,
if we grant the existence of the local Langlands correspondence, we in fact obtain
L-homomorphisms ϕv : LFv → L G for every place v. We could now ask: is there
an L-homomorphism ϕ : LF → L G which induces the various ϕv . The answer is“no” in general,65 even if G = GLn . But we have the following conjecture:
6.4.1. Conjecture (Langlands reciprocity). There is a bijection between irreducible
representations ϕ : LF → GLn and cuspidal automorphic forms π on GLn (AF ),via which ϕ and π correspond if ϕ|Fv = ϕv (the local L-homomorphism associatedto πv via the local Langlands correspondence) for all places v.
Why do we call this a reciprocity law? Because, according to Conjecture 5.9.2,
each motive over F gives rise to a representation ρ of LF . By the conjecture,each irreducible constituent of ρ then corresponds to a cuspidal automorphic representation π of some GLn (AF ). In particular, L(s, ρ) (which will coincide with the Lfunction of our original motive) is a product of standard L-functions L(s, π, standard),
and so has the anticipated analytic continuation and functional equation.
The conjecture also has an implication in the reverse direction: if π is a cuspidal
automorphic representation on GLn (AF ), whose factors at the infinite places are
algebraic,66 then the conjecture produces an irreducible representation ρ of LF ,which is algebraic in the sense of Conjecture 5.9.2, and so should correspond to acompatible family of `-adic Galois representations coming from a motive over F .
6.4.2. Reciprocity in terms of L-functions. A key point, which we have alreadyintimated, is that the reciprocity conjecture can be phrased in terms of L-functions,and hence stated (and studied!) independently of the problem of the existence of theLanglands group, at least in the algebraic case. Thus, suppose that π is an algebraic
cuspidal automorphic representation of GLn (AF ), conjecturally corresponding toan algebraic representation ϕ : LF → GLn (C). This algebraic representation inturn corresponds to a motive M over F , and the conjectured relationship between
π and ϕ can be rephrased as an equality(66)

L(s, π, standard) = L(s, ϕ) = L(s, M ).

Thus, phrased in terms of L-functions this way, we can study the conjecture ineither direction: we can begin with an algebraic cuspidal automorphic representation π and try to construct a motive M , or at least a compatible family of `-adic
65For G = GL , this is due to non-isobaric automorphic representations, in the sense of Langnlands’s paper [Lan79]. For more general G, it is then related to the representations that Langlands(in the same paper) calls anomolous. Arthur’s theory of non-tempered endoscopy [Art89] aims to
explain automorphic representations that appear in L2 G(F ) \ G(AF ) , and which are anomolous
in the sense of [Lan79], in O
terms of L-homomorphisms LF × SL2 → L G.
066Suppose that π =πv is an automorphic representation of some G(AF ). At eacharchimedean place v, the factor πv induces a morphism ϕv : LFv → L G via the local Langb We can then ask
lands correspondence [Lan89], which in turn restricts to a morphism C× → G.
b wherethat this morphism be algebraic, in the sense that it factor through a maximal torus Tb of G
it is then described by characters of the form z 7→ z p z q . If this condition holds, we say that π∞
is algebraic. This corresponds to the notion of L-algebraic, in the terminology of [BG14].

RECIPROCITY AND L-FUNCTIONS

55

Galois representations, satisfying (66). Conversely, we can begin with a (suitably
irreducible) motive M and try to construct a cuspidal automorphic representation πsatisfying (66).
6.4.3. Functoriality. 67 Langlands has made another fundamental conjecture related to L-groups and his L-functions, namely his functoriality conjecture. To
describe it, we suppose that G and H are two connected reductive linear algebraic
groups over F , and that ρ : L G → L H is an L-homomorphism of their L-groups.
Suppose also that π is an automorphic representation of G(AF ), giving rise to the
Frobenius–Hecke classes cp for good primes p. We may then consider the collectionof classes ρ(cp ) in L H, and Langlands conjectures that if H is furthermore quasisplit,68 then there is an automorphic representation Π of H(AF ) whose associatedFrobenius–Hecke classes are equal to the classes r(cp ) (for all but finitely many p).
In short, automorphic representations are functorial in the L-group.
We can describe the conjectured relationship between π and Π in terms of theirL-functions: if r : L H → GLn , then a direct consideration of the definitions showsthat (taking S to be sufficiently large)(67)

LS (s, π, r ◦ ρ) = LS (s, Π, r).

6.4.4. Example. One example of functoriality is the case when H = GLn . ThenL
H is simply the direct product of GLn with the Galois group, and so we may
ignore the Galois factor, so that ρ is just a homomorphism L G → GLn . In thiscase, taking r to be the standard representation, the formula (67) reduces to theequality(68)

LS (s, π, ρ) = LS (s, Π, standard).

Thus functoriality implies that all automorphic L-functions are standard L-functions,and so in particular have analytic continuation and satisfy a functional equation.
6.4.5. Example. Consider now the case when G is the trivial group! As Langlands has emphasized, although G is trivial, its L-group is not. If we take the
Galois form for L-groups, then can write its L-group as Gal(K/F ) for some finite
extension K, and ρ : Gal(K/F ) → GLn is simply a representation of the kindconsidered in Subsection 5.1. There is only one automorphic representation of thetrivial group, namely the trivial representation, and its Frobenius–Hecke classes
are just the Frobenius conjugacy classes Frobp ∈ Gal(K/F ). Now apply functoriality, in particular (68). The L-function of the trivial representation is precisely theArtin L-function of ρ. Thus functoriality predicts the existence of an automorphic
representation Π of GLn (A) for which(69)

L(s, ρ) = L(s, Π, standard)).

If ρ is irreducible then we anticipate that Π should be cuspidal, in which case Artin’s
Conjecture 5.1.1 would hold for ρ. In this way Artin’s conjecture is situated in themuch more general context of functoriality.
Of course, since the representation ρ is a particular example of a motive, theexistence of Π satisfying (69) is also a special case of the reciprocity conjecture.
In fact, assuming that the Langlands group LF exists, we could use the LF -formof the L-group. In this case, if G is the trivial group, then L G = LF , and giving
67For an elaboration on this topic, see Arthur’s chapter in this volume [Art21].
68In (6.4.6) below we explain the significance of this assumption.

