VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES
MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

1. Introduction
Let ρ̄ : GQ → GL2 (k) be an absolutely irreducible modular Galois representation
over a finite field k of characteristic p. Assume further that ρ̄ is p-ordinary andp-distinguished in the sense that the restriction of ρ̄ to a decomposition group at p
is reducible and non-scalar. The Hida family H(ρ̄) of ρ̄ is the set of all p-ordinary
p-stabilized newforms f with mod p Galois representation isomorphic to ρ̄. (If ρ̄is unramified at p, then one must also fix an unramified line in ρ̄ and require thatthe ordinary line of f reduces to this fixed line.) These newforms are a dense set ofpoints in a certain p-adic analytic space of overconvergent eigenforms, consisting of
an intersecting system of branches (i.e. irreducible components) T(a) indexed by
the minimal primes a of a certain Hecke algebra.
To each modular form f ∈ H(ρ̄) one may associate the Iwasawa invariantsµan (f ), λan (f ), µalg (f ) and λalg (f ). The analytic (resp. algebraic) λ-invariants
are the number of zeroes of the p-adic L-function (resp. of the characteristic power
series of the dual of the Selmer group) of f , while the µ-invariants are the exponentsof the powers of p dividing the same objects. In this paper we prove the following
results on the behavior of these Iwasawa invariants as f varies over H(ρ̄).
Theorem 1. Fix ∗ ∈ {alg, an}. If µ∗ (f0 ) = 0 for some f0 ∈ H(ρ̄), then µ∗ (f ) = 0
for all f ∈ H(ρ̄). (We then write simply µ∗ (ρ̄) = 0.)
It is conjectured by Greenberg that if a p-ordinary modular form f of weight
two has a residually irreducible Galois representation, then µan (f ) = µalg (f ) =0; Theorem 1 thus shows that this conjecture is equivalent to the corresponding
conjecture for modular forms of arbitrary weight.
Theorem 2. Fix ∗ ∈ {alg, an} and assume that µ∗ (ρ̄) = 0. Let f1 , f2 ∈ H(ρ̄) lie
on the branches T(a1 ), T(a2 ) respectively. ThenX
(1.1)
λ∗ (f1 ) − λ∗ (f2 ) =e` (a2 ) − e` (a1 );`|N1 N2

here the sum is over all primes dividing the tame level of f1 or f2 and e` (aj ) is acertain explicit non-negative invariant of the branch T(aj ) and the prime `.
The first (resp. second) theorem corresponds to Theorems 3.7.5 and 4.3.3 (resp.
Theorems 3.7.7 and 4.3.4) with trivial twist by the mod p cyclotomic character.
Note that the right-hand side of (1.1) is identical both algebraically and analytically. In particular, using work of Kato on the main conjecture of Iwasawa theory
for modular forms, we obtain the following result.
Corollary 1. Assume that µalg (ρ̄) = µan (ρ̄) = 0. If the main conjecture holds forsome f0 ∈ H(ρ̄), then it holds for all f ∈ H(ρ̄).
1

2

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

The corollary in particular reduces the main conjecture to the case of modular forms of weight two, together with the conjecture on the vanishing of theµ-invariant. We also obtain the following result on the variation of λ-invariants in
a Hida family.
Corollary 2. Fix ∗ ∈ {alg, an} and assume that µ∗ (ρ̄) = 0.
(1) λ∗ (·) is constant on branches of H(ρ̄).
(2) λ∗ (·) is minimized on the branches of H(ρ̄) of minimal tame level.
Although we give an extensive summary of our methods in the following sections,
we give a brief indication now of our approach. On the analytic side, our results are
obtained by specialization of a two-variable p-adic L-function. Our construction is
inspired by those of [22, 28], although we are able to obtain somewhat better control
over the canonical periods. This control of canonical periods in families and overvarying tame level (see Section 3 especially Theorem 3.6.2) is the key technicalinput needed for our Iwasawa theoretic applications. On the algebraic side we usea theory of residual Selmer groups to relate the Iwasawa invariants of the modular
forms in H(ρ̄).
This work was motivated by the paper [15] of Greenberg and Vatsal where Theorems 1 and 2 were obtained for p-ordinary modular forms of weight two. Our results
here help to illuminate the results of [15]; indeed, if two congruent elliptic curves
have different λ-invariants, it follows from these results that they must lie on two
branches of the associated Hida family with different ramification behavior. One
may thus think of the change of λ-invariants in terms of “jumps” as one moves fromone branch to another at crossing points (which necessarily occur at non-classical1
eigenforms).
1.1. Iwasawa theory. The papers [12] of Greenberg and [27] of Mazur introduce
the following point of view on Iwasawa theory: If X is a p-adic analytic family of
p-adic representations of GQ interpolating a collection of motivic representations,then there should exist an analytic function (or perhaps a section of an invertible
sheaf) L, which we call a p-adic L-function, defined on X and a coherent sheaf H
on X, such that the zero locus of L coincides with the codimension one part ofthe support of H (thought of as a Cartier divisor on X). The function L should
p-adically interpolate the (suitably modified) values at s = 1 of the classical Lfunctions attached to the Galois representations at motivic points of X, while atsuch a motivic point x ∈ X, the fiber Hx should coincide with an appropriately
defined Selmer group of the motive attached to x.
The L-function considered above is what is usually known as the analytic p-adic
L-function in classical Iwasawa theory; the equation of the codimension one part ofthe support of H is usually known as the algebraic p-adic L-function. The equalityof the zero locus of L with the divisorial part of the support of H is thus a statementof the main conjecture of Iwasawa theory for the family of Galois representations
X.
In this paper, we restrict to the case of two dimensional nearly ordinary modular representations, in which case the space X can be described via Hida theory.
Consider a nearly ordinary residual representation; such a representation may be
written in the form ρ̄ ⊗ ω i : GQ → GL2 (k), where ρ̄ is as above, ω denotes the
1To avoid circumlocutions, we adopt the convention that weight one is a non-classical weight.

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

3

mod p cyclotomic character and 0 ≤ i ≤ p − 1. As ρ̄ is ordinary, we may (and do)
fix a p-stabilization of ρ̄ ⊗ ω i : a choice of one dimensional Gp -invariant quotient ofρ̄ ⊗ ω i .
Consider the universal deformation space YΣn.ord parameterizing all nearly ordinary deformations of the p-stabilized representation ρ̄ ⊗ ω i which are unramified atprimes in Σ ∪ {p} for some fixed finite set of primes Σ. Any such deformation isof the form ρ ⊗ ω i ⊗ χ, where ρ is an ordinary deformation of ρ and χ is a character of Γ := 1 + pZp . (We regard χ as a Galois character by composing it withthe projection onto Γ of the p-adic cyclotomic character.) If we let YΣord denotethe universal deformation space parameterizing all ordinary deformations of the
p-stabilized representation ρ̄ which are unramified at primes in Σ ∪ {p}, then YΣn.ord
is equal to the product (as formal schemes) of YΣord and Spf (Zp [[Γ]]) . By results ofWiles and Taylor-Wiles [37, 34], as strengthened by Diamond [6], the deformation
spaces YΣn.ord can be identified with various local pieces of the universal ordinary
Hecke algebra constructed by Hida (at least under mild hypotheses on ρ̄).
If we allow Σ to vary, the spaces YΣn.ord form a formal Ind-scheme over W (k) parameterizing nearly ordinary deformations of ρ̄ of arbitrary conductor. Let Y n.ord
be an irreducible component of this formal Ind-scheme; we may write Y n.ord as
the product of Y ord (an irreducible component of the the formal Ind-scheme parameterizing ordinary deformations of ρ̄) and Spf (Zp [[Γ]]). The motivic points aredense on Y n.ord and all have the same tame conductor, which we will denote bydenotes the new-quotient of the Hida Hecke algebra of level N , thenN . If Tnew
Nord
= Spf (Tnew
Y ord corresponds to a minimal prime a of Tnew
N /a) andN ; we set X
ord
n.ord
new
is nearly the same as Y ord
X= Spf (TN /a [[Γ]]). (We point out that X– and hence X n.ord is nearly the same as Y n.ord . Indeed, X ord is a finite cover ofY ord and maps isomorphically to it after inverting p.)
The generalities of Iwasawa theory discussed above apply to this space X n.ord .
In Section 3 we give a construction of the analytic p-adic L-function on X n.ord ;as is usual, we regard it as a “function of two-variables” – one variable moving
along X ord and the other along Spf (Zp [[Γ]]) . An important feature of the situation
we consider is that the nearly ordinary deformation space contains many different
irreducible components. The main goal of this paper is to describe how to pass
Iwasawa theoretic information (such as knowledge of the main conjecture) between
components.
In our discussion of Selmer groups we do not go so far as to construct a coherent
sheaf of Selmer groups over the space X n.ord . For this, the reader should refer torecent and forthcoming work of Ochiai [30] in which a “two-variable main conjecture” is formulated for each fixed tame level. Moreover, Ochiai shows that thistwo-variable main conjecture is equivalent to the one-variable main conjecture foreach classical modular form in the family. In particular, he deduces our Corollary 1
for modular forms of a fixed tame level. His method has the advantage that hedoes not need any assumption on the µ-invariant, but has the disadvantage that it
only applies to forms on a fixed branch of the Hida family.
1.2. Two-variable p-adic L-functions. To establish the analytic parts of Theorems 1 and 2, we make use of a two-variable p-adic L-function (playing the roleof L above). As already remarked upon in 1.1, one variable is a parameter of wild
character space (i.e. the standard cyclotomic variable) and the second variable runs

4

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

through the Hida family (which is a finite cover of weight space) of some fixed residual representation ρ̄. Of the many constructions of two-variable p-adic L-functions(see [14], [33], [31]), our construction follows most closely that of Mazur [28] andKitigawa [22], with two main differences.
The first difference is that by using the fact that the Hecke algebras are known to
be Gorenstein (as in [37]) we may construct these L-function with fewer assumptionson ρ̄. The second difference is that we do not limit ourselves to working solely ona part of the Hida family parameterizing newforms of some fixed tame level.
For each branch Y of the Hida family we construct a two-variable L-function
along Y , which at each classical point specializes to the p-adic L-function of thecorresponding newform computed with respect to its canonical period. (Actually,
the two-variable L-function is defined not on Y , but on the partial normalization X
of Y discussed above in 1.1.) However, for any finite set of primes Σ not containing
p, we also construct a two-variable L-function along YΣ (the Hida family of ρ̄ withramification only at the primes of Σ). If we specialize this two-variable p-adic Lfunction at a classical point, we obtain a non-primitive p-adic L-function attachedto the corresponding newform (i.e. the usual p-adic L-function stripped of its Euler
factors at primes of Σ). In fact, we prove that this non-primitivity occurs in families:
the p-adic L-function on the Hida family YΣ , restricted to some branch Y , is equal
to the p-adic L-function of that branch stripped of its two variable Euler factors at
the primes of Σ.
With this construction in hand, the analytic part of Theorem 1 follows immediately. Indeed, the vanishing of the analytic µ-invariant for any particular form f in
the Hida family is equivalent to the vanishing of the µ-invariant of the two-variable
p-adic L-function along the branch Y containing f , which in turn is equivalent tothe vanishing of the µ-invariant of the non-primitive two-variable L-function on YΣ
for any sufficiently large choice of Σ (i.e. such that YΣ contains Y ). This last condition is independent of f and so is equivalent to the vanishing of the µ-invariant
for every modular form in the family.
The analytic part of Theorem 2 also follows from this construction. The analytic
λ-invariant of a modular form f is equal to the λ-invariant of the two-variable Lfunction attached to the branch containing f . To compare the λ-invariants of two
branches, we choose Σ large enough so that YΣ contains both of these branches. We
then relate each λ-invariant to the λ-invariant of the two-variable p-adic L-function
attached to YΣ . As explained above, the difference of the two λ-invariants is thenrealized in terms of the λ-invariants of certain Euler factors. The quantity e` (ai )appearing in the formula of Theorem 2 is precisely the λ-invariant of the Euler
factor at ` along the branch corresponding to ai .
1.3. Residual Selmer groups. Let ρ̄ : GQ → GL2 (k) be as above. For any
f ∈ H(ρ̄), say with Fourier coefficients in the finite extension K of Qp , Greenberg
has defined the Selmer group Sel(Q∞ , ρf ) as the subgroup of H 1 (Q∞ , Af ) cut out
by local conditions, all of which are as strong as possible except for the condition
at p; here Q∞ is the cyclotomic Zp -extension of Q and Af is (K/OK )2 with Galois
action via ρf . If one defines Sel(Q∞ , ρ̄) in the analogous way, one is confronted
with the fundamental problem that the π-torsion on Sel(Q∞ , ρf ) may be larger
than Sel(Q∞ , ρ̄); this is, of course, precisely the reason why congruent modular
forms need not have isomorphic Selmer groups.

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES5

In [15] this issue is overcome by introducing non-primitive Selmer groups of ρ̄;essentially, if f1 , f2 ∈ H(ρ̄) have tame levels N1 and N2 respectively, then the
π-torsion on Sel(Q∞ , ρf1 ) and Sel(Q∞ , ρf2 ) can both be compared to the nonprimitive Selmer group of ρ̄ obtained by ignoring the local conditions at primes
dividing N1 N2 .
Although this approach can also be made to work in the higher weight case, we
proceed somewhat differently. Following Mazur [26], we allow non-strict, but notnecessarily vacuous, local conditions S on the cohomology of ρ̄, resulting in a familyof residual Selmer groups SelS (Q∞ , ρ̄). We show that for any f ∈ H(ρ̄) there is alocal condition S(f ) such thatSelS(f ) (Q∞ , ρ̄) ∼= Sel(Q∞ , ρf )[π].
It follows that the Iwasawa invariants of f can be recovered from the residual Selmer
group SelS(f ) (Q∞ , ρ̄).
In fact, if the tame level of f coincides with the conductor of ρ̄, then we showthat SelS(f ) (Q∞ , ρ̄) agrees with the naive residual Selmer group Sel(Q∞ , ρ̄). Sincesuch f always exist by level lowering, we are then able to use duality results toshow that the difference
dimk SelS(f ) (Q∞ , ρ̄) − dimk Sel(Q∞ , ρ̄)
is precisely given by the dimensions of the local conditions S(f ). The main algebraic
results follow from this.
We note that, unlike on the analytic side, Hida theory plays little overt rolein these algebraic computations. In particular, in [36], these methods have been
applied more generally for algebraic groups other than GL2 . The key inputs areautomorphic descriptions of the Galois representation at ramified primes, level lowering results and the fact that the Selmer groups of interest are Λ-cotorsion; theremainder of the argument is essentially formal.
1.4. L-functions modulo p. In the paper [25], Mazur raises the question ofwhether one can define a mod p L-function attached to the residual representation ρ̄ and a choice of tame conductor N (divisible by the tame conductor of ρ̄).
The construction discussed in 1.2 gives a positive answer to this question, with the
caveat that the appropriate extra data is not simply a choice of tame conductor N ,but rather the more precise data of a component Y of the universal deformation
space of ρ̄. We may then specialize the p-adic L-function at the closed point of
the partial normalization X of Y on which it is defined, so as to obtain a mod p
L-function attached to ρ̄ and Y .
We also show that the dimension of SelS(f ) (Q∞ , ρ̄) over k depends only on thecomponent Y of the Hida family that contains f ; we thus simply write λalg (Y ) todenote this dimension. Assuming that both the analytic and algebraic µ-invariantof ρ̄ vanish, Theorem 2 then shows that the main conjecture for any member ofH(ρ̄) is equivalent to the following mod p main conjecture (for one, or equivalently
every, choice of Y ):Mod p Main Conjecture. Let Y be a branch of H(ρ̄). Then the mod p L-function
Lp (Y ) of Y is non-zero, λalg (Y ) is finite andλ Lp (Y ) = λalg (Y ).

6

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

1.5. Overview of the paper. In the following section, we recall the Hida theory
used in this paper. In particular, we discuss the Hida family attached to a residual
representation ρ̄, its decomposition into irreducible components and the various
Galois representations attached to this family with an emphasis on the integral
behavior.
In the third section, we construct two-variable p-adic L-functions on the Hida
family of ρ̄ and on each of its irreducible components. By relating these two constructions to each other and to the construction of classical p-adic L-functions, we
obtain proofs of the analytic parts of Theorems 1 and 2.
In the fourth section, we prove the algebraic parts of Theorems 1 and 2 via thetheory of residual Selmer groups and the use of level lowering to reduce to theminimal case.
In the final section we give applications to the main conjecture. We discuss some
explicit examples to illustrate the general theory, including the congruence modulo
11 between the elliptic curve X0 (11) and the modular form ∆.
1.6. Acknowledgments. The authors wish to thank Ralph Greenberg, Masato
Kurihara, Barry Mazur and Chris Skinner for helpful conversations.
Notation. We fix an algebraic closure Q̄ of Q and write GQ to denote the Galois
group of Q̄ over Q. If Σ is a finite set of places of Q, we write QΣ for the maximal
extension of Q in Q̄ unramified away from Σ. We fix an odd prime p and let Q∞
denote the cyclotomic Zp -extension of Q in Q̄. For each prime ` of Q (resp. placev of Q∞ ) fix a decomposition group G` ,→ GQ (resp. Gv ,→ GQ∞ ) with inertia
group I` (resp. Iv ) such that Gv ⊆ G` whenever v divides `; note that I` = Iv fora place v dividing a prime ` 6= p. Let vp denote the unique place of Q∞ above p.
We write ε : GQ → Z×
p for the p-adic cyclotomic character. We let Γ denote the×group of 1-units in Zp ; the cyclotomic character induces a canonical isomorphism∼=

Gal(Q∞ /Q) −→ Γ. If O is a Zp -algebra, we let ΛO denote the completed group×ring O[[Γ]]; we simply write Λ for ΛZp . We denote the natural map Z×
p → Zp [[Zp ]]×
by γ 7→ hγip ; this restricts to the natural map Γ → Λ .
Recall that Λ is a complete regular local ring of dimension two, non-canonicallyisomorphic to the power series ring Zp [[T ]]. (Such an isomorphism is obtained bychoosing a topological generator γ of Γ and mapping hγip to T + 1.) We let L
denote the field of fractions of Λ.
We let ∆ denote the group of cyclotomic units of Zp ; there is then an isomorphism
×
Γ×∆∼
= Z×
p . We let ω denote the inclusion of ∆ in Zp , so thati
Hom(∆, Z×p ) = {ω | 0 ≤ i ≤ p − 2}.

If Zp,(i) denotes Zp regarded as a ∆-module via the character ω i , then there is a
Qp−2
natural isomorphism of Zp [∆]-algebras Zp [∆] ∼= i=0 Zp,(i) . Combining this with∼
the natural isomorphism Zp [[Z×p ]] = Zp [∆] ⊗Zp Λ, we obtain an isomorphism of×
Zp [[Zp ]]-algebras∼
Zp [[Z×p ]] =p−2YΛ(i) ,

i=0

where Λ(i) denotes a copy of Λ, enhanced to a Zp [[Z×p ]]-algebra by having ∆ act
through ω i .

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

7××
If N is a natural number prime to p, then we will write Z×
p,N := (Z/N ) × Zp .
×
We will have occasion to consider the completed group ring Zp [[Zp,N ]]. In thissituation, we will extend the diamond bracket notation introduced above and write
××
x 7→ hxiN p to denote the natural map Z×p,N → Zp [[Zp,N ]] .

2. Hida theory
2.1. The universal ordinary Hecke algebra. Fix a positive integer N relatively
prime to p. We begin by briefly recalling the basic properties of TN , the universal
ordinary Hecke algebra of tame level N (denoted by h 0 (N , Zp ) in [17]) acting
on ordinary p-adic modular forms of tame level N , arbitrary p-power level andarbitrary weight. We first fix some notation.
For each integer k ≥ 2, let Sk (N p∞ , R) denote the union over all r ≥ 0 of
the spaces of weight k cusp forms on Γ1 (N pr ) whose Fourier coefficients lie in a×
fixed Zp -algebra R. We make Sk (N p∞ , R) into a Zp [[Z×p ]]-module by letting Zp
k
act via the product of the nebentypus action and the character γ 7→ γ . Let
Sk (N p∞ , R)ord be the subspace of these cusps forms on which Up acts invertibly.
If κ is a homomorphisms from Γ = 1 + pZp to R× , we let Sk (N p∞ , R)ord [κ] denote
the R-submodule of Sk (N p∞ , R)ord on which Γ acts by the character κ.
If ℘ is a height one prime of Λ, then we set O(℘) := Λ/℘. Let κ℘ : Γ → O(℘)×
be the character induced by the embedding Γ → Λ× . We say that ℘ is classicalof weight k ≥ 2, if it is of residue characteristic zero and if there is a finite index
subgroup Γ0 of Γ such that κ℘ , when restricted to Γ0 , coincides with the character×γ 7→ γ k ∈ Z×p ⊂ O(℘) .
More generally, if ℘ is a height one prime ideal of a finite Λ-algebra T,Tthen
we again write O(℘) := T/℘. We say ℘ is classical of weight k, if ℘0 := ℘ Λ isclassical of weight k and write κ℘ := κ℘0 : Γ → O(℘0 )× ⊂ O(℘)× .
The algebra TN is defined to be commutative algebra of endomorphisms of
Sk (N p∞ , R)ord generated over Zp [[Z×p ]] by the Hecke operators T` (resp. U` ) for
primes ` - N p (resp. primes ` | N p), together with the diamond operators hai for×a ∈ (Z/N )× . By definition TN is a Zp [[Z×→ T×p ]]-algebra. The map (Z/N )N
defined via a 7→ hai allows us to extend this to a structure of Zp [[Z×]]-algebra
on
p,N
TN .
Theorem 2.1.1.
(1) The algebra TN is free of finite rank over Λ = Zp [[Γ]].
(2) If ℘ is a classical height one prime in Λ of weight k, then TN /℘TN is
canonically identified with the the quotient of the full Hecke algebra thatacts faithfully on Sk (N p∞ , O(℘))ord [κ℘ ].
Proof. See [18, Thm. 3.1, Cor. 3.2] and [17, Thm. 1.1, Thm. 1.2].



Remark 2.1.2. It follows from part (2) of the above theorem, together with theusual duality between spaces of modular forms and Hecke algebras, that classicalheight one primes in TN are in one-to-one correspondence with Galois conjugacy
classes of p-ordinary normalized Hecke eigenforms of tame level N and weight k ≥ 2(defined over an algebraic closure of Q̄p ). For such an eigenform f , we denote thecorresponding prime ideal by ℘f and for a classical height one prime ideal ℘ in TN ,we denote the corresponding eigenform by f℘ .

8

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

The new quotient Tnew
of TN is defined to be the quotient of TN that actsNfaithfully on the space of newforms. The following theorem of Hida summarizes its
basic properties.
Theorem 2.1.3.
(1) Tnewis a finite and reduced torsion free Λ-algebra.
N(2) The classical height one primes of Tnew
correspond (under pull-back) toNthose classical height one primes ℘ of TN for which the corresponding normalized eigenform f℘ is of tame conductor N .
(3) If ℘ is a classical height one prime ideal of Λ, then Tnew
N ⊗Λ Λ℘ is a finiteétale extension of the discrete valuation ring Λ℘ .
Proof. Parts (1) and (2) follow from the results of [18]. See, in particular, Corollaries 3.3 and 3.7. Part (3) is a rewording of [17, Cor. 1.4].

2.2. Galois representations. In this section we recall the basic facts regarding
Galois representations attached to Hida families. As above, we fix a tame level N
prime to p. Recall that L denotes the field of fractions of Λ.
Theorem 2.2.1. There is a continuous Galois representation
ρ : GQ → GL2 (TnewN ⊗Λ L),characterized by the following properties:(1) ρ is unramified away from N p.
(2) If ` is a prime not dividing N p then ρ(Frob` ) has characteristic polynomial
equal to X 2 − T` X + h`iN p `−1 ∈ Tnew
N [X]. (Recall that h`iN p denotes theunit in Zp [[Z×]]correspondingtotheelement ` ∈ Z×p,Np,N .)The representation ρ satisfies the following additional properties:(3) ρ is absolutely irreducible.
(4) The determinant of ρ is equal to the following character:
GQ → Ẑ× → Z×p,Nx7→hxiN p x−1
p

−→

×
new ×
Zp [[Z×p,N ]] ⊂ (TN ) .

(Here the first arrow is the full cyclotomic character, and xp denotes the×image of x under the projection Z×p,N → Zp .)(5) The space of Ip -coinvariants of ρ is free of rank one over Tnew
N ⊗Λ L and.
Frobp acts on this space through the eigenvalue Up ∈ Tnew
N
Proof. See [17, Thm. 2.1] and [14, Thm. 2.6].



Remark 2.2.2. The representation ρ encodes all of the representations attachedto the classical modular forms occurring in the Hida family determined by Tnew
N .
newIndeed, if T̂newdenotesthenormalization
of
T
,
then
we
may
descend
ρ
to
N
N
new
a two dimensional representation defined over T̂N (see [18, Theorem 2.1]). If℘ is a classical height one prime ideal in Tnew
N , then we have an isomorphism
new
∼
(Tnew
)
(
T̂
)
,
by
part
(3)
of
Theorem
2.1.3, and consequently we may de=℘℘NNscend ρ to a two dimensional representation over (TnewN )℘ . Reducing this representation modulo the maximal ideal of this local ring, we obtain a representationρ̄℘ : GQ → GL2 (O(℘)[1/p]). Part (2) of Theorem 2.2.1 shows that this is theusual absolutely irreducible two dimensional Galois representation attached to thenewform f℘ .

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

9

From the Galois representation ρ, we may also construct various related Galois
newrepresentations corresponding to the minimal and maximal primes of TQ
N .
new
∼
We consider first the case of minimal primes. Note that TN ⊗Λ L = a (TnewN )awhere the product is taken over all minimal primes of Tnew
.
For
a
a
fixed
minimalN
prime ideal of Tnew
N , let ρa denote the representation
ρa : GQ → GL2 ((TnewN )a )
newobtained by composing ρ with the projection Tnew
N ⊗Λ L → (TN )a .
Since ρa takes values in a field, we may define its tame conductor by the usual
formula. That is, if ρa acts on the two dimensional (TnewN )a -vector space V, thenthe exponent of a prime ` 6= p in the tame conductor is equal toZ ∞dim V − dim VI` +dim V − dim VI`u du,0

where {I`u }u≥0denotes the usual filtration of I` by higher ramification groups,
indexed by the upper numbering. (Note that this formula is often expressed interms of invariants under the ramification groups, rather than coinvariants. Oneuobtains the same value with either formulation, since dim VI`u = dim V I` for any
value of u.)

Proposition 2.2.3. If a is any minimal prime of Tnew
N , then the tame conductorof ρa is equal to N .
Proof. Write T̂ to denote the normalization of Tnew
N /a; this is the component of the
cut
out
by
the
minimalprime
a. Since it is Cohen-Macaulay
normalization of Tnew
N(being normal and of dimension two) and finite over Λ, it is in fact finite flat over
Λ by [24, Thm. 23.1].
Let V denote a two dimensional vector space over the fraction field of Tnew
N /a onwhich ρa acts and let M be a free rank two T̂-lattice in V invariant under GQ . If
containing a, then as was observed above,℘ is a classical height one prime of TnewN
the injection(TnewN /a)℘ → T̂℘is an isomorphism. Moreover, GQ acts on the O(℘)[1/p]-vector space V (℘) :=(M/℘M )[1/p] via the usual Galois representations ρ̄℘ attached to newform f℘ oftame conductor N .
The ratio of the tame conductor of ρa and the tame conductor of ρ̄℘ (that is, N )is equal toY(2.1)` dim V (℘)I` −dim VI` ;`6=pfurthermore, each of the exponents appearing in this expression is non-negative(i.e. dim V (℘)I` bounds dim VI` from above). (See [23, §1]). Thus to prove theproposition, we must show that dim V (℘)I` = dim VI` for each prime ` 6= p.
If dim V (℘)I` = 2, then ` does not divide N . In this case, ρa is unramified
at `, and so dim VI` = 2 as well. Conversely, if dim VI` = 2, then the same istrue of dim V (℘)I` (since the latter dimension bounds the former dimension fromabove). If dim V (℘)I` = 0, then also dim VI` = 0 (again, since the latter dimension
is bounded above by the former). Thus it remains to show that if dim V (℘)I` > 0,then dim VI` > 0. This we now do.

10

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

Note that the formula (2.1) (and the fact that the tame conductor of V (℘)
is equal to N , and thus is independent of the particular choice of ℘ containing a)shows that dim V (℘)I` is independent of the choice of the classical height one prime℘ containing a. Thus we may assume that dim V (℘)I` is positive for every classicalheight one prime ℘ containing a. There is a natural map I` → Zp (1), given by
projection onto the p-Sylow subgroup of the tame quotient of I` . Let J` denote thekernel of this projection. Since J` has order prime to p, it acts on M through a finite
quotient of order prime to p and so certainly dim VJ` = dim V (℘)J` for any classicalheight one prime ℘. Let σ denote a topological generator of Zp (1) and consider
the matrix ρa (σ) − I acting on VJ` . (Here I denotes the identity matrix.) By
assumption, the determinant of this matrix (which lies in T̂) lies in ℘T̂ for everyclassical height one prime ℘ of T̂. Thus it lies in ℘0 T̂ for every classical height
one prime ℘0 of Λ. (Recall that these primes are unramified in T̂, by part (3) of
Theorem 2.1.3.) Since T̂ is finite flat over Λ, we see that this determinant vanishes
and thus ρa (σ) admits a non-zero coinvariant quotient of VJ` . This proves thatdim VI` > 0 as required.

As a byproduct of the proof of the preceding proposition, we also obtain thefollowing useful result.
and let L
Proposition 2.2.4. Let ℘ be a classical height one prime ideal in Tnew
N
denote a choice of two dimensional (Tnew)-lattice
on
which
ρ
acts.
(In
particular,℘
N
L/℘L is a two dimensional O(℘)[1/p]-vector space on which ρ̄℘ acts.) If ` is any
prime distinct from p, then LI` is a free (TnewN )℘ -module and there are natural
isomorphisms LI` ⊗Λ L → (L ⊗Λ L)I` and LI` /℘LI` → (L/℘L)I` .
Proof. We let a denote the minimal ideal contained in ℘ (unique since (TnewN )℘ isa discrete valuation ring) and employ the notation introduced in the proof of thepreceding proposition. The lattice L is a free rank two (TnewN )℘ -module, which we
may regard as being embedded in a GQ -equivariant fashion in V.
For general reasons, the composite surjection L → L/℘L → (L/℘L)I` induces
an isomorphism LI` /℘LI` ∼= (L/℘L)I` . Thus a minimal set of generators for LI`)-modulecontains
at most dim(L/℘L)I` elements. On the other hand
as a (Tnew℘N
the surjection V → VI` induces a surjection LI` ⊗Λ L → VI` and hence LI` ⊗Λ
L is of dimension at least dim VI` . The proof of Proposition 2.2.3 shows thatdim(L/℘L)I` = dim VI` and hence (taking into account the fact that (TnewN )℘ is adiscrete valuation ring) we may conclude that LI` is free over (Tnew
)
of
rank
equal
℘
N
to dim VI` . Thus the surjection LI` ⊗Λ L → VI` is an isomorphism as claimed. 
We now turn to the Galois representations attached to maximal ideals. If m is a
new
new
and
maximal ideal of Tnew
N , then the localization (TN )m is a direct factor of TNnewnewso (TN )m ⊗Λ L is a direct factor of TN ⊗Λ L. We let ρm denote the representation
ρm : GQ → GL2 ((TnewN )m ⊗Λ L)
newobtained by composing ρ with the projection Tnew
N ⊗Λ L → (TN )m ⊗Λ L.
The next result describes the residual representation ρ̄m attached to a maximal
ideal of Tnew
N .

Theorem 2.2.5. If m is a maximal ideal of Tnew
N , then attached to m is a semisimple representation ρ̄m : GQ → GL2 (Tnew/m),
uniquely determined by the propN
erties:VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

11(1) ρ̄m is unramified away from N p.
(2) If ` is a prime not dividing N p then ρ̄m (Frob` ) has characteristic polynomial
equal to X 2 − T` X + h`iN p `−1 mod m ∈ (Tnew
N /m)[X].
Furthermore, ρ̄m satisfies the following condition:(3) The restriction of ρ̄m to Gp has the following shape (with respect to a suitable choice of basis):


χ ∗
,
0 ψ
×
where χ and ψ are (Tnew
N /m) -valued characters of Gp such that ψ is unramified and ψ(Frobp ) = Up mod m.
Proof. The representation ρ̄m is constructed in the usual way, by choosing an integral model for ρ over T̂new
N , reducing this model modulo a maximal ideal m̂ lifting
newm, semi-simplifying and then descending (if necessary) from T̂new
N /m̂ to TN /m.
The stated properties follow from the corresponding properties of ρ.
Proposition 2.2.6. If m is a maximal ideal of Tnew
for which the associatedN
residual representation ρ̄m is irreducible, then ρm admits a model over (TnewN )m(which we denote by the same symbol)
ρm : GQ → GL2 ((TnewN )m ),
unique up to isomorphism.
Proof. This follows from the irreducibility of ρ̄m and the fact that the traces of ρmlie in (TnewN )m . (See [2].)Definition 2.2.7. If k is a field of characteristic p and ρ̄ : GQ → GL2 (k) is a continuous representation for which ρ̄|Gp is reducible, then we say that ρ̄ is p-distinguished
if the semisimplification of ρ̄|Gp is non-scalar (i.e. if the two characters appearing
as Jordan-Hölder factors of the reducible representation ρ̄|Gp are distinct).
×
Let hεip : GQ → Zp [[Z×denote the character obtained by composing the
p ]]
×
× ×
p-adic cyclotomic character ε : GQ → Z×
p with the map h ip : Zp → Zp [[Zp ]] .

for which the associatedProposition 2.2.8. Let m be a maximal ideal of Tnew
N
residual representation ρ̄m is irreducible and let L denote a choice of a free rank
two (Tnew
N )m -module on which the representation ρm acts. (Such an L exists by
Proposition 2.2.6.) Let L0 denote the maximal submodule of L on which Ip acts
through the character hεip ε−1 . If ρ̄m is p-distinguished, then each of L0 and L/L0
is free of rank one over (TnewN )m and the Gp -action on L/L0 is unramified.
Proof. Theorem 2.2.1 shows that L0 ⊗Λ L is a one-dimensional direct summand ofL⊗Λ L and that Gp acts on this space and on (L⊗Λ L)/(L0 ⊗Λ L) through a character.
Thus Gp acts on each of L0 and L/L0 through a character. By assumption, L/mL
is a p-distinguished representation of Gp and so (L/L0 )/m(L/L0 ) must be onedimensional over TN /m. The proposition now follows by Nakayama’s Lemma. 
Suppose that m is a maximal ideal of Tnew
satisfying the hypothesis of theNpreceding proposition. As in the statement of that proposition, let L denote a
choice of free rank two (TnewN )m -module on which the representation ρm acts and let
L0 be the maximal submodule of L on which Ip acts through the character hεip ε−1 .
The group GQ then acts on the quotient L/mL via the residual representation ρ̄m .

12

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

Since the space L/L0 is a free rank one quotient of L on which the Gp -action isunramified (by Proposition 2.2.8), we see that (L/L0 )/m(L/L0 ) is a one dimensional
unramified quotient of L/mL.
Definition 2.2.9. If k is a finite field of characteristic p and ρ̄ : GQ → GL2 (k) is
a continuous Galois representation acting on a two dimensional k-vector space V ,then a p-stabilization of ρ̄ is a choice of a one dimensional quotient of V on which
the Gp -action is unramified.
Definition 2.2.10. The discussion preceding Definition 2.2.9 shows that if m is a
maximal ideal of Tnew
N for which ρ̄m is irreducible and p-distinguished, then the quotient (L/L0 )/m(L/L0 ) (in the notation of that discussion) forms a p-stabilization ofρ̄m . We refer to this as the canonical p-stabilization of ρ̄m attached to the maximal
ideal m.
2.3. The reduced Hida algebra.
Definition 2.3.1. For any level N , we let T0N denote the Λ-subalgebra of TN
generated by the Hecke operators T` for ` prime to N p, together with the operator
Up .
Since T0N is a subalgebra of the finite flat Λ-algebra TN , it is certainly finite
and torsion free over Λ. It turns out that T0N is also reduced. (Being reduced isa standard property of Hecke algebras in which we omit the operators indexed by
the primes dividing the level.) In fact, we can be somewhat more precise.
If M is a divisor of N , then restricting the action of the prime-to-N Hecke
operators to p-ordinary forms of level dividing M yields a surjective map of Λalgebras T0N → T0M . Composing this with the composite T0M ⊂ TM → Tnew
Myields a map(2.2)

T0N → Tnew
M .