56

M. EMERTON

an L-homomorphism ρ as above amounts to giving a morphism ρ : LF → GLn .
Applying functoriality in this context, we obtain an automorphic representation
Π of GLn (AF ) for which L(s, ρ) = L(s, Π, standard). Thus functoriality, when interpreted in a sufficiently broad manner, implies the “motivic to automorphic”
direction of reciprocity!
6.4.6. Example. If G is any connected linear reductive group over F , we can take H
to be the quasi-split inner form of G, and let ρ be the equality L G = L H. We thus
expect to have a functorial transfer of automorphic representations from G(AF ) toH(AF ). It is typically not surjective — automorphic representations for the quasisplit group H can’t always be “descended” to the inner form H. This is why, inthe statement of functoriality, we assume that H is quasi-split.
In the case when H = GL2 , this transfer was established by Jacquet and Langlands in their book [JL70]. They did this via an application of the trace formula,and Langlands has emphasized the importance of the trace formula as a tool tostudy functoriality in general.
6.4.7. Example. There are many other examples of functoriality that we could mention in this context, and which will play a role in the discussion of the next section.
Among them are base change, automorphic induction, symmetric power functorialc 2 → GLn+1 = GLc n+1 ),ity (corresponding to the representation Symn : GL2 = GL
and endoscopic functoriality. Another important example occurs when G is a clasb is again classical,
sical (i.e. special orthogonal or symplectic) group. In this case GLb
and we may consider ρ : G → GLn arising from the standard representation of G.
This case of functoriality (subject to some splitness/quasi-splitness assumptions
on G) has been proved by Arthur [Art13], as part of his theory of non-tempered
endoscopy and stabilization of the trace formula [Art89].
7. Progress
The proof of class field theory — abelian reciprocity — can be thought of as
consisting of two steps: (i) constructing the ray class fields, for which one has
an explicit reciprocity law (i.e. their Galois groups, and the Frobenius elements
inside them, are described in automorphic terms), and (ii) showing that any abelian
extension is contained in a ray class field. In practice, in modern proofs of classfield theory, this order of argumentation is not necessarily observed (e.g. one canprove Artin reciprocity for arbitrary abelian extensions before proving the existence
theorem for ray class fields), but it provides a helpful framework for viewing thedominant contemporary approaches to studying reciprocity.
The analogue of ray class fields, then, will be certain varieties (or motives) that
admit an explicit reciprocity law, i.e. for which the Galois action on their `-adic
cohomology can be described in automorphic terms. These are the Shimura varieties. Although not all motives over number fields can be constructed out of them,
they play a fundamental role in the study of reciprocity.
7.1. Shimura varieties. These are a class of varieties attached to certain reductive
groups over Q, originally studied by Shimura, and subsequently by Langlands (whocoined their name) and a host of other researchers. The simplest examples arise
from the groups GL2 over Q, for which the associated Shimura varieties are modular
curves, and the groups ResE
Q Gm for imaginary quadratic extensions E of Q (which

RECIPROCITY AND L-FUNCTIONS

57

are forms of the group R×>0 U (1) ⊂ GL2 (C)), whose associated Shimura varieties
are finite sets of “singular” j-invariants (i.e. j-invariants of CM elliptic curves).
The second example above is of course famously related to the class field theoryof quadratic imaginary fields (Kronecker’s Jugendtraum). The first example wasalso intensively studied throughout the 19th and 20th centuries. In the 1950’s, itcame to be realized (by Eichler [Eic54] and Shimura [Shi58] particularly, and by
Taniyama as well) that the ζ-functions of modular curves admitted a description interms of the L-functions of weight 2 cuspforms. Exploiting the universal family of
elliptic curves over the modular curve (or, in an alternative formulation, allowing
non-constant coefficients over the modular curves), Ihara [Iha67] found that higher
weight cuspforms were also related to motivic ζ-functions.
Shimura introduced his eponymous varieties as generalizations of the modular
curves, and (exploiting the 20th century outgrowths of Kronecker’s Jugendtraum,i.e. class field theory and the theory of CM abelian varieties) he constructed theircanonical models,69 over an appropriate reflex field.70 Langlands proposed the problem of studying their ζ-functions (with both constant and twisted coefficients), as anatural generalization of Kronecker’s Jugendtraum [Lan76]. He further developed
this suite of ideas in [Lan79] and [LR87].
There is an enormous amount of subsequent research on this problem, and it has
proved to be one of the most stimulating directions of research in contemporary
number theory and arithmetic geometry. (As just one example, Langlands’s beginning investigations into this question led to his discovery of endoscopy.) We won’t
begin to attempt to describe the current state of the field. We do mention that
Arthur has given a precise conjectural description of the ζ-function of the intersection cohomology of the minimal compactification of a Shimura variety in [Art89,§9]. These conjectures imply that these ζ-functions can be expressed in terms ofautomorphic L-functions. We also mention that these are typically not exhibited as
standard L-functions, but as various L-functions attached to G and its endoscopic
groups (if G is the group giving rise to the Shimura variety).71
We also mention Langlands’s conjecture on conjugation of Shimura varieties [Lan79].
This conjecture proposes that the collection of Shimura varieties is stable under theaction of Gal(Q/Q), even though any particular Shimura variety is defined only overits reflex field E (and so stable only by Gal(Q/E)). In fact, if we think of Shimura’s
results (the so-called Shimura reciprocity law) as describing the action of Gal(Q/E)
on H 0 of a particular Shimura variety, then Langlands’s proposed extension describes the action of Gal(Q/E) on the collection of H 0 ’s taken over all Shimuravarieties. Langlands’ reciprocity law was proved (in most cases) in [DMOS82]. As
far as I know, no one has posited an extension of Langlands’s conjecture to thegeneral H i of Shimura varieties, although there should be such an extension, generalizing the conjecture of [Art89] for the action of Gal(Q/E) on the cohomology
69In many cases; the general case is treated in [Mil83].
70Shimura actually worked with what now called connected Shimura varieties, and showed thatthey are defined over ray class fields of the reflex field, the conductor of the field depending on thelevel. We follow Deligne’s formulation of the theory [Del71b, Del79], and consider (disconnected,
i.e. what are now called) Shimura varieties, which are defined over the reflex field, no matter what
the level.
71The representation ρ of the L-group that appears in these L-functions is related to thedatum originally used to define the Shimura variety. It first appeared in Langlands’s letter toLang [Lanb], and is also described in [Lan79].