Taking the product of these maps over all divisors M of N , we obtain a map
Y
(2.3)
T0N →
Tnew
M .
M |N

Proposition 2.3.2.
(1) The map (2.3) is injective and induces an isomorphism after localizing at
any classical height one prime ideal of Λ (and hence also after tensoringover Λ with its fraction field L).
(2) If ℘ is a classical height one prime ideal of Λ, then T0N ⊗Λ Λ℘ is a finiteétale extension of the discrete valuation ring Λ℘ .
Proof. Part (1) follows from the theory of newforms for ordinary families developedin [18]. Part (2) then follows from part (1) together with part (3) of Theorem 2.1.3.
Taking the product of the Galois representations
GQ → GL2 (Tnew
M ⊗Λ L)
given by Theorem 2.2.1, as M ranges over all divisors of N , and taking into accountpart (1) of the preceding proposition, we obtain a Galois representation
ρ : GQ → GL2 (T0N ⊗Λ L)

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

13

satisfying the analogue of Theorem 2.2.1.
Just as in the preceding section, we may reduce ρ modulo a height one prime
ideal ℘ or a maximal ideal m of T0N . We denote the corresponding residual representations by ρ̄℘ and ρ̄m respectively. Similarly, we may localize T0N at any maximal
ideal m and obtain a corresponding representation
ρm : GQ → GL2 ((T0N )m ⊗Λ L).
If ρ̄m is irreducible, then the analogue of Proposition 2.2.6 holds (by the same
appeal to the results of [2]) and so we obtain a uniquely determined representation
ρm : GQ → GL2 ((T0N )m ).
2.4. The reduced Hida algebras attached to ρ̄. If k is a finite field of characteristic p and ρ̄ : GQ → GL2 (k) is a continuous two dimensional Galois representation
defined over k, then we say that ρ̄ is p-ordinary if ρ̄ restricted to Gp has an unramified quotient of dimension one over k. Clearly ρ̄ admits a p-stabilization, inthe sense of Definition 2.2.9, if and only if ρ̄ is p-ordinary. If ρ̄ is furthermore pdistinguished (in the sense of Definition 2.2.7), then ρ̄ admits at most two choicesof p-stabilization (and does admit two such choices precisely when the determinantof ρ̄ is unramified at p).
Let us now fix such a representation ρ̄ : GQ → GL2 (k) and let V be a twodimensional k-vector space on which ρ̄ acts. We assume that ρ̄ is irreducible, odd,
p-ordinary and p-distinguished and we fix a choice of p-stabilization of ρ̄. We
assume that k is equal to the field generated by the traces of ρ̄. (If it were not,we could replace k by this field of traces and descend ρ̄ to the smaller field – see[29, Section 6].) Finally, we suppose that ρ̄ is modular (i.e. that it arises as theresidual representation attached to a modular form of some weight and level definedover Q̄p ). Our goal in this section is to define, for Σ a finite set of primes, thereduced ordinary Hecke algebra TΣ (ρ̄) attached to a modular p-ordinary residual
representation and to describe its basic properties.
We let N (ρ̄) denote the tame conductor of ρ̄. If ` 6= p is prime, then definem` = dimk V I`
and for any finite set of primes Σ that does not contain p, writeY
N (Σ) = N (ρ̄)
`m` .
`∈Σ

Theorem 2.4.1. There is a unique maximal ideal m of T0N (Σ) such that ρ̄m , with
its canonical p-stabilization, is isomorphic to ρ̄, with its given p-stabilization.
Proof. The uniqueness is clear. The existence follows from the assumption that ρ̄is modular and the results of [5].
Proposition 2.4.2. If m denotes the maximal ideal of T0N (Σ) of the preceding
theorem, then there is a unique maximal ideal n of TN (Σ) satisfying the following
conditions:(1) n lifts m.
(2) T` ∈ n for each ` ∈ Σ.
(3) The natural map of localizations (T0N (Σ) )m → (TN (Σ) )n is an isomorphismof Λ-algebras.

14

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

In particular, (T0N (Σ) )m is a finite flat Λ-algebra. Also, the image of T` in thelocalization (TN (Σ) )n vanishes for each ` ∈ Σ.
Proof. This is a variant of [37, Prop. 2.1.5] and is proved in an analogous manner.
The second to last claim follows from the rest of the proposition together withpart (1) of Theorem 2.1.1.
Definition 2.4.3. We let TΣ (ρ̄) (or simply TΣ , if ρ̄ is understood) denote thelocalization of T0N (Σ) at the maximal ideal whose existence is guaranteed by Theorem 2.4.1. We let ρΣ : GQ → GL2 (TΣ ) denote the Galois representation attachedto this local factor of T0N (Σ) as discussed in Section 2.3. Recall that ρΣ is characterized by the following property: If ` is a prime not dividing N (Σ)p, then ρΣ (Frob` )has trace equal to T` ∈ TΣ .
Taking into account Proposition 2.4.2, we see that TΣ is a reduced and finite
flat Λ-algebra. Note that if Σ ⊂ Σ0 , then N (Σ) | N (Σ0 ) and the natural surjection
T0N (Σ0 ) → T0N (Σ) induces a surjection TΣ0 → TΣ . The Galois representations ρΣ0and ρΣ are evidently compatible with this surjection.
We refer to Spec TΣ as the universal p-ordinary family of newforms, or sometimes
simply “the Hida family” attached to ρ̄ and our chosen p-stabilization that is minimally ramified outside Σ. Localizing ρΣ over Spec TΣ , we obtain a two dimensional
vector bundle on which GQ acts, which we refer to as the universal family of Galois
representations over Spec TΣ . If Σ ⊂ Σ0 , then the surjection TΣ → TΣ0 induces
a closed embedding Spec TΣ → Spec TΣ0 and the universal family of Galois representations on the target pulls back to the universal family of Galois representations
on the source. If we consider all Σ simultaneously, then we obtain an Ind-scheme“lim” Spec TΣ−→Σand a family of two dimensional Galois representations lying over it. We refer tothis Ind-scheme as the universal p-ordinary family of newforms, or simply “the Hida
family” attached to ρ̄ and its chosen p-stabilization.
The Hida family corresponding to Σ = ∅ will play a special role; we refer to it
as the minimal Hida family attached to ρ̄ and our chosen p-stabilization.
When ρ̄ is irreducible after restriction to the quadratic field of discriminant(−1)(p−1)/2 p, the results of Wiles and Taylor–Wiles [37, 34], as strengthened by Diamond [6], in fact allow us to identify the rings TΣ , and their accompanying Galois
representations ρΣ , with certain universal deformation rings, and their accompanying universal Galois representations, attached to the residual representation ρ̄.
Specifically, let RΣ denote the universal deformation ring parameterizing lifts of ρ̄
which are p-ordinary and whose tame conductor coincides with that of ρ̄ at primes
not in Σ ∪ {p}. The representation ρΣ induces a map RΣ → TΣ , which by [6] is
an isomorphism after tensoring with the quotient of Λ by any classical height one
prime; it follows that in fact RΣ → TΣ is an isomorphism, as claimed.
2.5. Branches. We now prove some results concerning the irreducible componentsof the Hida family attached to ρ̄.
Definition 2.5.1. If a is a minimal prime ideal in TΣ for any Σ as above, thenwe write T(a) := TΣ /a. Note that T(a) is a local domain, finite over Λ. We

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

15

write K(a) to denote the fraction field of T(a). (Thus there is an isomorphism
K(a) ∼= T(a) ⊗Λ L.) We let ρ(a) denote the Galois representation
ρ(a) : GQ → GL2 (T(a))
induced by ρΣ .
Proposition 2.5.2. If a is a minimal prime of TΣ for any Σ as above, then there
is a unique divisor N (a) of N (Σ) and a unique minimal prime a0 ⊆ Tnew
N (a) sitting
over a such thatQ
/ T0N (Σ)/ M |N (Σ) Tnew
TΣ
N (a)

TΣ /a

=


0
/ Tnew
N (a) /a/ T(a)

commutes.
Proof. Since TΣ is finite over Λ, the minimal primes of TΣ are in bijection withthe local components of TΣ ⊗Λ L. Since TΣ is a local factor of T0N (Σ) , these local
components are included in the local components of T0N (Σ) ⊗Λ L. By part (1) of
Proposition 2.3.2, we have thatY
T0N (Σ) ⊗Λ L ∼Tnew=
M ⊗Λ L.
M |N (Σ)

Q

new
M |N (Σ) TM ⊗Λ L are in one-to-one correspondence withThe local components of
its minimal primes. Thus our given minimal prime a gives rise to a minimal primeof this ring. However, any such minimal prime corresponds to a minimal prime a0in Tnew
N (a) for some N (a) | N (Σ), which establishes the proposition.
Definition 2.5.3. For a minimal prime ideal a of TΣ , we refer to N (a) (of thepreceding proposition) as the tame conductor attached to a (or to the irreducible◦0
component of Spec TΣ corresponding to a). Moreover, set T(a) := Tnew
N (a) /a .
Remark 2.5.4. Proposition 2.5.2 gives rise to an embedding of local domains◦◦
T(a) → T(a) . (Note that T(a) is local since it is a complete finite Λ-algebra andhence a product of local rings. Being a domain, it must be local.)We next observe that at the generic point of the a-component of Spec TΣ , theuniversal Galois representation has conductor equal to the conductor of a.
Corollary 2.5.5. The tame conductor of the Galois representation ρ(a) equals
N (a). (Here we define the tame conductor by regarding ρ(a) as a Galois representation defined over the field of fractions K(a) of T(a).)
Proof. This follows from Propositions 2.2.3 and 2.5.2.

The next result deals with the classical height one primes in TΣ .
Proposition 2.5.6. Let ℘ be any classical height one prime ideal of TΣ .
(1) The ring TΣ is étale over Λ (and so regular) in a neighborhood of ℘; consequently, ℘ contains a unique minimal prime a of TΣ and the natural mapof localizations (TΣ )℘ → T(a)℘ is an isomorphism.

16

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON◦(2) Thinking of ℘ as a height one prime of T(a), the map T(a) → T(a) is
an isomorphism in a neighborhood of ℘. Consequently, there is a unique◦height one prime ℘0 of T(a) that pulls back to ℘ under this map and the◦map of localizations T(a)℘ → T(a)℘0 is an isomorphism.
Proof. Both claims follow directly from Proposition 2.3.2.

◦In the situation of the preceding proposition, we write O(℘) := O(℘0 ); this isa finite extension of O(℘). Recall that we have defined the classical newform f℘0◦
attached to the height one prime ideal ℘0 of T(a) (thought of as a classical heightnew
one prime of TN (a) ). We write f℘ := f℘0 ; part (2) of Theorem 2.1.3 implies that◦f℘ lies in Sk (N (a)p∞ , O(℘) )ord
new [κ℘ ].
2.6. Hecke eigenvalues. In this section, we will give two lemmas that describe
Hecke eigenvalues at primes dividing the level in terms of the corresponding Galois
representation. The first lemma treats the case of a single modular form and thesecond discusses the case of a Hida family.
Let f denote a classical newform over Q̄p of some weight and of tame level N .
Let ρf denote the p-adic Galois representation attached to f , let Vf denote its
underlying space and let a` (f ) denote its Hecke eigenvalues. If ` does not divide
N , then ρf is unramified at ` and a` (f ) is equal to the trace of ρf (Frob` ). The
following lemma describes a` (f ) for ` dividing N in terms of ρf .
Lemma 2.6.1. If ` is a prime dividing N , then the following are equivalent:(1) a` (f ) is a non-unit.
(2) a` (f ) = 0.
(3) VI` = 0.
If these equivalent conditions do not hold, then we have that VI` is one dimensional
and a` (f ) is equal to the eigenvalue of Frob` acting on this line.
Proof. This is standard.



Fix a tame level N and a maximal ideal n in Tnew
N . Let V denote the free rank⊗
L-module
on
which
ρ
acts.
The
followinglemma describes T` for `two TnewΛnN
dividing N in terms of ρn .
Lemma 2.6.2. If ` is a prime dividing N , then the following are equivalent:(1) T` lies in the maximal ideal n.
(2) T` ≡ 0 mod ℘ for some classical height one prime ideal ℘ of (Tnew
N )n .
(3) T` ≡ 0 mod ℘ for every classical height one prime ideal ℘ of (Tnew
N )n .
(4) T` = 0 in (Tnew).
nN(5) VI` = 0.
If these equivalent conditions do not hold, then we have that VI` is free of rank one
over (Tnew
N )n ⊗Λ L and T` is equal to the eigenvalue of Frob` acting on this freerank one module.
Proof. It is clear that (4) =⇒ (3) =⇒ (2) =⇒ (1). Also, Condition (1) implies
that a` (f℘ ) is contained in the maximal ideal of O(f℘ ) = O(℘) for each classicalheight one prime ℘ of (Tnew
N )n . By Lemma 2.6.1, a` (f℘ ) = 0 for each such prime andthus (1) implies (3). Proposition 2.2.4, together with Lemma 2.6.1, shows that (3)and (5) are equivalent. Lastly, Lemma 2.6.3 (below), together with Theorem 2.1.3,
establishes that (3) implies (4).

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

17

Suppose now that these equivalent conditions do not hold. Then, Proposition 2.2.4 and Lemma 2.6.1 show that the space VI` is free of rank one over
new
(Tnew
N )n ⊗Λ L as claimed. Now let A` ∈ (TN )n ⊗Λ L denote the eigenvalue ofFrob` acting on this module. By Proposition 2.2.4, A` ∈ (TnewN )℘ for each classicalheight one prime ℘ of Tnew
.
By
Lemma
2.6.1,
weseethatT` − A` lies in theNmaximal ideal of this ring for each such prime ℘. Then, by Lemma 2.6.3 (below)
and Theorem 2.1.3, T` = A` .

Lemma 2.6.3. Let T be a reduced finite free Λ-algebra such that T ⊗Λ Λ℘ is anétale extension of Λ℘ for every classical height one prime ℘ of Λ. If x ∈ T ⊗ Llies in the maximal ideal of T℘0 for every classical height one prime ℘0 of T, thenx = 0.
Proof. Let ℘ be a classicalT height one prime of Λ. Since T ⊗Λ Λ℘ is étale over Λ,
we see that ℘T ⊗Λ Λ℘ = ℘0 ℘0 T℘0 , where the intersection runs over all height one
primes ℘0 of T lying over ℘. Our assumption on x thus implies that x ∈ ℘T ⊗Λ Λ℘
for all classical height
T one primes ℘ of Λ. As T is free over Λ, the lemma follows
once we note that ℘ ℘Λ℘ = 0 (the intersection taking place in L). Indeed, thenumerator of any element in this intersection lies in an infinite number of height
one primes of Λ and so vanishes.

2.7. Λ-adic modular forms and Euler factors. For each minimal prime a of◦TΣ , we can define a formal q-expansion along the partial normalization Spec T(a)of the component Spec T(a) of Spec TΣ that interpolates the q-expansions of the
newforms f℘ obtained from the classical primes ℘ in Spec T(a). Namely, if we write◦◦0
T (a) = Tnew
N (a) /a , as in Definition 2.5.3, then we define f (a, q) ∈ T(a) [[q]] viaXf (a, q) =(Tn mod a0 )q n .
n≥1(Here we have written Tn rather than Upr Tn0 , when n is of the form n = n0 pr with
(n0 , p) = 1, for the sake of uniformity of notation.)◦
An alternative way of describing this formal q-expansion along Spec T(a) is todescribe the corresponding Euler factors.
◦
Definition 2.7.1. Let a be a minimal prime of TΣ . As above, write T(a) ∼=0newTN (a) /a , For each prime ` 6= p, define the reciprocal Euler factor E` (a, X) ∈◦T(a) [X] via the usual formula:(
1 − (T` mod a0 )X + h`iN (a)p `−1 X 2 if ` is prime to N (a)E` (a, X) :=otherwise.
1 − (T` mod a0 )X

For the sake of completeness, define Ep (a, X) = (1 − (Up mod a0 )X).
In terms of these reciprocal Euler factors, the formal q-expansion f (a, q) is characterized by the fact that its “formal Mellin transform” is equal to the formal
Dirichlet series
YE` (a, `−s )−1 .
`We will now give a Galois theoretic description of these reciprocal Euler factors.

18

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

Proposition 2.7.2. Let a be a minimal prime of TΣ and let V denote a twodimensional K(a)-vector space on which the Galois representation ρ(a) acts. If` 6= p is prime, then the Euler factor E` (a, X) ∈ K(a)[X] is equal to the determinantdet(Id − Frob` X | VI` ).
Proof. This follows from Lemma 2.6.2.

◦

Since T(a) ⊂ T(a)[1/p], we may in particular regard the reciprocal Euler factors E` (a, X) and the formal q-expansion f (a, q) as varying over Spec T(a)[1/p].
Let us close this section by signaling a phenomena which will be fundamental to
all that follows: the reciprocal Euler factors E` (a, X) (or equivalently the formal
q-expansions f (a, q)) do not extend in a well-defined fashion over Spec TΣ [1/p].
In general, if ℘ is a (necessarily non-classical) height one prime lying in the intersection of two different components of Spec TΣ [1/p], say Spec T(a1 )[1/p] andSpec T(a2 )[1/p], then E` (a1 , X) and E` (a2 , X) (and hence f (a1 , q) and f (a2 , q))
may have different specializations in O(℘)[1/p][X] (resp. O(℘)[1/p][[q]]) for certain
values of ` dividing N (a1 ) or N (a2 ).
3. Two-variable p-adic L-functions
3.1. Canonical periods. For any level M , we let X1 (M ) denote the closed modular curve of level Γ1 (M ) and let C1 (M ) denote its set of cusps. Let Lk (R) denotethe set of polynomials of degree less than or equal to k − 2 with coefficients in R
and let L̃k (R) denote the associated local system on X1 (M ).
For N prime to p, let T∗N,r,k (resp. TN,r,k ) denote the Hecke algebra corresponding to p-ordinary modular forms (resp. cusp forms) on Γ1 (N pr ) of weight k. To
ease notation, we write
H1 (N pr ; L̃k (Zp )) := H1 (X1 (N pr ); L̃k (Zp ))and
H1 (N pr , {cusps}; L̃k (Zp )) := H1 (X1 (N pr ); C1 (N pr ); L̃k (Zp )).
We then have a short exact sequence of T∗N,r,k -modules:
0 → H1 (N pr ; L̃k (Zp )) → H1 (N pr , {cusps}; L̃k (Zp )) → H̃0 (C1 (N pr ); L̃k (Zp )) → 0.
(This is a part of the relative homology sequence of the pair (X1 (N pr ), C1 (N pr ));the tilde over the H0 denotes reduced homology.)
If we localize the above sequence at a maximal ideal m in T∗N,r,k corresponding toan irreducible residual GQ -representation, then the localization of H̃0 (C1 (N pr ); Zp )will vanish and the two H1 terms will become isomorphic. Passing to ordinary parts(an exact functor) yields an isomorphism(3.1)
H1 (N pr ; L̃k (Zp ))ord ∼= H1 (N pr , {cusps}; L̃k (Zp ))ordmmof (TN,r,k )m ∼= (T∗N,r,k )m -modules. We write (MN,r,k )m to denote these isomorphic
modules.
There is an action of complex conjugation on X1 (N pr ) which induces an action on these homology group. We will use a superscript of ± to indicate the±-eigenspaces for this action; since p is odd, we have (MN,r,k )m = (MN,r,k )+m ⊕(MN,r,k )−.
m
The following proposition is fundamental to the construction of the canonicalperiod.

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

19

Proposition 3.1.1. Let m be a maximal ideal of TN,r,k whose residual representation is irreducible and p-distinguished. Then (MN,r,k )±m is a free (TN,r,k )m -moduleof rank 1. In particular, there exists an isomorphism of (TN,r,k )m -modules(3.2)(MN,r,k )±m

αN,r,k∼=(TN,r,k )m(although αN,r,k is only well-defined up to a p-adic unit).
Proof. We consider first the case k = 2 and r = 1. Set M := MN,1,2 and T :=TN,1,2 . Let J1 (N p) denote the Jacobian of X1 (N p). By [37, Thm. 2.1], the ordinary
m-torsion J1 (N p)(Q̄)[m]ord is free of rank 2 over T/m. Since
J1 (N p)(Q̄)[m] ∼= H 1 (N p; Fp )[m],by dualizing, we have that (MN )m ⊗T/m is free of rank 2 over T/m. As (MN )m ⊗Qp
is free of rank 2 over Tm ⊗ Qp , it follows that (MN )m is free of rank 2 over Tm .
Passing to ±-subspaces establishes the proposition for k = 2.
The case k > 2 or r > 1 can be deduced from the case k = 2 and r = 1 by Hida
theory. We will give the details in Section 3.3 immediately after Proposition 3.3.1.

Fix a p-ordinary and p-distinguished eigenform f defined over Q̄p of weight k
on Γ1 (N pr ) whose corresponding maximal ideal in TN,r,k is m. By the duality
between spaces of cusp forms and Hecke algebras, the eigenform f gives rise to amorphism of Zp -algebras(3.3)(TN,r,k )m → O(f );here O(f ) is the finite extension of Zp in Q̄p generated by the Fourier coefficients off . Composing (3.3) with the fixed map αN,r,k of (3.2) then yields an O(f )-valued
functionalδf±

H1 (N pr ; L̃k (O(f )))±
m → O(f ).
On the other hand, integration yields a map
ωf±

H1 (N pr ; L̃k (C))± −→ C.
If we fix an isomorphism ı : Q̄p ∼= C, then this integration map factors through0
H1 (N pr ; L̃k (C))±
,
where
m
is
the
maximal ideal in TN,r,k ⊗Zp ,ı C obtained as them0kernel of the C-algebra map
TN,r,k ⊗Zp ,ı C → C
induced by composing (3.3) with ı. By the multiplicity one theorem, the functionals
δf± (extended to C) and ωf± differ by a non-zero constant and we have±ωf± = Ω±
f · δf±×with Ω±
f ∈ C . The constant Ωf is the canonical period of f . It is well-defined up
to a p-adic unit. (The ambiguity arises from the choice of the map αN,r,k in (3.2)).

Remark 3.1.2. The constructions given here are dual (but equivalent) to the ones
given by Vatsal in [35], since here we are working with homology while Vatsal used
cohomology.

20

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

3.2. One variable p-adic L-functions. The advantage of describing (MN,r,k )mas a relative homology group is that the corresponding relative homology classes
admit a description via modular symbols. More precisely, for fixed r, there is amap
P1 (Q) → H1 (N pr , {cusps}; L̃k (Zp ))defined by sending the element a ∈ P1 (Q) to the homology class corresponding
to the image of the path [∞, a] on the modular curve X1 (N p). By projecting toordinary parts and localizing at m, we obtain a map
P1 (Q) → (MN,r,k )m ,which we denote by a 7→ {∞, a}. This map allows us to define an (MN,r,k )m -valued
measure on Z×p in the usual way.
Definition 3.2.1. For any open subset a + pr Zp of Z×
p , we defineµk (a + pr Zp ) = Up−r {∞, a/pr } ∈ (MN,r,k )m .
Proposition 3.2.2. The function µk is a measure (i.e. it is additive).
Proof. This is standard.



For f an eigenform of weight k on Γ1 (N pr ), let ℘f denote the correspondingheight one prime ideal of TN,r,k and let m be the unique maximal ideal containing℘f . By decomposing µk into its ±-parts and applying the map αN,r,k of (3.2),±
we may view µ±
k as taking values in TN,r,k . Thus µk mod ℘f is O(f )-valued andmoreover, it is precisely the p-adic L-function attached to f computed with respect±
to the canonical period Ω±
f . That is, µk mod ℘f interpolates the special values±L(f, χ, 1)/Ωf where χ is a finite order character of Z×p such that χ(−1) = ±1.
3.3. Canonical periods in families. In [35], it is proven that if f and g are
p-ordinary eigenforms of weight k whose residual representations are isomorphic,
then their respective p-adic L-functions are congruent. More precisely, it is proventhat one can simultaneously choose canonical periods Ωf and Ωg so that the p-adic
L-functions computed with respect to these periods are congruent.
We will now describe how one can simultaneously choose canonical periods for
all modular forms in a Hida family so that the associated single variable p-adic
L-functions vary p-adic analytically in k. These constructions are originally due
to Mazur [28] and Kitigawa [22]. Using results of Wiles in [37], we have extended
these constructions to a wider class of modular forms.
Let TN be the universal ordinary Hecke algebra defined in Section 2 and let T∗N
be the corresponding Hecke algebra defined with respect to all ordinary modular
forms rather than just cusp forms. Let m ⊆ T∗N be a maximal ideal whose residual
representation is irreducible.
For each r ≥ 0, we have an isomorphismrord∼
H1 (N pr ; Zp )ord
m = H1 (N p , {cusps}; Zp )mand passing to the projective limit in r, we obtain a corresponding isomorphism of(TN )m ∼= (TN )∗m -modules∼lim
H1 (N pr ; Zp )ord
H1 (N pr , {cusps}; Zp )ord
m = lim
m .
←−
←−
r

r

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

21

We denote these isomorphic modules by (MN )m . As before, there is an action of−complex conjugation that yields a decomposition (MN )m = (MN )+m ⊕ (MN )m .
The following proposition is the Hida-theoretic analogue of Proposition 3.1.1.
Proposition 3.3.1. If ρ̄m is irreducible and p-distinguished, then (MN )±m is a free(TN )m -module of rank 1.
Proof. Let ℘2 denote the product of all classical height one primes ℘ of weight 2 inTN such that κ℘ is trivial. Then, by [17, Thm. 3.1],(3.4)(MN )m ⊗ TN /℘2 ∼= (MN,1,2 )m2

where m2 is the maximal ideal in TN,1,2 induced by m. By Proposition 3.1.1 (appliedin the case k = 2 and r = 1), we know that (MN,1,2 )m2 is a free TN,1,2 ∼= TN /℘2 module of rank 2. Since (MN )m ⊗ Qp is free of rank 2 over (TN )m ⊗ Qp , it follows
that (MN )m is free of rank 2 over (TN )m . Passing to ±-subspaces then establishes
the proposition.
Using Proposition 3.3.1, we now complete the proof of Proposition 3.1.1 forarbitrary k and r. (Note that the proof of Proposition 3.3.1 only relied upon
Proposition 3.1.1 in the already established case of k = 2 and r = 1 and so thesearguments are not circular.)
Proof of Proposition 3.1.1. Let ℘N,r,k denote the product of all classical height one
primes ℘ of weight k in TN such that κ℘ restricted to 1 + pr Zp is trivial. Then, by[19, Theorem 1.9],(3.5)(MN )m ⊗ TN /℘N,r,k ∼= (MN,r,k )mN,r,k

where mN,r,k is the maximal ideal in TN,r,k induced by m. By Proposition 3.3.1,(MN )m is free of rank two over TN . Thus (MN,r,k )mN,r,k is free of rank two overTN,r,k ∼= TN /℘N,r,k . Passing to ±-subspace then completes the proof.
We now assume that the hypotheses of Proposition 3.3.1 are satisfied. Thus wemay choose an isomorphism
α

(3.6)

N
(TN )m ∼= (MN )±
m.

Using the notation of the above proof, if we consider αN modulo ℘N,r,k , we get an
isomorphism
αN,r,k(TN,r,k )mN,r,k

∼
=

(M±
N,r,k )mN,r,k .

For a given modular form f of level N pr and of weight k in the Hida family corresponding to (TN )m , we then define its canonical period to be the one computedwith respect to αN,r,k .
Note that the usual ambiguity of the canonical period coming from the choice ofisomorphism αN,r,k is controlled across the entire Hida family by the single choiceof αN identifying (MN )±m with (TN )m . In the following section, we show that underthese choices of canonical periods, the p-adic L-functions enjoy good congruence
properties; that is, they interpolate into a two-variable p-adic L-function.

22

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

3.4. Two variable p-adic L-functions. Recall, that for each r ≥ 1, we have amap
P1 (Q) → H1 (N pr , {cusps}; Zp ).
This map is compatible with varying r and so projecting to ordinary parts, localizing
at m and passing to the limit, we obtain a map
P1 (Q) → (MN )m ,which we again denote by a 7→ {∞, a}.
Definition 3.4.1. For any open subset a + pr Zp of Z×
p , we defineµ(a + pr Zp ) = Up−r {∞, a/pr } ∈ (MN )m .
Proposition 3.4.2. The function µ is a measure (i.e. it is additive).
Proof. This is standard.



Recall that the completed group ring Zp [[Z×p ]] may naturally be regarded as thespace of Zp -valued measures on Z×.
We
maythus regard µ as defining an elementpˆ Zp Zp [[Z×L(m, N ) ∈ (MN )m ⊗p ]].
(Here the tensor product is completed with respect to the usual (i.e. profinite topology) on Zp [[Z×p ]] and the m-adic topology on (MN )m .) We may decompose L(m, N )under the action of complex conjugation to get a pair of elements×ˆ
L± (m, N ) ∈ (MN )±m ⊗Zp Zp [[Zp ]].

We now assume that the hypotheses of Proposition 3.3.1 are satisfied. Thus we
may identify (TN )m with (MN )±m and so regard×∼ˆ Zp Zp [[Z×L± (m, N ) ∈ (TN )m ⊗p ]] = (TN )m [[Zp ]].

We will refer to L± (m, N ) as the two-variable p-adic L-function attached to (TN )m .
Note that it is well-defined up to a unit in (TN )m .
If we fix a tame character ω i for some 0 ≤ i ≤ p − 2, then we may project±
L (m, N ) onto the “ω i -part” of Zp [[Z×p ]] and obtainˆ Zp Λ(i) .
L(m, N, ω i ) ∈ (TN )m ⊗(Here we choose the sign ± to equal (−1)i , since it is well-known, and easily checked,
that otherwise the projection is trivial.)
For any height one prime ℘ in (TN )m , we may reduce L± (m, N, ω i ) modulo ℘to obtain an element
ˆ Zp Λ(i) ∼ˆ Zp Λ(i) ,L± (m, N, ω i )(℘) ∈ ((TN )m /℘) ⊗= O(℘)⊗
which is well-defined up to a unit in O(℘). Also, we may reduce modulo m to obtain
an element
ˆ Zp Λ(i) ∼ˆ Zp Λ(i) ,L̄ ± (m, N, ω i ) ∈ ((TN )m /m) ⊗=k ⊗which is well-defined up to an element of k × .
The following proposition describes the manner in which the two-variable padic L-function L± (m, N, ω i ) interpolates the one-variable p-adic L-functions of
the modular forms in the Hida family corresponding to (TN )m .

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

23

Proposition 3.4.3. If ℘ is a classical height one prime ideal in (TN )m , thenˆ Zp Λ(i)L± (m, N, ω i )(℘) ∈ O(℘)⊗
is the usual analytic p-adic L-function attached to the corresponding normalized
eigenformf℘ ∈ Sk (N p∞ , O(℘))ord [κ℘ ](computed with respect to the canonical period described in Section 3.3).
Proof. This comparison follows immediately from our given constructions of one
and two-variable p-adic L-functions together with the isomorphism in (3.5).
Remark 3.4.4. Note that the normalized eigenform f℘ appearing in the statementof Proposition 3.4.3 need not be a newform and thus L± (m, N, ω i )(℘) may not be a“primitive” p-adic L-function. This issue will be explored in the following section.
Remark 3.4.5. The following remark answers a question asked by the referee; itwill not be used elsewhere in the paper.
Note that the formula of Definition 3.4.1 works equally well to define an (MN )m valued measure on Zp . Integrating the constant function 1 on Zp with respect
to this measure yields an element of (MN )m ; explicitly, this is just the element{∞, 1} ∈ (MN )m . This element in fact lies in (MN )+m , and thus (via our chosen
identification of (MN )+m with (TN )m ) gives rise to an element of (TN )m , which
S wedenote by L∗ (m, N, 1). Any easy computation (using the fact that Zp = Z×pZp )pshows that (1 − Up−1 )L∗ (m, N, 1) = L+ (m, N, 1). (Note that the ‘1’ in the thirdargument stands for the trivial character ω 0 .) Thus L∗ (m, N, 1) coincides with the“improved p-adic L-function” of [14], computed with respect to a canonical period.
3.5. Two variable p-adic L-functions on branches of the Hida family. Fix
an irreducible modular Galois representation ρ̄ : GQ → GL2 (k) that is both pordinary and p-distinguished. We choose also a p-stabilization on ρ̄ (which we will
suppress in our discussion and notation). Furthermore, fix a finite set of primes Σand consider the Hecke algebra TΣ associated to ρ̄ as in Section 2.4.
Proposition 2.4.2 yields an isomorphism TΣ ∼= (TN (Σ) )n for a certain maximal
ideal n of TN (Σ) . The construction of the preceding section thus defines a twovariable p-adic L-function
ˆ Zp Λ(i) .
L(n, N (Σ), ω i ) ∈ (TN (Σ) )n ⊗The isomorphism TΣ ∼= (TN (Σ) )n then yields elements
ˆ Zp Λ(i) ,LΣ (ρ̄, ω i ) ∈ TΣ ⊗LΣ (ρ̄, ω i )(℘) ∈ O(℘) ⊗Zp Λ(i) ,ˆ Zp Λ(i) ,LΣ (ρ̄, ω i ) ∈ k ⊗as defined in the previous section, which are well-defined up to units in TΣ , O(℘)
and k respectively.
We now note that there are two different ways that one can naturally attach
a modular form to a classical height one prime ideal ℘ of TΣ . First, via theisomorphism of Proposition 2.4.2, this prime corresponds to a classical height one
prime ideal of TN (Σ) which we again call ℘. By part (2) of Theorem 2.1.1, this prime
gives rise to a normalized eigenform f℘ of tame level N (Σ). On the other hand,by Proposition 2.5.6, ℘ also corresponds to a classical height one prime of Tnew
N (a)

24

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

which we denote by ℘new . (Here a is the unique minimal prime of TΣ contained in℘.) By part (2) of Theorem 2.1.3, this prime gives rise to a normalized newformf℘new := f℘new of tame level N (a).
In general, f℘ and f℘new are not equal. Indeed, they are eigenforms for different
Hecke algebras: f℘new is an eigenform for TN (a) while f℘ is an eigenform for TN (Σ) .
Proposition 2.4.2 makes it clear how they are related: f℘ is the normalized oldform
obtained from f℘new by “removing the Euler factor” at each of the primes ` ∈ Σ.
Note that Proposition 3.4.3 shows that LΣ (ρ̄, ω i ) interpolates the p-adic Lfunctions attached to the modular forms f℘ for ℘ a classical height one primeof TΣ . In particular, these are “non-primitive” p-adic L-functions.
We now seek to construct a two variable p-adic L-function for each minimal
prime a of TΣ , which interpolates the p-adic L-functions of the modular forms
f℘new as ℘new runs through the classical height one primes of T(a). For this, we◦0
recall the natural isomorphism T(a) ∼= Tnew
N (a) /a of Definition 2.5.3, which gives
rise to the composite surjection◦(3.7)
TN (a) → Tnew → Tnew /a0 ∼= T(a) .
N (a)

N (a)

If we let m denote the maximal ideal of TN (a) obtained as the preimage of the◦
maximal ideal of T(a) under (3.7), then the construction of the preceding section
ˆ Zp Λ(i) . The surjection (3.7) induces
yields an L-function L(m, N (a), ω i ) ∈ TN (a) ⊗a corresponding surjection
ˆ Zp Λ(i) → T(a)◦ ⊗ˆ Zp Λ(i) .
TN (a) ⊗We then denote the image of L(m, N (a), ω i ) under this surjection by◦ˆ Zp Λ(i)L(ρ̄, a, ω i ) ∈ T(a) ⊗and we refer to L(ρ̄, a, ω i ) as the two variable p-adic L-function attached to T(a).
The following proposition describes the interpolation property of L(ρ̄, a, ω i ).
Proposition 3.5.1. If ℘new is a classical height one prime ideal in T(a),ˆ Zp Λ(i)L(ρ̄, a, ω i )(℘new ) ∈ O(℘new )⊗equals the p-adic L-function of f℘new := f℘new computed with respect to a canonicalperiod.
Proof. This proposition follows immediately from Proposition 3.4.3.



3.6. Comparisons. We now have a two-variable p-adic L-function LΣ (ρ̄, ω i ) defined over the entire Hida algebra TΣ and, for each minimal prime a of TΣ , wehave a two-variable p-adic L-function L(ρ̄, a, ω i ) defined on the branch T(a). It isnatural then to compare L(ρ̄, a, ω i ) with LΣ (ρ̄, ω i ) modulo a.
◦ˆ Zp Λ(i) . The interpolation property
Each of these p-adic L-functions lies in T(a) ⊗satisfied by classical p-adic L-functions shows that for any height one prime ℘ of TΣ
containing a, the L-functions Lp (f℘new ) and Lp (f℘ ) agree up to multiplication bya certain product of Euler factors (reflecting the Euler factors removed from f℘new
to obtain f℘ ) and an element of O(℘)[1/p]× (reflecting the ratio of the canonicalperiod of f℘new and the canonical period of f℘ ).
◦We will show that in fact the ratio of these canonical periods lies in (O(℘) )× .
Hence, Lp (f℘new ) and Lp (f℘ ) agree up to multiplication by a product of Euler factors◦
and the inevitable ambiguity of a unit of O(℘) . Furthermore, we will show that

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

25

this occurs uniformly in Hida families; in other words, that LΣ (ρ̄, ω i ) mod a andL(ρ̄, a, ω i ) agree up to multiplication by a product of Euler factors and a unit in◦T(a) .
We begin by defining the relevant Euler factors. Recall the Euler factors E` (a, X)◦
in T(a) [X] defined in Section 2.7. Since ` is prime to p, we may regard ` as an×element of Z×p and so obtain a corresponding unit element h`ip ∈ Zp [[Zp ]]. Thus
◦ˆ
◦
× ∼
×
E` (a, h`i−1p ) ∈ T(a) [[Zp ]] = T(a) ⊗Zp Zp [[Zp ]].