58

M. EMERTON

of a particular Shimura variety. The paper [Tay12] establishes a particular resultof this type.
7.2. From automorphic representations to Galois representations. Shimuravarieties are endowed with large numbers of correspondences — “Hecke correspondences” — and so can be decomposed into motives according to the eigenvalues ofthe Hecke correspondences. Correspondingly, their ζ-functions will decompose into
the product of the L-functions of these motives, which (following e.g. the general
conjecture of [Art89]) one anticipates will be related to automorphic L-functions
associated to G or to its endoscopic groups. Reversing the order of this discussion,
one sees that if a particular automorphic representation “contributes” to the decomposition of the ζ-function, one can hope to find a motive which corresponds
to this automorphic representation, in the sense that the L-function of the motive
coincides with an L-function of the given automorphic representation.
The simplest case of this analysis is that of modular curves; in this case theupshot is the construction of a compatible family of `-adic representations [Del71a](and even a motive [Sch90]) associated to the automorphic representation of GL2 (AQ )generated by any cuspidal Hecke eigenform of weight k ≥ 2. (The weight 2 casewas treated first, by Shimura [Shi71].) This proves one direction of the reciprocity
conjecture for such automorphic representations.
The preceding construction extends to the case of Hilbert modular forms, but in
an indirect way: if F is totally real, then while GL2 (AF ) gives rise to a Shimura variety (essentially a Hilbert modular variety), the reflex field is Q, not F , and the cohomology of this Shimura variety won’t produce the 2-dimensional representations
that ought to be attached to cuspidal automorphic representations on GL2 (AF ).
(Rather, it produces their “tensor inductions” from Gal(Q/F ) to Gal(Q/Q).) Instead, one has to work with Shimura curves [Car86]; these are Shimura varieties
attached to a non-split inner form of GL2 over F , and so there are obstructions tofunctorially transferring automorphic representations from GL2 (AF ) to these inner
forms. The upshot is that one constructs compatible families of Galois representations for most, but not all, Hilbert modular eigenforms of weights ki ≥ 2.
In the case of modular eigenforms of weight k = 1, or Hilbert modular eigenforms
with one or more of its weights = 1, there is no direct connection between thesemodular forms (or the automorphic representations that they generate) and the
motives arising from modular curves or Shimura curves. One can still find associated families of `-adic representations in these cases, but they are obtained by `-adic
limiting processes from the Galois representations associated to higher weight forms(using the theory of congruences of modular forms) [DS74, RT83, Oht84, Jar97]. In
this case one works with a single ` at a time to construct the family; the compatibility comes at the end, by comparing the representations with varying ` to the fixed
modular eigenform. There are no obvious motives giving rise to these compatible
families. In the case of weight 1 modular forms, or parallel weight 1 Hilbert modular forms, one can prove a posteriori that the resulting Galois representations areof finite image, and hence are motivic. For partial weight 1 Hilbert modular forms,
the problem of showing that these Galois representation are motivic remains open
in general. The construction of compatible families associated to Hilbert modular
forms of weights ki ≥ 2 that cannot be transferred to Shimura curves also proceedsby `-adic limiting/congruence arguments. Again, the resulting compatible family is not evidently motivic. An alternative approach in this case is to find theseRECIPROCITY AND L-FUNCTIONS

59

compatible families in the cohomology of a higher dimensional Shimura variety via
endoscopy [BR93]; this approach produces a motive, at least if one grants the Tate
conjecture.
The problems of the preceding paragraphs compound if one tries to treat GLn (AF )for n > 2. The group GLn never gives rise to a Shimura variety if n > 2. If F istotally real or CM, one does have Shimura varieties associated to forms of GU (n)defined over F . These groups are outer twists of GLn . Roughly, the cuspidal automorphic representations of GLn (AF ) which satisfy an appropriate self-duality arise
as functorial transfers from automorphic representations on some form of GU (n).
One also has the happy fact that if the generalized unitary group under consideration has signature (1, n − 1) at one infinite place, and definite signature at allothers, than the most significant of the L-functions appearing in the description ofits ζ-function are in fact standard L-functions.
Applying appropriate extensions of all the methods described in the GL2 -case(cohomology of Shimura varieties, endoscopy, `-adic limiting arguments), as well
as a passage from dimension n to dimension 2n to impose self-duality in the nonessentially-self-dual case, leads to the following result.72
7.2.1. Theorem. If F is totally real or CM, and if π is a cuspidal automorphic
representation whose infinitesimal characters at all infinite places are regular, thenthere exists a compatible family of Galois representations {ϕ` } associated to π.
In many of the cases when π is self-dual, these are known to be motivic (either
unconditionally or subject to the Tate conjecture), since they are constructed in
the cohomology of Shimura varieties. But not always, and not at all if π is not
self-dual (because of the `-adic limiting arguments that are required).
7.3. From Galois representations to automorphic representations. We now
discuss the general problem of passing from a motive, or its associated compatible
family of Galois representations, to an associated automorphic representation. It
is helpful to consider this problem through the lens of various particular examplesof 2-dimensional Galois representations.
To begin with, suppose that K/Q is a degree two extension, and that ψ is an algebraic Hecke character for K. The construction of Subsection 4.6 gives rise to a compatible family {eρ` } of two-dimensional Galois representations, which, as we observedin Subsection 5.8, are motivic. As we also observed there, L(s, {eρ` }) = L(s, ψ), andso in this case the associated L-function does satisfy the expected analytic continuation and functional equation. If K is imaginary, and in certain cases if K is
real, one can then apply the Hecke–Weil converse theorem discussed in 6.2 to showthat L(s, ψ) = L(s, π, standard) for some cuspidal automorphic representation π ofGL2 (AQ ). In fact, using the relationship between quadratic fields and quadratic
forms, one can also directly use arguments with θ-series (so, ultimately, the Poissonsummation formula) to construct π (or, rather, to construct a modular eigenform fthat generates π).73 Maass [Maa49] made a similar construction in the remaining
real quadratic cases (and introduced Maass forms to this end).
72An enormous number of authors are involved in the complete proof of this result. In theessentially self-dual case, we should mention [Clo91], [Kot92], [HT01], [Shi11], and [CH13]. In the
not-necessarily self-dual case, the references are [HLTT16] and [Sch15].
73One can think of this as taking the θ-series arguments that go into proving the functional
equation for L(s, ψ), and using them directly to construct the required modular form.