For 0 ≤ i ≤ p − 2, we write h`ip,i to denote the projection of h`ip ∈ Zp [[Z×p ]] under−1
the surjection Zp [[Z×
]]
→
Λ
.
We
write
E
(a,
h`i
)
to
denotethecorresponding
`
(i)
p
p,i
◦ˆ
element of T(a) ⊗Zp Λ(i) .
Definition 3.6.1. If a is a minimal prime of TΣ , then we writeY◦ˆ×EΣ (a) :=E` (a, h`i−1p ) ∈ T (a) ⊗Zp Zp [[Zp ]]`∈ΣandEΣ (a, ω i ) :=Y◦ˆE` (a, h`i−1p,i ) ∈ T (a) ⊗Zp Λ(i) .

`∈ΣiSince LΣ (ρ̄, ω ) and L(ρ̄, a, ω i ) are constructed using modular symbols of levels N (Σ) and N (a) respectively, in order to compare them, we will need to beable to compare the corresponding Hecke algebras. Recall that by constructionTΣ = (T0N (Σ) )m0 for a certain maximal ideal m0 of T0N (Σ) and that n is a certain
maximal ideal of TN (Σ) lying over m0 with the property that the map (T0N (Σ) )m0 →(TN (Σ) )n is an isomorphism. We let m denote the preimage in TN (a) , under the◦surjection (3.7), of the maximal ideal of T(a) . Altogether, we have the following
diagram of maps between the various Hecke algebras:(3.8)(T0N (Σ) )m0(TN (a) )m∼=/ (TN (Σ) )n/ T(a)◦ .

By inverting the upper horizontal isomorphism, we obtain a map (TN (Σ) )n →◦T(a) . This is a homomorphism of T0N (Σ) -algebras when we equip the source andtarget with the T0N (Σ) -algebra structure provided by the diagram (3.8).
We write (MN (Σ) )n and (MN (a) )m to denote the modules of p-adic modular symbols constructed in Section 3.3 for the indicated choice of tame level and localized
at the indicated maximal ideal. The following result provides the key to comparing the L-functions LΣ (ρ̄, ω i ) mod a and L(ρ̄, a, ω i ). We postpone its proof to theSection 3.8.
◦

Theorem 3.6.2. There is an isomorphism of T(a) -modules◦◦∼
T(a) ⊗(T) (MN (Σ) )n = T(a) ⊗(T) (MN (a) )m ,N (Σ) nN (a) mcompatible with the action of complex conjugation.
Moreover, the induced map on the corresponding spaces of measures taking values
in these modules sends the measure L(n, N (Σ)), which takes values in the source of

26

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

the isomorphism, to the measure L(m, N (a))EΣ (a), which takes values in the targetof the isomorphism.
◦

Corollary 3.6.3. There is a unit u ∈ T(a) such thatLΣ (ρ̄, ω i ) ≡ u · L(ρ̄, a, ω i )EΣ (a, ω i )(mod a).

Proof. This follows immediately from the preceding theorem.



3.7. Iwasawa invariants. If we choose an isomorphism(3.9)
Λ∼= Zp [[T ]],ˆ Zp Λ(i) ∼and hence an isomorphism (TN )m ⊗= (TN )m [[T ]], we may then regardiL(m, N, ω ) as an element of Tm [[T ]]. We are, of course, interested in the Iwasawa
invariants of such power series. We begin this section by reviewing the definitions
and basic properties of such invariants.
Definition 3.7.1. If R is a ring and f (T ) ∈ R[[T ]] is a one-variable power series
with coefficients in R, then we define the content of f (T ) to be the ideal I(f (T )) ⊂ R
generated by the coefficients of f (T ).
The proof of the next lemma is straightforward.
Lemma 3.7.2. Let f (T ) ∈ R[[T ]].
(1) If α is any automorphism of R[[T ]], then I(f (T )) = I(α(f (T ))).
(2) If u(T ) ∈ R[[T ]]× , then I(u(T )f (T )) = I(f (T )).
(3) If f (T ) = g(T )h(T ) in R[[T ]], then if any two of these three power series
have unit content, so does the third.
(4) If φ : R → S is a morphism of rings then I(φ(f (T ))) = φ(I(f (T ))).
In particular, regarding L(m, N, ω i ) as a power series, we may consider its content I(L(m, N, ω i )), an ideal of (TN )m . Parts (1) and (2) of the preceding lemmashow that this ideal is in fact independent of the choice of the isomorphisms (3.6)
and (3.9).
For each height one prime ℘, we may likewise form the corresponding ideal
I(L(m, N, ω i )(℘)) in O(℘). By part (4) of Lemma 3.7.2, this ideal equals the imageof I(L(m, N, ω i )) under the surjection (TN )m → (TN )m /℘ ∼= O(℘). We also havethat I(L̄(m, N, ω i )) is an ideal of k and is thus either zero or all of k. Again,
we have that this ideal equals the image of I(L(m, N, ω i )) under the surjection(TN )m → (TN )m /m ∼= k.
The next result is an immediate consequence of the fact that (TN )m is a local
ring.
Proposition 3.7.3. Fix i such that 0 ≤ i ≤ p − 2. The following are equivalent:(1) I(L(m, N, ω i )) = (TN )m .
(2) I(L(m, N, ω i )(℘)) = O(℘) for some height one prime ℘ of (TN )m .
(3) I(L(m, N, ω i )(℘)) = O(℘) for every height one prime ℘ of (TN )m .
(4) I(L̄(m, N, ω i )) is non-zero (and hence equal to k).
We remark that if ℘ is a classical height one prime ideal in (TN )m , then thelength of the quotient O(℘)/I(L(m, N, ω i )(℘)) is related to the usual µ-invariantof the p-adic L-function L(m, N, ω i )(℘) ∈ O(℘)[[T ]]. In particular, this µ-invariant
vanishes if and only if I(L(m, N, ω i )(℘)) is the unit ideal of O(℘).

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

27

Analogously, we have well-defined ideals
I(LΣ (ρ̄, ω i )), I(LΣ (ρ̄, ω i )(℘)) and I(LΣ (ρ̄, ω i ))of TΣ , O(℘) and k respectively. The above result applies equally well to these
ideals since they are constructed out of I(L(m, N, ω i )) for some choice of m and N .
We now compute the content of the reciprocal Euler factor EΣ (a, ω i ) defined in
Section 3.6.
◦ˆLemma 3.7.4. The element EΣ (a, ω i ) of T (a) ⊗Zp Λ(i) has unit content.

Proof. Part (3) of Lemma 3.7.2 shows that it suffices to prove that each of thereciprocal Euler factors E` (a, h`i−1p,i ) has unit content. If γ is a topological generatornof Γ, then we may write `−1 ω(`) = γ up for some u ∈ Z×
p and some integer n ≥ 0.
If we choose our isomorphism (3.9) so that hγip 7→ 1 + T , then we see thatE` (a, h`i−1p,i ) =

n
0
−i
upn

+ h`iN (a)p ω −2i (`)`−1 (1 + T )2up )(1 − (T` mod a )ω (`)(1 + T )if ` is prime to N (a)n(1 − (T` mod a0 )ω −i (`)(1 + T )up ) otherwise.
In the second case either the constant term is a unit (if T` is not a unit) or elsenthe coefficient of T p is a unit (if T` is a unit). In the first case, we may computethe content after making the substitution (1 + T )u 7→ 1 + T , in which case one seesn
immediately that the coefficient of T 2p is a unit.
The following theorem establishes that if the µ-invariant vanishes for one form
in a Hida family, then it vanishes for every form in that family.
Theorem 3.7.5. The following are equivalent:(1) µ(Lp (f, ω i )) = 0 for some newform f in the Hida family of ρ̄.
(2) µ(Lp (f, ω i )) = 0 for every newform f in the Hida family of ρ̄.
(3) L(ρ̄, a, ω i ) has unit content for some irreducible component T(a) of the Hida
family of ρ̄.
(4) L(ρ̄, a, ω i ) has unit content for every irreducible component T(a) of the Hida
family of ρ̄.
Proof. Let T(a) be an irreducible component of the Hida family of ρ̄ and let ℘be a classical height one prime ideal of T(a). Since L(ρ̄, a, ω i ) modulo ℘ equals◦
Lp (f℘ , ω i ) and T(a) is a local ring, we deduce from Proposition 3.7.3 that thefollowing are equivalent: L(ρ̄, a, ω i ) has unit content; Lp (f℘ , ω i ) has unit content(i.e. trivial µ-invariant) for one classical height one prime ℘ of T(a); Lp (f℘ , ω i ) has
unit content for every classical height one prime ℘ of T(a).
To complete the proof of the proposition, it suffices to show that if T(a1 ) andT(a2 ) are two irreducible components of the Hida family of ρ̄, then L(ρ̄, a1 , ω i ) has
unit content if and only if the same holds true for L(ρ̄, a2 , ω i ). We may choose Σlarge enough so that each of T(a1 ) and T(a2 ) is an irreducible component of TΣ .
Corollary 3.6.3, together with Lemmas 3.7.2 (3) and 3.7.4, then shows that eachof L(ρ̄, a1 , ω i ) and L(ρ̄, a2 , ω i ) have unit content if and only if the same is true ofLΣ (ρ̄, ω i ).
28

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

If the equivalent conditions of the preceding theorem hold, then we writeµan (ρ̄, ω i ) = 0.
(As usual, our notation suppresses the choice of p-stabilization of ρ̄.) In the casewhen the µ-invariant vanishes we can further study the λ-invariant of these power
series. (Note that for a local ring which is not a discrete valuation ring one canonly define the λ-invariant for power series of unit content.)
Definition 3.7.6. If A is a local ring and f (T ) ∈ A[[T ]] is a power series having
unit content, then we define the λ-invariant λ(f (T )) to be the smallest degree inwhich f (T ) has a unit coefficient.
We remark that if φ : A → B is a local morphism of complete local rings and if
f (T ) is an element of A[[T ]] having unit content, then λ(f (T )) = λ(φ(f (T ))).
If T(a) is an irreducible component of the Hida family of ρ̄ with µan (ρ̄, ω i ) = 0,then Theorem 3.7.5 shows that L(ρ̄, a, ω i ) has unit content. We can therefore define
the analytic λ-invariant of a branch byλan (ρ̄, a, ω i ) = λ(L(ρ̄, a, ω i )).
Our main results on analytic λ-invariants in Hida families are as follows.
Theorem 3.7.7. Assume that µan (ρ̄, ω i ) = 0.
(1) For any given irreducible component T(a) of the Hida family of ρ̄, the λinvariant of Lp (f℘ , ω i ) takes on the constant value of λan (a, ω i ) as ℘ varies
over all classical height one primes of T(a).
(2) For any two irreducible components T(a1 ), T(a2 ) of the Hida family of ρ̄,
we have thatX
λan (a1 , ω i ) − λan (a2 , ω i ) =(e` (a2 , ω i ) − e` (a1 , ω i ))`6=pwhere e` (a, ω i ) = λ(E` (a, h`i−1p,i )).
Proof. The first part follows from Proposition 3.5.1 and Definition 3.7.6. For thesecond part, choose Σ large enough so that both T(a1 ) and T(a2 ) are irreducible
components of TΣ . Then, by Corollary 3.6.3, we have thatX
λ(LΣ (ρ̄, ω i )) = λan (aj , ωi ) +e` (aj , ω i )`∈Σ

for j = 1, 2. Our formula then follows since e` (a1 , ω i ) = e` (a2 , ω i ) for ` ∈/ Σ.



3.8. Proof of Theorem 3.6.2. In this section, we present the proof of Theorem 3.6.2. We will utilize the notation introduced prior to the statement of thetheorem and begin the proof by introducing some additional notation.
If M ≥ 1 is an integer, d is a divisor of M and d0 is a divisor of d, then we let
Bd,d0 : X1 (M ) → X1 (M/d) denote the map induced by the map τ 7→ d0 τ on theupper half-plane. This map induces a corresponding map(Bd,d0 )∗ : H1 (M, {cusps}; Zp ) → H1 (M/d, {cusps}; Zp )on relative homology groups.

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES29

If ` is any prime distinct from p, then we let e` denote the largest power of `dividing N (Σ)/N (a). We have the inequality 0 ≤ e` ≤ 2. Also, e` = 0 unless ` ∈ Σ,
while e` = 2 only if ` ∈ Σ and ` is prime to N (a). For each such prime `, we write
if e` = 01(`) := ((B`,1 )∗ − `−1 T` (B`,` )∗ )if e` = 1−1
−3((B`2 ,1 )∗ − ` T` (B`2 ,` )∗ + ` h`iN (a)p (B`2 ,`2 )∗ ) if e` = 2.
Now write Σ = {`1 , . . . , `n } and for any r ≥ 1 definer : H1 (N (Σ)pr , {cusps}; Zp )ord → H1 (N (a)pr , {cusps}; Zp )ordby r = (`n ) ◦ · · · ◦ (`1 ).
e`e
Let us explain this formula. For any i = 1, . . . , n, write Ni = N (Σ)/`1`1 · · · `i i .
In the formula for r , the map (`i ) is taken to be the map
H1 (Ni−1 pr , {cusps}; Zp )ord → H1 (Ni pr , {cusps}; Zp )ord
given by the stated formula for (`i ). (The symbol T`i in the formula for (`i ) is
understood to stand for the corresponding Hecke operator acting in level Ni pr .) It
is easily verified that the map r is in fact independent of the choice of ordering of
the elements of Σ.
For any tame level M we let (T∗M )0 denote the Λ-subalgebra of the ordinary
Hecke algebra T∗M generated by the Hecke operators prime to M . If we regard thesource and target of r as (T∗N (Σ) )0 -modules via the inclusion (T∗N (Σ) )0 ⊂ T∗N (Σ)
and the natural map (T∗N (Σ) )0 → (T∗N (a) )0 ⊂ T∗N (a) , then r is immediately seen tobe (T∗N (Σ) )0 -linear.
As r varies, the sources and targets of the maps r each form a projective systemand the maps r are evidently compatible with the projection maps on source andtarget. Thus, passing to the limit in r, we obtain a (T∗N (Σ) )0 -linear map
∞ : lim
H (N (Σ)pr , {cusps}; Zp )ord → lim
H (N (a)pr , {cusps}; Zp )ord .
←− 1
←− 1
r

r

We denote the source and target of this map by MN (Σ) and MN (a) respectively.
We may regard each of the maximal ideals m0 , n and m equally well as maximal
ideals of (T∗N (Σ) )0 , T∗N (Σ) and T∗N (a) . If we localize ∞ with respect to m0 , we obtaina map(MN (Σ) )m0 → (MN (a) )m0 .

(3.10)

Now n and m each pull back to m0 under the natural maps T0N (Σ) → TN (Σ) andT0N (Σ) → TN (a) and so the localizations (TN (Σ) )n and (TN (a) )m are local factors of
the complete semi-local rings (T0N (Σ) )m0 ⊗T0N (Σ) TN (Σ) and (T0N (Σ) )m0 ⊗T0N (Σ) TN (a) .
Thus the localizations (MN (Σ) )n and (MN (a) )m appear naturally as direct factorsof (MN (Σ) )m0 and (MN (a) )m0 respectively, and so the map (3.10) induces a map(MN (Σ) )n → (MN (a) )m .
◦Tensoring the source of this map with T(a) over (TN (Σ) )n and the target with◦◦T(a) over (TN (a) )m , we obtain a T(a) -linear map(3.11)◦◦

T(a) ⊗(TN (Σ) )n (MN (Σ) )n → T(a) ⊗(TN (a) )m (MN (a) )m .

We claim that this map satisfies the requirements of Theorem 3.6.2.