60

M. EMERTON

There is an enormous generalization of the preceding construction, known as
automorphic induction. If K/F is a cyclic Galois extension of degree n, if ψ isa Hecke character, and if ρ is the induction of ψ to WK/F , then we may find an
automorphic representation π of GLn (AF ) such that L(s, ρ) = L(s, π). In the casewhen ψ is algebraic, so that ρ gives rise to the motivic family {eρ` } (generalizing
the context of the preceding paragraph), we then have L(s, {eρ` }) = L(s, π), and wehave established reciprocity for the family {eρ` }.
When n = 2, the converse theorem of [JL70] can be used to prove this result. For
general n, a different argument is required. One considers the problem in the opticof (6.4.3), by working with the Weil form of the L-group. Automorphic inductionis then proved by an argument using the trace formula [AC89, Hen12].
Another 2-dimensional example comes from 2-dimensional Artin representations.
If these have solvable image in GL2 (C), then by analyzing the structure of these
representations (e.g. relating them to certain inductions), and using automorphic
induction, as well as cases of base change functoriality, they can be shown to satisfy
reciprocity — this is the celebrated Langlands–Tunnell theorem [Lan80, Tun81].
In particular, this establishes the Artin conjecture (Conjecture 5.1.1) for these
representations. In fact, in this case, the Artin conjecture is essentially equivalent
to reciprocity, by the Jacquet-Langlands converse theorem for GL2 .74
Yet another 2-dimensional example is given by H 1 (E) for an elliptic curve E
over Q. The conjecture on meromorphic continuation of ζ(s, E) is attributed toHasse by Weil [Wei52], while the reciprocity conjecture for H 1 (E) seems to havebeen discovered by Shimura and Taniyama. Taking into account the converse theorems for GL2 , this reciprocity is seen to be equivalent
to the holomorphic continua
tion and functional equation of L s, H 1 (E) ; and indeed, Weil proved his converse
theorem in part to establish this equivalence. In particular, if E has CM, then we
have seen that H 1 (E) arises from a Weil group representation, and so we are in thecase discussed at the beginning of this subsection — the converse theorem indeed
can be applied to deduce reciprocity for H 1 (E). For elliptic curves without CM, thecompatible family H 1 (E) does not arise from a Weil group character, and there isno evident way to directly prove this analytic continuation and functional equation.
Indeed, the only known proof of this analytic continuation and functional equation
is via reciprocity.
In this case, reciprocity was famously proved by Wiles [Wil95] and Taylor–Wiles [TW95] for semistable E, and then by Breuil–Conrad–Diamond–Taylor [BCDT01]
for general E. We will say the briefest amount about the methods. They are `-adic
in nature, exploiting the theory of congruences of modular forms. Very roughly, one
finds an automorphic representation π 0 which is known to have an associated `-adic
representation, and for which this `-adic representation, when reduced mod `, isisomorphic to the mod ` reduction of H 1 (E) (so, concretely, to the Galois representation on the group of `-division points on E). An automorphy lifting theorem(here “lifting” refers to lifting back from mod ` to the original `-adic representation H 1 (E, Q` )) then implies that H 1 (E, Q` ) is itself “automorphic”, i.e. the `-adic
representation associatedto an automorphic representation π. Now we are done:L(s, π) = L s, H 1 (E) (since we can check this by working at a single `).
74Tate once wryly noted that between the wars, both Artin and Hecke were in Hamburg, andboth were studying the problem of analytic continuation of L-functions. Nevertheless, it took
thirty-odd more years before Langlands made the connection between their respective viewpoints.

RECIPROCITY AND L-FUNCTIONS

61

As Wiles remarks in the introduction to [Wil95], this method of argument does
not need the entire compatible family of `-adic representations, other than thatsome ` may be easier to argue with than others. Having flexibility in choosing
which ` to work with turns out to be essential, though, because in implementing
the method, one must exhibit an existing automorphic representation π 0 which will
give rise to the desired automorphic representation π via automorphy lifting.
In the original argument of [Wil95], the choice of ` = 3 is made; since GL2 (F3 ) issolvable, one can use the Langlands–Tunnell result to construct a π 0 . For certain E,
the argument at ` = 3 doesn’t work, and one shifts to ` = 5 instead. For this one,has to find a π 0 matching with the 5-torsion of E. To this end, one finds anotherelliptic curve E 0 with E 0 [5] = E[5] (as Gal(Q/Q)-modules), but with better behaved
3-torsion, so that the Langlands–Tunnell result applies. The previous argumentshows that E 0 satisfies reciprocity, so now one has a π 0 from which to construct thedesired π via 5-adic automorphy lifting. The argument of [BCDT01] uses further
switching between the primes 3 and 5. These switches are possible because there
are an abundance of auxiliary E 0 available with prescribed 3 or 5-torsion, since themodular curves of level 3 and 5 are rational curves.
Once one tries to generalize this argument, the moduli schemes from which one
is “sampling” one’s auxiliary objects have essentially uncontrollable geometry, andthere is no guarantee that they have rational points. This necessitates having topass from Q, or whatever ground field one is working over, to an extension field, inorder to find the auxiliary automorphic representation π 0 that will effectuate theautomorphy lifting. One also needs a source of “automatic automorphy”, since ananalogue of Langlands–Tunnell is typically not available. Automorphic induction
provides the solution to this — mod ` Galois representations which are monomial
automatically arise from automorphic representations π 0 .
We should also mention that the automorphy lifting arguments depend verymuch on the “automorphic to Galois” direction of reciprocity. And of course there
are many other technical constraints which limit the precise arguments that can bemade and, consequently, the scope of the results that can be proved. Nevertheless,much progress has been made in the last 25 years. Here are some sample recent
results.
7.3.1. Theorem (Allen–Calegari–Caraiani–Gee–Helm–Le Hung–Newton–Scholze—
Taylor–Thorne [ACC+ 18]). The Sato–Tate conjecture holds for any elliptic curve
over a CM field.75
7.3.2. Theorem (Boxer–Calegari–Gee–Pilloni [BCGP18]). If C is a genus two
curve over a totally real field, then ζC (s) admits a meromorphic continuation to
the entire complex plane, and satisfies the expected functional equation.
Both results are about meromorphic continuation — the second evidently, andthe first because, as we explained above, the proof of the Sato–Tate conjecture
requires us to continue each L s, Symn H 1 (E) to the line s ≥ 1 + n2 , and to provethat it is zero-free along the edge of this region. Both results are established byproving a potential version of reciprocity; i.e. by constructing the relevant automorphic representation π that controls the given L-function, not over F , but over some(uncontrollably large) extension K of F . (As explained above, this is necessary to
75As indicated in footnote 57, the case of elliptic curves over totally real fields was proved
earlier.

62

M. EMERTON

find the required auxiliary points on some moduli scheme.) A variant
 of Brauer’s
argument (Theorem 5.1.3) then shows that each L(s, Symm H 1 (E) , in the context of the Theorem 7.3.1, or L(s, H 1 (C)), in the context of Theorem 7.3.2, has
the required meromorphic continuation. We also remark that the the automorphy
lifting argument used to prove Theorem 7.3.2 takes place in the context of automorphic representations of a (generalized) symplectic group, and the authors rely
on Arthur’s work on functoriality for classical groups [Art13] (mentioned in (6.4.7))
to compare these automorphic representations with automorphic representations
on the group GL4 — for example, in order to manipulate the associated Galois
representations, and in order to apply base-change arguments such as those usedin the Brauer-type argument for meromorphic continuation.
In his Shaw Prize address [Lan11], Langlands observes that the Sato–Tate conjecture, analogous as it is to Čebotarev’s theorem, is closer to Brauer’s Theorem 5.1.3
than to Artin’s Conjecture 5.1.1; this is a valid point, and indeed, the potential
reciprocity that is being applied here to deduce it is weaker than true reciprocity.
On the other hand, from the point of view of current technique, reciprocity andpotential reciprocity don’t seem all that different; to establish one rather than theother, one just has to find a method for constructing the auxiliary automorphic representation π 0 without making an unwanted field extension. One recent instance of
this is the paper [AKT19] of Allen–Khare–Thorne, in which the authors are able
to improve on some of the results of [ACC+ 18] by finding a large class of elliptic
curves over CM fields for which reciprocity holds genuinely, not just potentially.
We also mention the paper [CCG20], which employs the arguments of [BCGP18]
to give examples of abelian surfaces over Q for which reciprocity holds.
In the same address, Langlands also discusses the significance of establishing
results such as Sato–Tate, which are consequences of functoriality, by using thetechnique of reciprocity instead. The same ten authors who proved Theorem 7.3.1
also established entirely new cases of the Ramanujan conjecture for cuspidal automorphic representations on GL2 (AF ) (F a CM field). This is another applicationof reciprocity to establish a consequence of functoriality.
As we noted in (6.4.3), from a certain viewpoint reciprocity can be entirely
subsumed into functoriality. In any event, the two problems seem to be just as,
or even more, intertwined than ever before. Whether this is merely a contingent
phenomenon, or the reflection of something deeper, remains to be seen.
References[AC89][ACC+ 18]

[AGV72][AKT19][AL70]

James Arthur and Laurent Clozel, Simple algebras, base change, and the advanced
theory of the trace formula, Annals of Mathematics Studies, vol. 120, Princeton
University Press, Princeton, NJ, 1989. MR MR1007299 (90m:22041)
Patrick B. Allen, Frank Calegari, Ana Caraiani, Toby Gee, David Helm, Bao V. Le
Hung, James Newton, Peter Scholze, Richard Taylor, and Jack A. Thorne, Potential
automorphy over cm fields, 2018.
M. Artin, A. Grothendieck, and J. Verdier, Théorie des topos et cohomologie étale desschémas. Tome 2, Lecture Notes in Mathematics, Vol. 270, Springer-Verlag, BerlinNew York, 1972, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA
4), Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de
N. Bourbaki, P. Deligne et B. Saint-Donat. MR 0354653 (50 #7131)
Patrick B. Allen, Chandrashekhar Khare, and Jack A. Thorne, Modularity ofGL2 (Fp )-representations over cm fields, 2019.
A. O. L. Atkin and J. Lehner, Hecke operators on Γ0 (m), Math. Ann. 185 (1970),
134–160. MR 268123