30

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

We first observe that this map satisfies the claimed property with regard to the
L-functions. This follows from an explicit calculation of the effect of the maps ` onmodular symbols.
X In fact, one easily shows that for any character χ of conductor
pn and α =χ(a) {a/pn , ∞}, we have thata∈(Z/pn )×r (α) =Y(1 − χ−1 (`)`−1 T` )i s.t. e`i =1

Y(1 − χ−1 (`)`−1 T` + χ−2 (`)`−3 h`iN (a)p ) · α.

i s.t. e`i =2(In the left hand side of this equation, the modular symbols {a/pn , ∞} are regarded
as lying in H1 (N (Σ)pr , {cusps}; Zp ). On the right hand side, they are regarded aslying in H1 (N (a)pr , {cusps}; Zp ).) Passing to the limit in r, and taking into accountthe fact that χ is an arbitrary Dirichlet character of p-power conductor, we conclude
the the isomorphism (3.11) has the required effect on L-functions.
We now turn to showing that (3.11) is an isomorphism. Note that Proposi◦tion 3.3.1 shows that both source and target are free of rank two over T(a) . Thus
to see that this map is an isomorphism, it suffices to check that it induces a surjection after being reduced modulo the maximal ideal of T(a)◦ . To do this, we firstmod out by a classical prime of weight two and then by the full maximal ideal.
Let ℘ denote the classical height one prime in Λ of weight two for which κ℘ istrivial. If we tensor each side of (3.11) by Λ/℘ over Λ, we obtain the map(3.12)◦◦

ord
T(a) ⊗(TN (Σ) )n H1 (N (Σ)p; Zp )ord
n → T(a) ⊗(TN (a) )m H1 (N (a)p; Zp )m

induced by localizing the source and target of 1 at n and m respectively and◦then extending scalars to T(a) . (Here, as in Proposition 3.3.1, we are using [17,Thm. 3.1].)◦
Thus the reduction modulo the maximal ideal of T(a) of (3.11) coincides withthe map(3.13)TN (a) /m ⊗TN (Σ) /n H1 (N (Σ)p; Zp )ord /nH1 (N (Σ)p; Zp )ord
→ H1 (N (a)p; Zp )ord /mH1 (N (a)p; Zp )ordof TN (a) /m-vector spaces induced by (3.12).
Rather than showing directly that (3.13) is surjective, we will show that thecorresponding dual map(3.14)H 1 (N (a)p; Fp )ord [m] → TN (a) /m ⊗TN (Σ) /n H 1 (N (Σ)p; Fp )ord [n]is injective. (In writing the dual of (3.13) in this form, we have implicitly fixed
an isomorphism of one dimensional TN (a) /m-vector spaces between TN (a) /m andits TN (Σ) /m0 -linear dual. We will suppress this choice of isomorphism here andbelow.) This map may be written as a composite
H 1 (N (a)p; Fp )ord [m] → TN (a) /m ⊗TN (Σ) /n H 1 (N (a)p; Fp )ord [m0 ]→ TN (a) /m ⊗TN (Σ) /n H 1 (N (Σ)p; Fp )ord [m0 ](3.15)
→ TN (a) /m ⊗TN (Σ) /n H 1 (N (Σ)p; Fp )ord [n] .
To explain this, we first recall that since the localization (TN (Σ) )n is a local factor of
the tensor product (T0N (Σ) )m0 ⊗T0N (Σ) TN (Σ) for which the natural map (T0N (Σ) )m0 →VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

31(TN (Σ) )n is an isomorphism, the residue field TN (Σ) /n is a local factor of the Artin
local ring TN (Σ) /m0 TN (Σ) ; write∼ TN (Σ) /n × A,
TN (Σ) /m0 TN (Σ) =where A denotes the product of the remaining local factors. This decomposition
induces a corresponding decomposition
B(3.16)B∼= B[n] × A ⊗T
/m0 T
N (Σ)

N (Σ)

0

for any TN (Σ) /m TN (Σ) -module B. The third arrow of (3.15) is precisely projection
onto the the first of the two factors in (3.16) with

B = TN (a) /m ⊗TN (Σ) /n H 1 (N (Σ)p; Fp )ord [m0 ] .
The second arrow is induced by dualizing the reduction modulo m0 of the map (3.10)
and the first arrow is induced from the obvious inclusion, or if the reader prefers,
is obtained by dualizing the surjection
TN (a) /m ⊗TN (Σ) /n H1 (N (a)p; Zp )ord /m0 H1 (N (a)p; Zp )ord
→ H1 (N (a)p; Zp )ord /mH1 (N (a)p; Zp )ord .
Lemma 3.8.1. The map (3.14) is injective if and only if the composite of the firsttwo arrows of (3.15) is injective.
Proof. The only if statement is clear. In order to prove the other statement, wefirst observe that the second arrow of (3.15) is given by the cohomological versionof the map 1 . More precisely, if for each ` ∈ Σ we define
if e` = 01∗∗∗` := B`,1
− B`,``−1 T`
if e` = 1

 ∗
∗
−1
∗
−3
B`2 ,1 − B`2 ,` ` T` + B`2 ,`2 ` h`iN (a)p if e` = 2∗(where Bd,d0 denotes the map on cohomology induced by the degeneracy map Bd,d0 )and define∗ = ∗`1 ◦ · · · ◦ ∗`n : H 1 (N (a)p; Fp )ord → H 1 (N (Σ)p; Fp )ord ,then the second arrow of (3.15) is obtained from the map ∗ by passing to thekernel of m0 in the source and target and then extending scalars from TN (Σ) /n to
TN (a) /m.
One then checks that any element in the image of ∗ is annihilated by the Hecke
operators T` , for those ` ∈ Σ. On the other hand, Proposition 2.4.2 shows thatthe maximal ideal n is uniquely characterized by the property of containing these
operators. Thus the image of the second arrow of (3.15) lies in the local factorTN (a) /m ⊗TN (Σ) /n H1 (N (Σ)p; Fp )ord [n]of
TN (a) /m ⊗TN (Σ) /n H1 (N (Σ)p; Fp )ord [m0 ] .
This proves the lemma.



By the preceding lemma, we are reduced to proving that the composite of thefirst two arrows of (3.15) is injective. It will be notationally easier to deal with eachof the maps `i separately and so we put ourselves in the following more generalsituation. We consider a natural number M prime to p and a prime ` distinct from

32

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

p. We define n` = 1 or 2 according to whether or not ` divides M and write
N := `n` M . We let m denote a maximal ideal of TM for which ρ̄m is irreducible
and p-distinguished and let m0 denote the pull back of m under the natural map
T0N → TM . The map ∗` defined in the proof of Lemma 3.8.1 induces a TM /m-linearmap
(3.17)H 1 (M p; Fp )ord [m] → (TM /m) ⊗T0N /m0 H 1 (N p; Fp )ord [m0 ] .
Lemma 3.8.2. The map (3.17) is injective.
The proof of Lemma 3.8.1 shows that elements in the image of (3.17) are annihilated by T` and hence are in fact annihilated by a maximal ideal of the full
Hecke algebra TN . Thus we may apply Lemma 3.8.2 inductively to establish theinjectivity of the composite of the first two arrows of (3.15) and thereby complete
the proof of the theorem.
Proof of Lemma 3.8.2. It will be convenient to bring the residue field TM /m inside
the coefficients of cohomology. To do this, we choose a finite field k containing
TM /m and tensor the source and target of (3.17) with k over Fp to obtain amap of k ⊗Fp TM /m-modules. The chosen inclusion of TM /m in k determines a
projection
k ⊗Fp TM /m → k,

(3.18)

which realizes k as a local factor of k ⊗Fp TM /m. Projecting onto this local factor,
we recover our original map (3.17), but rewritten as(3.19)H 1 (M p; k)ord [mk ] → H 1 (N p; k)ord [m0k ].

Here we regard the source as a W (k) ⊗Zp TM -module and the target as a W (k) ⊗Zp
T0N -module. Also, we have written mk to denote the the maximal ideal of W (k)⊗Zp
TM that is the kernel of the composite
W (k) ⊗Zp TM → k ⊗Fp TM /m → k(where the second arrow is given by the projection (3.18)) and m0k to denote themaximal ideal of W (k) ⊗Zp T0N that is the kernel of the composite
W (k) ⊗Zp T0N → k ⊗Fp T0N /m0 → k ⊗Fp TM /m → k(where the second arrow is obtained by tensoring the injection T0N /m0 → TM /mby k over Fp and the third arrow is given by the projection (3.18)). We must showthat the map (3.19) is injective.
Our argument will rely on the results of [37, §2.2] which extend a well-known
result of Ihara. Recall that the standard practice for analyzing the map (3.19) is
to factor the map ∗` and so to write (3.19) as a compositeβ`α`
H 1 (M p; k)ord [mk ] −→(H 1 (M p; k)ord [mk ])n` +1 −→ H 1 (N p; k)ord [m0k ],where((1, −`−1 T` )α` =(1, −`−1 T` , `−3 h`iM p )if n` = 1
if n` = 2

and
(
∗
∗
B`,1
π1 + B`,`
π2
β` =
∗
∗
B`2 ,1 π1 + B`2 ,` π2 + B`∗2 ,`2 π3

if n` = 1
if n` = 2

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

33

with πi denoting projection onto the ith factor of the product(H 1 (N p; k)ord [mk ])n` +1 .
The map α` is manifestly injective and, in the case n` = 2, Wiles has shownthat the map β` is also injective. (See in particular the discussion at the top of [37,p. 497] or the discussion of Wiles’ results provided by [7, §§4.3] and [3, §§4.4, 4.5].)
Thus when n` = 2 the lemma is proved.
However, if n` = 1, then the situation is more complicated. Lemma 2.5 of [37]provides an exact sequence
β`

γ`

1
ord
1
ord 2
H 1 ((M/`)p; k)ord
m0 −→ (H (M p; k)m0 ) −→ H (N p; k)m0 ,
k

k

k

∗
∗
where γ` = (B`,`, −B`,1). (Here the subscript m0k denotes that we have localized at

this maximal ideal.)
Let x ∈ H 1 (M p; k)[mk ] be an element in the kernel of (3.19). Then (3.8) shows∗∗that we may find y ∈ H 1 ((M/`)p; k) such that (x, −T` x) = (B`,`y, −B`,1y). In∗particular, B`,` y = x is annihilated by mk and so is an eigenvector for the full∗
Hecke algebra TM . The following result shows that B`,`y = 0 and hence thatx = 0. This completes the proof of the lemma.
Lemma 3.8.3. Let D be a natural number prime to p, let ` be a prime distinct from∗
: H 1 (Dp; F̄p )ord → H 1 (D`p; F̄p )ord . If y is a class in
p and consider the map B`,`∗y is an eigenvector for the full Hecke ringthe domain with the property that B`,`TD` , corresponding to a maximal ideal for which the attached Galois representation∗y = 0.
into GL2 (F̄p ) is irreducible and p-distinguished, then B`,`∗on cohomology classes
Proof. We will prove the lemma by comparing the map B`,`
with the corresponding map on modular forms mod p. For such modular forms, theanalogue of the lemma follows immediately from a consideration of q-expansions.
For the comparison with modular forms mod p, we follow the discussion in the
proof of [37, Thm. 2.1].
Let us make some initial reductions in the situation of the lemma. We letm denote the maximal ideal in F̄p ⊗Fp TD` describing the action of the Hecke∗y. Let m0 denote the intersection of m with
operators on the eigenvector B`,`the subring F̄p ⊗Fp T0D` of F̄p ⊗Fp TD` . We may regard each of H 1 (Dp; F̄p )ordand H 1 (D`p; F̄p )ord as F̄p ⊗Fp T0D` -modules (the former via the natural map∗
is F̄p ⊗Fp T0D` -linear. If ym0 deF̄p ⊗Fp T0D` → F̄p ⊗Fp T0D ) and the map B`,`1
ord
notes the projection of y onto the localization H 1 (Dp; F̄p )ordat
m0 of H (Dp; F̄p )
0
∗
∗
∗
0
m , then B`,` ym = B`,` y (since B`,` y is assumed to be annihilated by m and so inparticular by m0 ). Also, there is a natural isomorphismY∼
H 1 (Dp; F̄p )ord
H 1 (Dp; F̄p )ordn ,m0 =n⊃m0 (F̄p ⊗Fp TD )where n ranges over all maximal ideals of F̄p ⊗Fp TD containing m0 (F̄p ⊗Fp TD ).
∗∗Thus to show that B`,`y = 0, it suffices to show that B`,`yn = 0 for each suchmaximal ideal n, where yn denotes the projection of y onto H 1 (Dp; F̄p )ord
n . Thus
for the duration of the proof we assume that y ∈ H 1 (Dp; F̄p )ord
,
for
some
maximalnideal n of F̄p ⊗Fp TD lying over m0 .

34

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

As in the proof of [37, Thm. 2.1], we consider two cases: that in which the modp diamond operators have non-trivial image in (F̄p ⊗Fp TD` )/m and that in which
they have trivial image.
In order to treat the first case, we begin by noting that the commutative diagram(3.20)H 1 (Dp; F̄p )ord
n

∗
B`,`

m

∼
=

∼
=


(F̄p ⊗Fp J1 (Dp)[p]ord )n/ H 1 (D`p; F̄p )ord

∗
B`,`


/ (F̄p ⊗F J1 (D`p)[p]ord )m ,p

in which the lower horizontal arrow is induced by the map of Jacobians arisingby Picard functoriality applied to the degeneracy map B`,` : X1 (D`p) → X1 (Dp),allows us to replace curves with coefficients in F̄p with a consideration of p-torsion
in the corresponding Jacobians (tensored by F̄p over Fp ). (The superscript ord on
J1 (Dp)[p] and J1 (D`p)[p`] denotes the localization of these p-torsion modules atthe p-ordinary part of the Hecke algebra, or equivalently, the image of the p-torsion
modules under Hida’s idempotent eord .)Let ρ̄ : GQ → GL2 (F̄p ) denote the residual Galois representation attached toeach of m and n. From [37, Thm. 2.1] and its proof we conclude that there are
isomorphisms of Galois modules(F̄p ⊗Fp TD )n /p(F̄p ⊗Fp TD )n ⊗F̄p ρ̄ ∼= (F̄p ⊗Fp J1 (Dp)[p]ord )nand(F̄p ⊗Fp TD` )m /p(F̄p ⊗Fp TD` )m ⊗F̄p ρ̄ ∼= (F̄p ⊗Fp J1 (D`p)[p]ord )m .
Also, passing to p-torsion in the short exact sequence [37, (2.2)] for each of the
maximal ideals m and n yields short exact sequences
0 → (F̄p ⊗Fp J1 (Dp)[p]ord )0n → (F̄p ⊗Fp J1 (Dp)[p]ord )n →(F̄p ⊗Fp J1 (Dp)[p]ord )En →0and
0 → (F̄p ⊗Fp J1 (D`p)[p]ord )0m → (F̄p ⊗Fp J1 (D`p)[p]ord )m →(F̄p ⊗Fp J1 (D`p)[p]ord )Em → 0.
∗The lower horizontal arrow B`,`of (3.20) is GQ -equivariant and also induces amorphism between these short-exact sequences.
∗
If B`,`y is non-zero, then, since ρ̄ is an irreducible GQ -module, we may find σ ∈∗GQ such that σ(B`,`y) has non-zero image in (F̄p ⊗Fp J1 (D`p)[p]ord )E
m . Replacing
y by the image of σ(y) in (F̄p ⊗Fp J1 (Dp)[p]ord )E,
weare
reducedtoproving thenfollowing claim.
∗
ord E
Claim: If y ∈ (F̄p ⊗Fp J1 (Dp)[p]ord )E)nn is such that B`,` y ∈ (F̄p ⊗Fp J1 (D`p)[p]∗is annihilated by m, then B`,` y vanishes.

Passing to the special fiber of the Néron model of each of J1 (Dp) and J1 (D`p)
over Zp [ζp ] (where ζp denotes a primitive pth root of unity), the discussion in theVARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

35

proof of [37, Thm. 2.1] yields a commutative diagram in which the horizontal arrows
are isomorphisms:/ H 0 (Σ1 (D)µ , Ω1 ) ⊕ H 0 (Σ1 (D)ét , Ω1 )ord(F̄p ⊗Fp J1 (Dp)[p]ord )E
n
n
∗
B`,`


(F̄p ⊗Fp J1 (D`p)[p]ord )E
m

∗
B`,`


/ H 0 (Σ1 (D`)µ , Ω1 ) ⊕ H 0 (Σ1 (D`)ét , Ω1 )ord .
m(For any M prime to p, we denote by Σ1 (M )µ and Σ1 (M )ét the base-change from
Fp to F̄p of the two components of the special fiber of the canonical model ofX1 (M p) over Zp [ζp ].) Thus to prove the claim, it suffices to show that if z is a nonord∗
zero element of H 0 (Σ1 (D)µ , Ω1 ) ⊕ H 0 (Σ1 (D)ét , Ω1 ) n , then B`,`cannot lie in

ord
0
µ
1
0
ét
1
the m-eigenspace of H (Σ1 (D`) , Ω ) ⊕ H (Σ1 (D`) , Ω ) m . This follows from∗the q-expansion principal; more precisely, the q-expansion of B`,`z at the cusp ∞`involves only powers of q and so cannot be an eigenform for the full Hecke algebraTD` .
The second case (when the mod p diamond operators are trivial modulo m) is
treated similarly, using the corresponding results from the proof of [37, Thm. 2.1].

4. Algebraic Iwasawa invariantsP
4.1. Selmer groups of modular forms. Let f =an q n be a p-ordinary andp-stabilized newform of weight k ≥ 2, tame level N and character χ. Let K denotethe finite extension of Qp generated by the Fourier coefficients of f and let O denote
the ring of integers of K; we write k for the residue field and fix also a uniformizer
π of O. Let
ρf : GQ → GL2 (K)be the corresponding Galois representation, characterized by the fact that the characteristic polynomial under ρf of an arithmetic Frobenius at a prime ` - N p is
X 2 − a` X + χ(`)`k−1 .
By [14, Thm. 2.6] the restriction of ρf to Gp is of the form
 k−1 −1

ε
χϕ
∗
∼
(4.1)
ρ|Gp =
0ϕwith ϕ : Gp → O× the unramified character sending an arithmetic Frobenius to ap .
We assume that the semisimple residual representationρ̄f : GQ → GL2 (k)is absolutely irreducible. It follows that, up to conjugation by GL2 (O), there is aunique integral modelρf : GQ → GL2 (O)of ρf , which we now fix. For 0 ≤ i ≤ p − 2, let Af,i denote a cofree O-module ofcorank 2 with GQ -action via ρf ⊗ ω i . We obtain from (4.1) and [16, Prop. 12.1] an
O[Gp ]-equivariant exact sequence(4.2)

0 → (K/O)(εk−1 χω i ϕ−1 ) → Af,i → (K/O)(ω i ϕ) → 0.

36

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

We write A0f,i (resp. A00f,i ) for the submodule (resp. quotient module) of Af,i in theabove sequence.
For a place v of Q∞ define(
H 1 (Q∞,v , Af,i )v 6= vp ;1

Hs (Q∞,v , Af,i ) =im H 1 (Q∞,vp , Af,i ) → H 1 (Ivp , A00f,i ) v = vp .
Following [10], the Selmer group of Af,i is defined byYSel(Q∞ , Af,i ) = ker H 1 (Q∞ , Af,i ) →Hs1 (Q∞,v , Af,i )v
1= ker H (QΣ /Q∞ , Af,i ) →Y
Hs1 (Q∞,v , Af,i )v∈Σfor any finite set of places Σ containing vp , all archimedean places and all places
dividing N . We regard Sel(Q∞ , Af,i ) as a ΛO -module via the natural action of Γ.
Recall that µalg (f, ω i ) (resp. λalg (f, ω i )) is defined to be the largest power of
π dividing (resp. the number of zeroes of) the characteristic power series of theΛO -dual of Sel(Q∞ , Af,i ) (assuming that this Selmer group is ΛO -cotorsion).
Theorem 4.1.1. Let f be a p-ordinary and p-stabilized newform with ρ̄f absolutely
irreducible. Then Sel(Q∞ , Af,i ) is co-finitely generated, ΛO -cotorsion and has noproper ΛO -submodules of finite index. Furthermore, µalg (f, ω i ) vanishes if and only
if Sel(Q∞ , Af,i )[π] is finite. If this is the case, then Sel(Q∞ , Af,i ) is O-divisible
andλalg (f, ω i ) = dimk Sel(Q∞ , Af,i )[π].
Proof. It is shown in [10, Prop. 6] that Selmer groups are always co-finitely generated. The fact that they are also ΛO -cotorsion for modular forms is proven in [20].
The equivalence of the vanishing of µalg (f, ω i ) and the finiteness of Sel(Q∞ , Af,i )[π]is now an immediate consequence of the structure theory of ΛO -modules.
The proof of [13, Prop. 4.14] easily adapts to show that Sel(Q∞ , Af,i ) has noproper ΛO -submodules of finite index. (As Af,i need not be self-dual, this also
requires the fact that Sel(Q∞ , A∗f,i ) is ΛO -cotorsion, which follows since A∗f,i is also
modular.) When Sel(Q∞ , Af,i )[π] is finite, the maximal O-divisible ΛO -submoduleof Sel(Q∞ , Af,i ) has finite index, so that it must coincide with Sel(Q∞ , Af,i ); that
is, Sel(Q∞ , Af,i ) is divisible. It now follows again from the structure theory ofΛO -modules that as O-modulesalgiSel(Q∞ , Af,i ) ∼= (K/O)λ (f,ω ) ,which proves the last statement.