RECIPROCITY AND L-FUNCTIONS[Art24]63

E. Artin, Quadratische Körper im Gebiete der höheren Kongruenzen. I, II, Math.
Z. 19 (1924), no. 1, 153–206, 207–246. MR 1544652[Art27]
Emil Artin, Beweis des allgemeinen Reziprozitätsgesetzes, Abh. Math. Sem. Univ.
Hamburg 5 (1927), no. 1, 353–363. MR 3069486[Art31]
E. Artin, Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren, Abh.
Math. Sem. Univ. Hamburg 8 (1931), no. 1, 292–306. MR 3069563[Art89]James Arthur, Unipotent automorphic representations: conjectures, Astérisque 171172 (1989), 13–71, Orbites unipotentes et représentations, II. MR 1021499[Art02], A note on the automorphic Langlands group, Canad. Math. Bull. 45 (2002),
no. 4, 466–482, Dedicated to Robert V. Moody. MR MR1941222 (2004a:11120)[Art13], The endoscopic classification of representations, American MathematicalSociety Colloquium Publications, vol. 61, American Mathematical Society, Providence, RI, 2013, Orthogonal and symplectic groups. MR 3135650, An introduction to Langlands functoriality, 2021, Genesis of the Langlands[Art21]
program.
[BCDT01] Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001),
no. 4, 843–939 (electronic). MR 1839918[BCGP18] George Boxer, Frank Calegari, Toby Gee, and Vincent Pilloni, Abelian surfaces overtotally real fields are potentially modular, 2018.
[BG14]
Kevin Buzzard and Toby Gee, The conjectural connections between automorphic
representations and Galois representations, Automorphic forms and Galois representations. Vol. 1, London Math. Soc. Lecture Note Ser., vol. 414, Cambridge Univ.
Press, Cambridge, 2014, pp. 135–187. MR 3444225[BJ79]
A. Borel and H. Jacquet, Automorphic forms and automorphic representations, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII,
Amer. Math. Soc., Providence, R.I., 1979, With a supplement “On the notion of an
automorphic representation” by R. P. Langlands, pp. 189–207. MR 546598[BLGHT11] Tom Barnet-Lamb, David Geraghty, Michael Harris, and Richard Taylor, A familyof Calabi-Yau varieties and potential automorphy II, Publ. Res. Inst. Math. Sci. 47(2011), no. 1, 29–98. MR 2827723[BR93]Don Blasius and Jonathan D. Rogawski, Motives for Hilbert modular forms, Invent.
Math. 114 (1993), no. 1, 55–87. MR 1235020[Bra47a]Richard Brauer, On Artin’s L-series with general group characters, Ann. of Math.
(2) 48 (1947), 502–514. MR 20105, On the zeta-functions of algebraic number fields, Amer. J. Math. 69 (1947),[Bra47b]
243–250. MR 20597[Car86]Henri Carayol, Sur les représentations l-adiques associées aux formes modulaires deHilbert, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 409–468.
[CCG20]Frank Calegari, Shiva Chidambaram, and Alexandru Ghitza, Some modular abeliansurfaces, Math. Comp. 89 (2020), no. 321, 387–394. MR 4011548[CH13]
Gaëtan Chenevier and Michael Harris, Construction of automorphic Galois representations, II, Camb. J. Math. 1 (2013), no. 1, 53–73. MR 3272052[Che40]C. Chevalley, La théorie du corps de classes, Ann. of Math. (2) 41 (1940), 394–418.
MR 2357[CHLN11] Laurent Clozel, Michael Harris, Jean-Pierre Labesse, and Bao-Châu Ngô (eds.), On
the stabilization of the trace formula, Stabilization of the Trace Formula, Shimura
Varieties, and Arithmetic Applications, vol. 1, International Press, Somerville, MA,
2011. MR 2742611[CHT08]Laurent Clozel, Michael Harris, and Richard Taylor, Automorphy for some l-adic
lifts of automorphic mod l Galois representations, Pub. Math. IHES 108 (2008),
1–181.
[Clo91]
Laurent Clozel, Représentations galoisiennes associées aux représentations automorphes autoduales de GL(n), Inst. Hautes Études Sci. Publ. Math. (1991), no. 73,
97–145. MR 1114211[Ded]
R. Dedekind, Supplement XI to Vorlesungen über Zahlentheorie, by L. Dirichlet, 4thed., 1894.

64[Del71a][Del71b][Del73][Del74][Del79][Dir][Dir37][DMOS82][DS74][Dwo56][Dwo60][Eic54][Eula][Eulb][Eulc][Euld][Eule][Eulf][Fla79][GJ72][Har77][Has36]

M. EMERTON

Pierre Deligne, Formes modulaires et représentations l-adiques, Séminaire Bourbaki,
Exposés 347–363, Lecture Notes in Mathematics, vol. 179, Springer-Verlag, 1971,
pp. 139–172.
, Travaux de Shimura, Séminaire Bourbaki, 23ème année (1970/71), Exp. No.
389, 1971, pp. 123–165. Lecture Notes in Math., Vol. 244. MR 0498581
P. Deligne, Les constantes des équations fonctionnelles des fonctions L, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp,
Antwerp, 1972), 1973, pp. 501–597. Lecture Notes in Math., Vol. 349. MR 0349635
Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. (1974),
no. 43, 273–307. MR 340258, Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, Automorphic forms, representations and L-functions(Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc.
Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 247–
289. MR 546620
L. Dirichlet, Recherches sur diverses applications de l’analyse infinitésimale a lathéorie des nombres, J. Reine Angew. Math., 19 (1839), 324–369, 21 (1840), 1–12,
134–155.
, Beweis des satzes, dass jede unbegrenzte arithmetische progression, deren
erstes glied und differenz ganze zahlen ohne gemeinschaftlichen factor sing, unendlich viele primzahlen erhält, Abhandlungen der Königlich Preussischen Akademie
der Wissenschaften (1837), 45–81.
Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih, Hodge cycles,
motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900, SpringerVerlag, Berlin-New York, 1982. MR 654325
Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids 1, Ann. Sci. Ec.
Norm. Sup. 7 (1974), 507–530.
B. Dwork, On the Artin root number, Amer. J. Math. 78 (1956), 444–472. MR 82476
Bernard Dwork, On the rationality of the zeta function of an algebraic variety, Amer.
J. Math. 82 (1960), 631–648. MR 140494
Martin Eichler, Quaternäre quadratische Formen und die Riemannsche Vermutungfür die Kongruenzzetafunktion, Arch. Math. 5 (1954), 355–366. MR 63406
L. Euler, De seriebus quibusdam considerationes, Opera Omnia: Series 1, Volume
14, pp. 407 – 462.
, De summa seriei ex numeris primis formatae 1/3–1/5 + 1/7 +1/11–1/13–1/17+1/19+1/23–1/29+1/31 etc. ubi numeri primi formae 4n–1 habentsignum positivum, formae autem 4n + 1 signum negativum, Opera Omnia: Series 1,
Volume 4, pp.146–162.
, De summis serierum reciprocarum, Opera Omnia: Series 1, Volume 14,
pp.73–89.
, Introductio in analysin infinitorum, volume 1, Opera Omnia: Series 1, Volume 8.
, Remarques sur un beau rapport entre les series desepuissances tant directes
que reciproques, Opera Omnia: Series 1, Volume 15, pp. 70 – 90.
, Variae observationes circa series infinitas, Opera Omnia: Series 1, Volume
14, pp. 217 – 244.
D. Flath, Decomposition of representations into tensor products, Automorphic forms,
representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ.,
Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Amer. Math.
Soc., Providence, R.I., 1979, pp. 179–183. MR 546596
Roger Godement and Hervé Jacquet, Zeta functions of simple algebras, Lecture Notes
in Mathematics, Vol. 260, Springer-Verlag, Berlin-New York, 1972. MR 0342495
Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977,
Graduate Texts in Mathematics, No. 52. MR 0463157
Helmut Hasse, Zur Theorie der abstrakten elliptischen Funktionenkörper I, II, III,
J. Reine Angew. Math. 175 (1936), 55–62, 69–88, 193–208. MR 1581496

RECIPROCITY AND L-FUNCTIONS[Has54][Has67][Hec17][Hec20][Hec27][Hec36][Hec37][Hen00][Hen12][hl][HLTT16][HSBT10][HT01][Iha67][Jar97][JL70][Kat75][Kle94][Kot84][Kot92][Kud03][Lana][Lanb][Lanc][Lan03]65, Artinsche Führer, Artinsche L-Funktionen und Gaussche Summen über
endlich-algebraischen Zahlkörpern, Acta Salmanticensia. Ciencias: Sec. Mat. 1954
(1954), no. 4, viii+113. MR 0064816, History of class field theory, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 266–279.
MR 0218330
Erich Hecke, Über die L-funktionen und den Dirichletschen Primzahlsatz für einenbeliebigen Zahlkörper, Nachr. Ges. Wiss. Göttingen (1917), 299–318.
E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung
der Primzahlen, Math. Z. 6 (1920), no. 1-2, 11–51. MR 1544392, Zur Theorie der elliptischen Modulfunktionen, Math. Ann. 97 (1927), no. 1,
210–242. MR 1512360, Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung,Math. Ann. 112 (1936), no. 1, 664–699. MR 1513069, Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I, II, Math. Ann. 114 (1937), no. 1, 1–28, 316–351. MR 1513122
Guy Henniart, Une preuve simple des conjectures de Langlands pour GL(n) sur uncorps p-adique, Invent. Math. 139 (2000), no. 2, 439–455. MR 1738446, Induction automorphe globale pour les corps de nombres, Bull. Soc. Math.
France 140 (2012), no. 1, 1–17. MR 2903769
Franz
Lemmermeyer(https://mathoverflow.net/users/3503/franzlemmermeyer), History of the analytic class number formula, MathOverflow,
URL:https://mathoverflow.net/q/242603 (version: 2016-06-19).
Michael Harris, Kai-Wen Lan, Richard Taylor, and Jack Thorne, On the rigid cohomology of certain Shimura varieties, Res. Math. Sci. 3 (2016), Paper No. 37, 308.
MR 3565594
Michael Harris, Nick Shepherd-Barron, and Richard Taylor, A family of Calabi-Yau
varieties and potential automorphy, Ann. of Math. (2) 171 (2010), no. 2, 779–813.
MR 2630056
Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001, With an appendix by Vladimir G. Berkovich.
MR MR1876802 (2002m:11050)
Yasutaka Ihara, Hecke Polynomials as congruence ζ functions in elliptic modularcase, Ann. of Math. (2) 85 (1967), 267–295. MR 207655
Frazer Jarvis, On Galois representations associated to Hilbert modular forms, J.
Reine Angew. Math. 491 (1997), 199–216. MR 1476093
H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes inMathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR 0401654
Nicholas M. Katz, p-adic L-functions via moduli of elliptic curves, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif.,1974), 1975, pp. 479–506. MR 0432649
Steven L. Kleiman, The standard conjectures, Motives (Seattle, WA, 1991), Proc.
Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 3–20.
MR 1265519
Robert E. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J.
51 (1984), no. 3, 611–650. MR 757954, On the λ-adic representations associated to some simple Shimura varieties,
Invent. Math. 108 (1992), no. 3, 653–665. MR 1163241
Stephen S. Kudla, Tate’s thesis, An introduction to the Langlands program(Jerusalem, 2001), Birkhäuser Boston, Boston, MA, 2003, pp. 109–131. MR 1990377
R. P. Langlands, Letter to André Weil, January 1967, published in “Emil Artinand beyond—class field theory and L-functions”, European Mathematical Society(EMS), Zürich, 2015.
Robert P. Langlands, Letter to Lang, December 5, 1970.
, On the functional equation of the Artin L-functions.
Edmund Landau, Neuer Beweis des Primzahlsatzes und Beweis des Primidealsatzes,
Math. Ann. 56 (1903), no. 4, 645–670. MR 1511191

66[Lan70][Lan71][Lan76][Lan79][Lan80][Lan89][Lan11][LR87][Maa49][Maz77][Mil83][Mil16][Oht84][Rie][RT83][Sat63][Sch90][Sch15][Ser60][Ser65][Ser68][Ser70]M. EMERTON