We close this section with a useful result on the local invariants H 0 (Gv , Af,i ) for
places v dividing primes ` 6= p. Let cond` (ρ̄f ) denote the exponent of the highest
power of ` that divides the conductor of ρ̄f .
Lemma 4.1.2. Let v be a place of Q∞ dividing a prime ` 6= p. If cond` (ρ̄f ) =ord` (N ), then H 0 (Gv , Af,i ) is O-divisible for all i.
Proof. By the invariance of the Swan conductor under reduction (see [23, §1]), we
have
cond` (ρf ) − cond` (ρ̄f ) = dimk Af [π]I` − dimK VfI`

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

37

where Vf is a two dimensional K-vector space with GQ -action via ρf . Since
cond` (ρf ) equals ord` (N ) by [1], we see that the hypothesis of the lemma is equivalent to the equality
dimk Af [π]I` = dimK VfI` .
It follows easily from this that AIf` = AIfv is O-divisible. As the Gv /Iv -invariants ofan O-divisible Gv /Iv -module are again O-divisible, we conclude that H 0 (Gv , Af )is divisible, as claimed. Since ω is unramified at `, the above argument works for
Af,i as well.

4.2. Residual Selmer groups. Let Ā denote a two-dimensional k-vector spaceequipped with a continuous k-linear action of GQ and a k[Gp ]-equivariant exact
sequence(4.3)

0 → Ā0 → Ā → Ā00 → 0with Ā0 and Ā00 one-dimensional. We make the following assumptions on this data.
(1) Ā is absolutely irreducible as a k[GQ ]-module;(2) The k-vector space Ā00 is unramified as a Gp -module;(3) Ā is p-distinguished in the sense that the representation of Gp on Ā is
non-scalar;(4) Ā is modular: there exists a totally ramified extension K 0 of the field of
fractions of the Witt vectors of k, a p-stabilized newform f ∈ O0 [[q]] of
weight k ≥ 2 and a k[GQ ]-isomorphism Ā ∼= Af [π 0 ] identifying Ā0 with
00
Af [π ] in the notation of the previous section; here π 0 is a uniformizer ofthe ring of integers O0 of K 0 .
We will study Selmer groups of Ā and its cyclotomic twists Ā ⊗ ω i with respect tovarious local conditions.
Definition 4.2.1. A finite/singular structure S on Ā⊗ω i is a choice of k-subspaces
(4.4)

1
Hf,S
(Q∞,v , Ā ⊗ ω i ) ,→ H 1 (Q∞,v , Ā ⊗ ω i )for each place v of Q∞ , subject to the restrictions:
1(Q∞,v , Ā ⊗ ω i ) = 0 for almost all v;(1) Hf,S
1(Q∞,vp , Ā ⊗ ω i ) = ker H 1 (Q∞,vp , Ā ⊗ ω i ) → H 1 (Ivp , Ā00 ⊗ ω i ) .
(2) Hf,S(Note that we are not allowing any variation in the choice of condition at vp ). We
1
define Hs,S
(Q∞,v , Ā ⊗ ω i ) as the cokernel of (4.4). The S-Selmer group of Ā ⊗ ω i
is
Y

1
Hs,S
(Q∞,v , Ā ⊗ ω i )SelS (Q∞ , Ā ⊗ ω i ) = ker H 1 (Q∞ , Ā ⊗ ω i ) →v
1i= ker H (QΣ /Q∞ , Ā ⊗ ω ) →

Y


1
Hs,S
(Q∞,v , Ā ⊗ ω i )v∈Σ

for any finite set of places Σ containing p, all archimedean places, all places at1which Ā is ramified and all places for which Hf,S
(Q∞,v , Ā ⊗ ω i ) does not vanish.
We will be especially interested in two kinds of finite/singular structures. First,
the minimal structure Smin on Ā ⊗ ω i is given by
1
Hf,S(Q∞,v , Ā ⊗ ω i ) = 0
minfor v 6= vp .

38

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

Next let f be a newform as in (4) above. We define the induced structure S(f, i)
on Ā ⊗ ω i by setting1
Hf,S(f,i)(Q∞,v , Ā ⊗ ω i ) = ker H 1 (Q∞,v , Ā ⊗ ω i ) → Hs1 (Q∞,v , Af,i )for all places v of Q∞ . Note that by the definition of Hs1 (Q∞,vp , Af,i ) we have that1
Hf,S(f,i)(Q∞,vp , Ā ⊗ ω i ) is the kernel of the composite map
H 1 (Q∞,vp , Ā ⊗ ω i ) → H 1 (Ivp , Ā00 ⊗ ω i ) → H 1 (Ivp , A00f,i );in fact, since H 0 (Ivp , A00f,i ) is either A00f (for i = 0) or else zero (for i 6= 0), the latter
map is injective, so that1
(Q∞,vp , Ā ⊗ ω i ) = ker H 1 (Q∞,vp , Ā ⊗ ω i ) → H 1 (Ivp , Ā00 ⊗ ω i )(4.5) Hf,S(f,i)
as required.
Lemma 4.2.2. For v - p we have

1
i
1
v
(Q∞,v , Ā ⊗ ω i ) = im AG
Hf,S(f,i)f,i /π ,→ H (Q∞,v , Ā ⊗ ω )1
In particular, Hf,S(f,i)(Q∞,v , Ā ⊗ ω i ) = 0 if Ā ⊗ ω i is unramified at v 6= vp so that
S(f, i) is a finite/singular structure. The natural map

SelS(f,i) (Q∞ , Ā ⊗ ω i ) → Sel(Q∞ , Af,i )[π]is an isomorphism.
Proof. The lemma follows from various exact sequences in cohomology coming fromthe exact sequenceπ
0 → Ā ⊗ ω i → Af,i −→ Af,i → 0
together with Lemma 4.1.2. We leave the details to the reader.

Proposition 4.2.3. Let Ā be as above. Then there exists a newform f as in (4)above such that the finite/singular structures Smin and S(Af,i ) coincide under theisomorphism Ā ⊗ ω i ∼= Af,i [π 0 ] for any i.
Proof. Let N denote the tame conductor of the Galois representation Ā. As f is
p-distinguished, by [5, Thm. 6.4] there exists a finite totally ramified extension K 0of K and a (not necessarily unique) p-stabilized newform f ∈ K 0 [[q]] of tame level
N and weight 2 satisfying the condition of (4). (The final condition in (4) is infact already automatic from the more standard level lowering result [5, Thm. 1.1]
unless Ā is an unramified Gp -module, in which case the Selmer case of [5, Thm.
6.4] ensures the existence of such an f .)v
Since the tame conductor of f equals the conductor of Ā, by Lemma 4.1.2, AG
f,i isdivisible for any place v 6= vp . It then follows from Lemma 4.2.2 that the structure
S(Af,i ) on Af,i [π 0 ] ∼
= Ā agrees with Smin , as desired.
The next proposition, which follows from a result of Greenberg, is crucial to our
method.
Proposition 4.2.4. For f be as above there is an exact sequence
0 → SelS(f,i) (Q∞ , Ā ⊗ ω i ) → H 1 (QΣ /Q∞ , Ā ⊗ ω i ) →Y
1
Hs,S(f,i)(Q∞,v , Ā ⊗ ω i ) → 0v∈ΣVARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

39

for any finite set of places Σ containing vp , all archimedean places and all places
dividing the tame level of f .
Proof. Since Af,i is odd and Sel(Q∞ , Af,i ) is ΛO -cotorsion, by [15, Prop. 2.1] there
is an exact sequenceY
0 → Sel(Q∞ , Af,i ) → H 1 (QΣ /Q∞ , Af,i ) →Hs1 (Q∞,v , Af,i ) → 0v∈Σwith Σ as above. As Sel(Q∞ , Af,i ) is O-divisible, the π-torsion of this sequence isan exact sequenceY
0 → Sel(Q∞ , Af,i )[π] → H 1 (QΣ /Q∞ , Af,i )[π] →Hs1 (Q∞,v , Af,i )[π] → 0.
v∈ΣBy Lemma 4.2.2 and the definition of the induced structure, we see that this sequence identifies with that of the proposition. (Note that (4.5) shows that the local

conditions agree at the place vp .)Our main algebraic theorems are consequences of the next result.
Corollary 4.2.5. Let Ā be as above and let f be a p-stabilized newform of tamelevel Nf such that Af [π] ∼= Ā as in (4). Then the sequence
Y
v
(4.6)
0 → SelSmin (Q∞ , Ā ⊗ ω i ) → Sel(Q∞ , Af,i )[π] →AG
f,i /π → 0
v|Nf

is exact for any i.
Proof. The exactness of (4.6) follows from the definitions and Lemma 4.2.2 except
for the surjectivity of the last map. For this, by Proposition 4.2.3 there exists a
totally ramified extension K 0 /K and a p-stabilized newform f0 over K 0 such that(Ā ⊗ ω i , Smin ) ∼= (Af0 ,i [π 0 ], S(f0 , i)). The surjectivity is then a formal consequenceof the exact sequence of Proposition 4.2.4 applied to both Af0 ,i and Af,i with
Σ = {v | Nf p}.
4.3. Algebraic Iwasawa invariants. Let Ā be as in the previous section. We
write ρ̄ : GQ → GL2 (k) for the corresponding Galois representation. We now use
Corollary 4.2.5 to study the relations between Selmer groups of newforms in the
Hida family of ρ̄. We first consider the behavior of the corresponding finite/singularstructures on Ā.
Lemma 4.3.1. Let f be a newform on the branch T(a) of the Hida family of ρ̄.
1(Q∞,v , Ā ⊗ ω i ) depends only on the
Then for any place v, the dimension of Hf,S(f,i)
branch T(a) and i.
Remark 4.3.2. It was pointed out to the authors by Ralph Greenberg that although the dimension of the local condition is constant along branches of Hida
families, it is possible (when the branch is special at v) that the actual subspace
1
Hf,S(f,i)(Q∞,v , Ā ⊗ ω i ) varies as f varies along the branch T(a). It would beinteresting to determine if such an example actually exists.
Proof. Fix a place v 6= vp dividing the rational prime `. Consider first the unramified G` -representation (Vf )I` . By the proof of Proposition 2.2.3 the dimension ofthis space is constant on the branch T(a). It follows that, as a G` -representation,
the residual representation of (Vf )I` is independent of f on T(a).

40

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

Consider now the exact sequence
(4.7)

1
i
1
v
0 → AGf,i /π → H (Q∞,v , Ā ⊗ ω ) → H (Q∞,v , Af,i )[π] → 0.

By [15, Prop. 2.4], H 1 (Q∞,v , Af,i )[π] has k-dimension equal to the multiplicity ofω 1−i in the residual representation of the unramified G` -representation (Vf )I` . By
the above discussion this is independent of f on the branch T(a), as claimed. 
For a branch T(a) of the Hida family of ρ̄ and a place v 6= vp , we may now define
1v
δv (a, ω i ) := dimk Hf,S(f,i)(Q∞,v , Ā ⊗ ω i ) = dimk AG
f,i /πfor any form f lying on the branch T(a).
We say thatµalg (ρ̄, ω i ) = 0
if SelSmin (Q∞ , Ā ⊗ ω i ) is finite dimensional over k; if this is the case we define
λalg (ρ̄, ω i ) = dimk SelSmin (Q∞ , Ā ⊗ ω i ).
Theorem 4.3.3. Let ρ̄ be as above. Let f be any newform in the Hida family ofρ̄. Then µalg (ρ̄, ω i ) = 0 if and only if µalg (f, ω i ) = 0.
Proof. By Theorem 4.1.1, µalg (f, ω i ) vanishes if and only if Sel(Q∞ , Af,i )[π] is finite
dimensional. By Corollary 4.2.5 this is equivalent to the finite dimensionality of
SelSmin (Q∞ , Ā ⊗ ω i ), as claimed.

Theorem 4.3.4. Let ρ̄ be as above. Assume that µalg (ρ̄, ω i ) = 0.
(1) Let f be a newform of level Nf lying on the branch T(a) of the Hida familyof ρ̄. ThenX
λalg (f, ω i ) = λalg (ρ̄, ω i ) +δv (a, ω i ).
v|Nf
alg

i

In particular, λ (f, ω ) depends only on the branch T(a) of f ; we write
λalg (a, ω i ) for this value.
(2) Let T(a1 ) and T(a2 ) be two branches of the Hida family of ρ̄. ThenX
δv (a1 , ω i ) − δv (a2 , ω i ).
λalg (a1 , ω i ) − λalg (a2 , ω i ) =v6=vp

Proof. The first statement is immediate from Corollary 4.2.5 and the definition of
δv (a, ω i ). The second statement follows from the first.

We remark that in Section 5 we will see that the formulas of the preceding
theorem compare well with the formulas of Theorem 3.7.7.
5. Applications to the main conjecture and examples
5.1. The main conjecture. Let O be the ring of integers of a finite extension Kof Qp and let f ∈ O[[q]] be a p-ordinary eigenform such that the residual representai
tion ρ̄f is irreducible. Let Lalgp (f, ω ) ∈ ΛO denote a generator of the characteristic
power series of the ΛO -dual of the Selmer group Sel(Q∞ , Af,i ) of Section 4.1. LetiiLanp (f, ω ) ∈ ΛO denote the usual p-adic L-function of f ⊗ ω (computed with respect to some canonical period). The main conjecture of Iwasawa theory in thiscontext is the following; it is independent of the particular choice of coefficient field
K over which f is defined.

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

41

Conjecture 5.1.1. There is a unit u ∈ Λ×
O such that
i
an
i
Lalg
p (f, ω ) · u = Lp (f, ω ).

We remark that the Selmer group in the above conjecture is not the Bloch–Kato
Selmer group in which the local condition at p is defined via crystalline periods, butinstead is the Greenberg Selmer group in which the local condition at p is defined
via the ordinary filtration as in Section 4.1. The former group is always containedin the latter group and the quotient is trivial unless the analytic p-adic L-function
has a trivial zero, in which case it has corank one (see [10, pp. 108-109]). It is forthis reason that in our statement of the main conjecture we do not need to consider
separately the case of trivial zeroes.
The following deep theorem of Kato [20] establishes one divisibility of the main
conjecture.
Theorem 5.1.2 (Kato). There is a u ∈ ΛO ⊗ Qp such that
i
an
i
Lalg
p (f, ω ) · u = Lp (f, ω ).

In particular, to verify the main conjecture for f ⊗ ω i it suffices to check thatµalg (f, ω i ) = µan (f, ω i ) and λalg (f, ω i ) = λan (f, ω i ).
Our results in Sections 3 and 4 yield the following result.
Theorem 5.1.3. Let k be a finite field of characteristic p and let ρ̄ : GQ → GL2 (k)be an irreducible, modular, p-ordinary and p-distinguished representation; as always
we fix a choice of p-stabilization. Suppose thatµalg (f0 , ω i ) = µan (f0 , ω i ) = 0 and λalg (f0 , ω i ) = λan (f0 , ω i )for some f0 in the Hida family attached to ρ̄ and some i. Thenµalg (f, ω i ) = µan (f, ω i ) = 0 and λalg (f, ω i ) = λan (f, ω i )for every f in the Hida family attached to ρ̄.
Before giving a proof, we first state an immediate corollary of Theorem 5.1.3 andKato’s result.
Corollary 5.1.4. Let ρ̄ be as above and suppose that µalg (ρ̄, ω i ) = µan (ρ̄, ω i ) = 0
for some i. If the main conjecture holds for f0 ⊗ ω i for one form f0 in the Hida
family of ρ̄, then the main conjecture holds for f ⊗ ω i for every form f in the Hida
family of ρ̄.
Since every Hida family contains a form of weight two, this corollary in particularreduces the main conjecture to the case of weight two and to the conjecture on thevanishing of the µ-invariants.
The proof of Theorem 5.1.3 is based on the following lemma which relates theinvariants e` (a, ω i ) and δv (a, ω i ) of Theorem 3.7.7 and Section 4.3.
Lemma 5.1.5. Let a1 and a2 be minimal primes of TΣ . For any prime ` 6= p
Xδv (a1 , ω i ) − δv (a2 , ω i ) = e` (a2 , ω i ) − e` (a1 , ω i )v|`

where the sum is taken over all primes v of Q∞ over `.

42

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

Proof. To prove the lemma it suffices to see that for any minimal prime a of TΣ ,the sumXe` (a, ω i ) +δv (a, ω i )v|`

is independent of a. For this, fix a classical newform f on the branch T(a). Consider
first the group H 1 (Q∞,v , Af,i ). Since H 1 (Q∞,v , Af,i ) is divisible (as Gv has pcohomological dimension one) we havedimk H 1 (Q∞,v , Af,i )[π] = λ charΛO (H 1 (Q∞,v , Af,i )∨ )where M ∨ = Hom(M, Qp /Zp ).
By [15, Proposition 2.4] we havecharΛO ⊕v|` H 1 (Q∞,v , Af,i )∨ = E` (f, h`i−1p,i ) · ΛO .
−1
Since E` (f, h`i−1p,i ) is simply E` (a, h`ip,i ) mod ℘f , we conclude thatX
(5.1)
dimk H 1 (Q∞,v , Af,i )[π] = e` (a, ω i ).
v|`

Consider now the exact sequence
1i1v
0 → AGf,i /π → H (Q∞,v , Ā ⊗ ω ) → H (Q∞,v , Af,i )[π] → 0.

Since the first term has k-dimension δv (a, ω i ) and the second term is certainly
independent of a, the lemma now follows from (5.1).
Proof of Theorem 5.1.3. Let f be a form in the Hida family of ρ̄. The vanishing of
the µ-invariants of f is immediate from that for f0 and Theorems 4.3.3 and 3.7.5.
By Lemma 5.1.5 and Theorems 4.3.4 and 3.7.7 we see also that
λalg (f, ω i ) − λalg (f0 , ω i ) = λan (f, ω i ) − λan (f0 , ω i ).
Since we are assuming the main conjecture for f0 ⊗ ω i , we have that
λalg (f0 , ω i ) = λan (f0 , ω i ),so that it follows that
λalg (f, ω i ) = λan (f, ω i )as desired.