R. P. Langlands, Problems in the theory of automorphic forms, Lectures in modern
analysis and applications, III, 1970, pp. 18–61. Lecture Notes in Math., Vol. 170.
MR 0302614
Robert P. Langlands, Euler products, Yale University Press, New Haven, Conn.London, 1971, A James K. Whittemore Lecture in Mathematics given at Yale University, 1967, Yale Mathematical Monographs, 1. MR 0419366
R. P. Langlands, Some contemporary problems with origins in the Jugendtraum,
Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure
Math., Vol. XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974), 1976, pp. 401–
418. MR 0437500, Automorphic representations, Shimura varieties, and motives. Ein
Märchen, Automorphic forms, representations and L-functions (Proc. Sympos. Pure
Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math.,
XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 205–246. MR 546619
Robert P. Langlands, Base Change for GL(2), Annals of Math. Studies 96 (1980).
R. P. Langlands, On the classification of irreducible representations of real algebraic groups, Representation theory and harmonic analysis on semisimple Lie groups,
Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 101–
170. MR 1011897
Robert P. Langlands, Reflexions on receiving the Shaw Prize, On certain L-functions,
Clay Math. Proc., vol. 13, Amer. Math. Soc., Providence, RI, 2011, pp. 297–308.
MR 2767520
R. P. Langlands and M. Rapoport, Shimuravarietäten und Gerben, J. Reine Angew.
Math. 378 (1987), 113–220. MR 895287
Hans Maass, Über eine neue Art von nichtanalytischen automorphen Funktionen und
die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann.
121 (1949), 141–183. MR 31519
Barry Mazur, Book Review: Ernst Edward Kummer, Collected Papers, Bull. Amer.
Math. Soc. 83 (1977), no. 5, 976–988. MR 1566995
J. S. Milne, The action of an automorphism of f C on a Shimura variety and its
special points, Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser
Boston, Boston, MA, 1983, pp. 239–265. MR 717596
James S. Milne, The Riemann hypothesis over finite fields: from Weil to the present
day [Reprint of 3525903], ICCM Not. 4 (2016), no. 2, 14–52. MR 3635684
Masami Ohta, Hilbert modular forms of weight one and Galois representations, Automorphic forms of several variables (Katata, 1983), Progr. Math., vol. 46, Birkhäuser
Boston, Boston, MA, 1984, pp. 333–352. MR 763021
B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie, November 1859.
J. D. Rogawski and J. B. Tunnell, On Artin L-functions associated to Hilbert modular
forms of weight one, Invent. Math. 74 (1983), no. 1, 1–42. MR 722724
Ichirô Satake, Theory of spherical functions on reductive algebraic groups over p-adic
fields, Inst. Hautes Études Sci. Publ. Math. (1963), no. 18, 5–69. MR 195863
A. J. Scholl, Motives for modular forms, Invent. Math. 100 (1990), no. 2, 419–430.
MR 1047142
Peter Scholze, On torsion in the cohomology of locally symmetric varieties, Ann. of
Math. (2) 182 (2015), no. 3, 945–1066. MR 3418533
Jean-Pierre Serre, Analogues kählériens de certaines conjectures de Weil, Ann. of
Math. (2) 71 (1960), 392–394. MR 112163, Zeta and L functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue
Univ., 1963), Harper & Row, New York, 1965, pp. 82–92. MR 0194396, Abelian l-adic representations and elliptic curves, McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute, W. A.
Benjamin, Inc., New York-Amsterdam, 1968. MR 0263823, Facteurs locaux des fonctions zêta des varietés algébriques (définitions etconjectures), Séminaire Delange-Pisot-Poitou. 11e année: 1969/70. Théorie des nombres. Fasc. 1: Exposés 1 à 15; Fasc. 2: Exposés 16 à 24, Secrétariat Math., Paris,
1970, p. 15. MR 3618526

RECIPROCITY AND L-FUNCTIONS[Ser77][Sha21][Shi58][Shi71][Shi11][Tak20][Tan57][Tat65][Tat67][Tat79][Tat94][Tay08][Tay12][Tsc26][Tun81][TW95][Wei49][Wei51][Wei52][Wei56a][Wei56b][Wei67][Wei95][Wil95]67, Cours d’arithmétique, Presses Universitaires de France, Paris, 1977,
Deuxième édition revue et corrigée, Le Mathématicien, No. 2. MR 0498338
F. Shahidi, Automorphic l-functions, 2021, Genesis of the Langlands program.
Goro Shimura, Correspondances modulaires et les fonctions ζ de courbes algébriques,
J. Math. Soc. Japan 10 (1958), 1–28. MR 95173, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton
University Press, 1971.
Sug Woo Shin, Galois representations arising from some compact Shimura varieties,
Ann. of Math. (2) 173 (2011), no. 3, 1645–1741. MR 2800722
T. Takagi, Über eine theorie des relativ-abel’schen Zahlkörpers, J. Coll. Sci. imp.
Univ. Tokyo 41 (1920), no. 9, 1–133.
Yutaka Taniyama, L-functions of number fields and zeta functions of abelian varieties, J. Math. Soc. Japan 9 (1957), 330–366. MR 95161
John T. Tate, Algebraic cycles and poles of zeta functions, Arithmetical Algebraic
Geometry (Proc. Conf. Purdue Univ., 1963), Harper & Row, New York, 1965, pp. 93–
110. MR 0225778
J. T. Tate, Fourier analysis in number fields, and Hecke’s zeta-functions, Algebraic
Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington,
D.C., 1967, pp. 305–347. MR 0217026
J. Tate, Number theoretic background, Automorphic forms, representations and Lfunctions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977),
Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I.,
1979, pp. 3–26. MR MR546607 (80m:12009)
John Tate, Conjectures on algebraic cycles in l-adic cohomology, Motives (Seattle,
WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI,
1994, pp. 71–83. MR 1265523
Richard Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations. II, Pub. Math. IHES 108 (2008), 183–239.
, The image of complex conjugation in l-adic representations associated toautomorphic forms, Algebra Number Theory 6 (2012), no. 3, 405–435. MR 2966704
N. Tschebotareff, Die Bestimmung der Dichtigkeit einer Menge von Primzahlen,welche zu einer gegebenen Substitutionsklasse gehören, Math. Ann. 95 (1926), no. 1,
191–228. MR 1512273
Jerrold Tunnell, Artin’s Conjecture for representations of octahedral type, Bull. AMS
5 (1981), 173–175.
Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Annals of Mathematics 142 (1995), 553–572.
André Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math.
Soc. 55 (1949), 497–508. MR 29393, Sur la théorie du corps de classes, J. Math. Soc. Japan 3 (1951), 1–35.
MR 44569, Number-theory and algebraic geometry, Proceedings of the International
Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, Amer. Math. Soc.,
Providence, R. I., 1952, pp. 90–100. MR 0045416, Abstract versus classical algebraic geometry, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, Erven P. Noordhoff
N.V., Groningen; North-Holland Publishing Co., Amsterdam, 1956, pp. 550–558.
MR 0092196, On a certain type of characters of the idèle-class group of an algebraic
number-field, Proceedings of the international symposium on algebraic number
theory, Tokyo & Nikko, 1955, Science Council of Japan, Tokyo, 1956, pp. 1–7.
MR 0083523, Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen,Math. Ann. 168 (1967), 149–156. MR 207658, Fonction zêta et distributions, Séminaire Bourbaki, Vol. 9, Soc. Math.
France, Paris, 1995, pp. Exp. No. 312, 523–531. MR 1610983
Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Annals of Mathematics 142 (1995), 443–551.

68

M. EMERTON[Wym72]B. F. Wyman, What is a reciprocity law?, Amer. Math. Monthly 79 (1972), 571–586;
correction, ibid. 80 (1973), 281. MR 308084

Email address: emerton@math.uchicago.edu
Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago,
IL 60637, USA