5.2. Raising the level. We conclude our general discussion with some results on
branches of Hida families. It is well-known (see [8, 9]) which levels can occur among
forms in the Hida family of ρ̄. In the next proposition we list the cases in which
the invariant δv (·, ω i ) increases.
Proposition 5.2.1. Let ρ̄ : GQ → GL2 (k) be an irreducible, p-ordinary and pdistinguished modular Galois representation; assume further that ρ̄ is ramified at p.
Let Σ be some finite set of primes not containing p and let ` 6= p be a prime at whichρ̄ is unramified. Set Σ0 = Σ ∪ {`} and let TΣ , TΣ0 be the Hida algebras associatedto ρ̄. Let a` = Trace ρ̄(Frob` ) and c` = det ρ̄(Frob` ), both viewed as elements of k.
Then for every branch T(a) of TΣ :(1) If there is an i such that a` = `−i + `1−i and c` = `1−2i , then there is abranch T(b) of TΣ0 such that
N (b) = N (a)` and δv (b, ω i ) = 1
for all places v dividing `.

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

43(2) If ` ≡ 1 (mod p) and a` = c` + 1, then there is a branch T(b) of TΣ0(distinct from the branch provided by (1)) such that
N (b) = N (a)` and δv (b, ω i ) = 1
for all places v dividing ` and all i.
(3) If ` ≡ 1 (mod p), a` = 2 and c` = 1, then there are distinct branches T(b1 )and T(b2 ) of TΣ0 such thatN (b1 ) = N (b2 ) = N (a)`2 and δv (b1 , ω i ) = δv (b2 , ω i ) = 2
for all places v dividing ` and all i.
(4) If ` ≡ −1 (mod p), a` = 0 and c` = −1, then there is a branch T(b) ofTΣ0 such that
N (b) = N (a)`2 and δv (b, ω i ) = 1
for all places v dividing ` and all i.
Note in particular that if (3) holds, then (1) and (2) also hold, so that the
proposition exhibits four distinct non-minimal branches of T0Σ for each branch ofTΣ .
Proof. If (1) holds, then one sees easily that the eigenvalues of Frob` on ρ̄ equal`1−i and `−i . It follows that there exists a ramified representationτ` : G` → GL2 (Zp )such that τ̄` ⊗ k ∼= ρ̄|G` and

τ` =

ω 1−i
0

∗
ω −i


.

By [9, Theorem 1] (which applies since ρ̄ is modular of weight 2 by [5, Theorem1.1]) there exists a newform g of tame level N ` in the Hida family of ρ̄ such that
ρg |I` ∼= τ` |I` . Since ρ̄ ⊗ ω i is unramified at ` with Frobenius eigenvalues ` and 1,
one computes easily that
δv (Ag , ω i ) = 1
for any place v dividing `. In particular, the branch a1 on which g lies satisfies therequirements of the proposition.
The proofs for (2) (principal series), (3) (either special or principal series) and(4) (supercuspidal) are entirely similar; the conditions on `, a` and c` are simply
those obtained by combining the conditions of [8, p. 435] for ramified lifts of theappropriate type to exist with the requirement that ρ̄(Frob` ) has a trivial eigenvalue.
Note also that ω|G` is trivial (resp. quadratic) in (2) and (3) (resp. (4)), so thatthe choice of i is irrelevant. Finally, the branches in (1) and (2) (resp. in (3)) are
distinct since one is special at ` and the other is principal series at `.
It would be interesting to determine when the branches of the preceding proposition intersect. The following proposition uses Λ-adic level raising to give someinsight into this question.
Proposition 5.2.2. We maintain the hypotheses of the previous proposition. Leta be a minimal prime of TΣ such that(T` mod a0 )2 − `−2 h`iN (a)p (` + 1)2

44

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

is not a unit in T(a)◦ . (Recall that a0 is a minimal prime of Tnew
N (a) sitting over a
0
/a
.)
Then
there
existssomeminimalprime
b of TΣ0 withand that T(a)◦ ∼= Tnew
N (a)N (b) = N (a)` and a height one prime ℘ of TΣ0 such that T(a) and T(b) cross at℘. (That is, both b and the pre-image of a under the natural map TΣ0 → TΣ arecontained in ℘.)Proof. Attached to the minimal prime a we have the Λ-adic modular formXf (a, q) =(Tn mod a0 )q n .
n≥1

By [4, Theorem
exists a finite extension R of T(a)◦ , a Λ-adic modular
P 6C], therenform g(q) = n≥1 bn q ∈ R[[q]] new of level N (a)`, and a height one prime ℘0 of
R such that
(5.2)

(Tn mod a0 ) ≡ bn(mod ℘0 )for each n relatively prime to N (a)`.
The Λ-adic form g(q) corresponds to some minimal prime ideal of TΣ . Indeed,0
new
there is a map Tnew
N (a)` → R that sends Tn to bn . Let b ⊆ TN (a)` denote the kernelof this map; it is necessarily a minimal prime since
Q both the source and the target
are finite extensions of Λ. The preimage of b0 in M |N (Σ0 ) Tnew
M is a minimal prime
and thus, by Proposition 2.3.2, corresponds to some minimal prime of T0N (Σ0 ) .
Since the residual representation attached to g(q) equals ρ̄, this minimal prime
yields a minimal prime of TΣ0 which we denote by b. Note that by construction
N (b) = N (a)`.
We thus have a map TΣ0 → R (sending Tn to bn ) with kernel b. If let ã denotes
the pre-image of a in TΣ0 , then we also have a map TΣ0 → R with kernel ã (given
by reducing mod ã and then embedding the image T(a) into R). By (5.2), thereduction of these two maps modulo ℘0 are the same. If we let ℘ denote the kernelof either of these maps modulo ℘0 , then ℘ is a height one prime containing both ofb and ã, as desired.

The ideal ℘ of the previous proposition could potentially be a prime ideal lying
over the principal ideal (p). It would be interesting to determine whether or notthese branches actually meet at a prime ideal of residue characteristic zero.
5.3. Examples.
Example 5.3.1. Set p = 11 and let f denote the weight 2 newform associated tothe elliptic curve X0 (11). It follows from the fact that f is the only newform of
weight 2 and level dividing 11 that T := TΣ (ρ̄f ) ∼= Λ for Σ = ∅. Thus for k ≥ 2,
there is a unique newform fk of weight k and level dividing 11 that is congruent tof modulo 11. For example, f12 is the 11-ordinary, 11-stabilized oldform of level 11
attached to the Ramanujan ∆-function.
The p-adic L-function of this family was studied in detail in [14]. Also, in [11],the congruence between X0 (11) and ∆ was exploited to gain information about the
Selmer groups of both forms. We review below what is known in these examples
and then go on to study the other branches of this Hida family with tame conductor
greater than one.
We first verify the main conjecture for f . Since X0 (11) has split multiplicative
reduction at p = 11, the p-adic L-function Lanp (f, T ) has a trivial zero at T = 0.

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

45

By the Mazur, Tate and Teitelbaum conjecture (proved by Greenberg and Stevens[14]) we have an explicit formula for the derivative of Lanp (f, T ) at 0; that is,
Lp (f ) L(f, 1)
d an
Lp (f, T )·.
=dT
logp (γ)ΩfT =0(Here, γ is a topological generator of 1+pZp that is implicitly chosen by writing thep-adic L-function in the T -variable; the appearance of the extra factor of logp (γ)
above is accounted for by the fact that we have written this formula in the T -variable
rather than the more standard s-variable.) One checks that Lp (f ) and logp (γ) areexactly divisible by 11 while L(f, 1)/Ωf is an 11-adic unit. Thus Lanp (f, T ) is a unit
multiple of T so that λan (f ) = 1 and µan (f ) = 0.
Since L(f, 1)/Ωf is an 11-adic unit, by an Euler system argument of Kolyvagin,
the classical 11-adic Selmer group Selp (X0 (11)/Q) of the elliptic curve X0 (11) vanishes. It thus follows from a refined control theorem of Greenberg ([13, Proposition3.7]) that Selp (X0 (11)/Q∞ ) vanishes as well. The latter group agrees with the
Bloch–Kato Selmer group of f ; as f has a trivial zero, it follows that the Greenberg Selmer group Selp (Q∞ , Af ) is simply Qp /Zp with trivial Galois action. (Thisexample is worked out in detail in [11, Example 3].) Thus Lalgp (f, T ) is a unit mulalg
alg
tiple of T , so that λ (f ) = 1 and µ (f ) = 0. In particular, this verifies the main
conjecture for X0 (11) at p = 11.
Corollary 5.1.4 now shows that the main conjecture holds at p = 11 for eachform fk with k ≥ 2, and Theorem 5.1.3 shows thatλan (fk ) = λalg (fk ) = 1 for k ≥ 2.
Note that the p-adic L-function corresponding to X0 (11) is the only p-adic Lfunction in the Hida family with a trivial zero. Nonetheless, the existence of thistrivial zero forces every p-adic L-function in the family to have at least one zero.
For k ≡ 2 (mod 10), the unique zero of Lan
p (fk ) can be explained by the functional equation of the p-adic L-function. Namely, the sign of this functional equation is −1 for such k and thus Lanp (fk ) vanishes at the character that sends x tok−2

x 2 . Using this observation, we can determine the two-variable p-adic L-function
attached to ρ̄f (which we denote by L(ρ̄f )). By Theorem 3.7.7, we have that
λ(L(ρ̄f )) = 1 and hence, under the identification of (3.9),
L(ρ̄f ) = (T − a0 ) · u(T )with a0 ∈ T and u(T ) ∈ T[[T ]] a unit. Since L(ρ̄f ) specializes at weight k to Lp (fk )(Proposition 3.4.3), we know that a0 specializes at weight k to the unique zero ofk−2
Lp (fk ). In particular, for k ≡ 2 (mod 10), we have that a0 specializes to γ 2 .
1Thus, a0 = γ −1 hγip2 since this element has the correct specialization for infinitely
many k.
So far we have only applied our results to minimal lifts of ρ̄f ; however, Corollary 5.1.4 applies to every modular form lifting this representation. For instance,
we can use Proposition 5.2.1 to explicitly find primes ` to add to the level that willproduce modular forms that have higher λ-invariants and for which we still know
the main conjecture.
The first prime that satisfies condition (3) (and thus (1) and (2)) of Proposition
5.2.1 is ` = 1321. If we set Σ = {1321}, then TΣ will have four distinct branches
T(a1 ), . . . , T(a4 ) with λ(a1 ) = λ(a2 ) = 2 and λ(a3 ) = λ(a4 ) = 3. (Here and in

46

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

what follows, we do not distinguish between the analytic and algebraic λ-invariants
in cases where the main conjecture is known to be true.) Note that 1321 is inert inthe cyclotomic Z11 -extension of Q and so there is a unique prime v of Q∞ sitting
over 1321.
The prime `0 = 2113 is also inert in Q∞ /Q and satisfies condition (3) of Proposition 5.2.1. If Σ0 = {1321, 2113}, we may apply Proposition 5.2.1 to each of the fivebranches we have exhibited in TΣ to see that TΣ0 possesses at least 20 additional
branches with λ-invariants ranging between 3 and 5.
Example 5.3.2. We conclude by considering an example of Greenberg and Vatsalin [15]. Consider the elliptic curves
E1 : y 2 = x3 + x − 10

E2 : y 2 = x3 − 584x + 5444,

of conductors 52 and 364 respectively, which are ordinary and congruent mod 5.
One computes that λan (E1 ) = µan (E1 ) = 0 so that Kato’s divisibility (Theorem 5.1.2) yields the main conjecture for E1 at p = 5. The main conjecture for E2
at p = 5 follows and in this case we have λ(E2 ) = 5. (This example is discussed indetail in [15, pp. 22, 44].)
We now examine this congruence from the point of view of Hida theory. Letfi denote the newform of weight 2 associated to Ei . One checks that f1 is notcongruent modulo 5 to any other modular forms of weight two and level dividing
22 · 5 · 132 (using [32], for example). It follows that we have TΣ ∼= Λ for Σ = {2, 13}.
For consistency of notation, let us denote the unique irreducible component of
this space as T(a1 ). Moreover, since µ(f1 ) = λ(f1 ) = 0, the two-variable p-adic
L-function attached to TΣ is a unit and λ(a1 ) = 0.
The prime ` = 7 satisfies condition (1) of Proposition 5.2.1 (with i = 0). Thus,
if Σ0 = {2, 7, 13}, then TΣ0 contains an irreducible component T(a2 ) not containedin TΣ . Moreover, since f2 is the unique normalized newform of weight 2 congruent
modulo 5 to f1 with level dividing 22 · 5 · 72 · 132 , it follows that TΣ0 has rank 2 overΛ. Hence, T(a2 ) is the only branch of TΣ0 not coming from TΣ . By Theorems 3.7.7
and 4.3.4, we have that λ(a2 ) = 5. Since f2 sits on the branch T(a2 ), it follows
that λ(f2 ) = 5.
We close with some questions. Proposition 5.2.2 establishes that the branches
T(a1 ) and T(a2 ) must cross at some (non-classical) height one prime, but does not
exclude the possibility that they cross at the prime (p). Do they in fact cross at aprime of residue characteristic zero and if so could one compute p-adic approximations of this prime? How many such crossing points do these two branches share?
It appears at present that little is known about the shape of these Hida families
when multiple branches appear (even in any particular case).
References[1] Henri Carayol, Sur les représentations l-adiques associées aux formes modulaires de Hilbert,
Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 409–468.
, Formes modulaires et représentations galoisiennes à valeurs dans un anneau local[2]
complet, p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA,
1991), Contemp. Math., vol. 165, Amer. Math. Soc., Providence, RI, 1994, pp. 213–237.
[3] Henri Darmon, Fred Diamond and Richard Taylor, Fermat’s last theorem, Elliptic curves,modular forms & Fermat’s last theorem (Hong Kong, 1993), Internat. Press, Cambridge,
MA, 1997, pp. 2–140.
[4] Fred Diamond, On congruence modules associated to Λ-adic forms, Compositio Math. 71(1989), 49–83.

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

[5]

47

, The refined conjecture of Serre, Elliptic curves, modular forms, & Fermat’s last
theorem (Hong Kong, 1993), Ser. Number Theory, I, Internat. Press, Cambridge, MA, 1995,
pp. 22–37.
[6], On deformation rings and Hecke rings, Ann. of Math. (2) 144 (1996), no. 1, 137–
166.
[7] Fred Diamond and Kenneth A. Ribet, l-adic modular deformations and Wiles’s “mainconjecture”, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, New
York, 1997, pp. 357–371.
[8] Fred Diamond and Richard Taylor, Nonoptimal levels of mod l modular representations,
Invent. Math. 115 (1994), no. 3, 435–462.
[9], Lifting modular mod l representations, Duke Math. J. 74 (1994), no. 2, 253–269.
[10] Ralph Greenberg, Iwasawa theory for p-adic representations, Algebraic number theory, Adv.
Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 97–137.
[11], Iwasawa theory for motives, in L-functions and arithmetic (Durham, 1989), 211–
233, Cambridge Univ. Press, Cambridge, 1991.
[12], Iwasawa theory and p-adic deformations of motives, Motives (Seattle, WA, 1991),
Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 193–223.
[13], Iwasawa theory for elliptic curves, Arithmetic theory of elliptic curves (Cetraro,1997), Lecture Notes in Math., vol. 1716, Springer, Berlin, 1999, pp. 51–144.
[14] Ralph Greenberg and Glenn Stevens, p-adic L-functions and p-adic periods of modular
forms, Invent. Math. 111 (1993), no. 2, 407–447.
[15] Ralph Greenberg and Vinayak Vatsal, On the Iwasawa invariants of elliptic curves, Invent.
Math. 142 (2000), no. 1, 17–63.
[16] Benedict Gross, A tameness criterion for Galois representations associated to modular
forms (mod p), Duke Math. J. 61 (1990), no. 2, 445–517.
[17] Haruzo Hida, Galois representations into GL2 (Zp [[X]]) attached to ordinary cusp forms,
Invent. Math. 85 (1986), no. 3, 545–613.
[18], Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm.
Sup. (4) 19 (1986), no. 2, 231–273.
[19], Modules of congruence of Hecke algebras and L-functions associated with cusp
forms, Amer. J. Math. 110 (1988), no. 2, 323–382.
[20] Kazuya Kato, p-adic Hodge theory and values of zeta functions of modular forms, preprint.
[21] Nicholas Katz, Higher congruences between modular forms, Ann. of Math. (2) 101 (1975),
332–367.
[22] Koji Kitigawa, On standard p-adic L-functions of families of elliptic curves, in p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), 81–110, Contemp. Math., 165, Amer. Math. Soc., Providence, RI, 1994.
[23] Ron Livné, On the conductors of mod l Galois representations coming from modular forms,
J. Number Theory 31 (1989), no. 2, 133–141, Amer. Math. Soc., Providence RI, 1994,
81–110.
[24] Hideyuki Matsumura, Commutative ring theory, second ed., Cambridge Studies in Advanced
Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989, Translated from theJapanese by M. Reid.
[25] Barry Mazur, On the arithmetic of special values of L-functions, Invent. Math. 55 (1979),
no. 3, 207–240.
[26], Euler systems in arithmetic geometry, notes from a course at Harvard University(1998), available at http://www.math.amherst.edu/∼taweston.
[27], The theme of p-adic variation, Mathematics: frontiers and perspectives, Amer.
Math. Soc., Providence, RI, 2000, pp. 433–459.
[28], Two-variable p-adic L-functions, unpublished notes.
[29], An introduction to the deformation theory of Galois representations, in Modular
forms and Fermat’s last theorem (Boston, MA, 1995), 243–311, Springer, New York, 1997.
[30] Tadashi Ochiai, On the two-variable Iwasawa main conjecture for Hida deformations, in
preparation.
[31] Alexei Panchishkin, Two-variable p-adic L-functions attached to eigenfamilies of positive
slope, Inventiones Math. 154, no. 3, 551–615.
[32] William Stein, The modular forms explorer, http://modular.fas.harvard.edu/mfd/mfe/.
[33] Glenn Stevens, Families of overconvergent modular symbols, preprint.

48

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON[34] Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann.
of Math. (2) 141 (1995), no. 3, 553–572.
[35] Vinayak Vatsal, Canonical periods and congruence formulae, Duke Math. J. 98 (1999),
no. 2, 397–419.
[36] Tom Weston, Iwasawa invariants of Galois deformations, preprint.
[37] Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141(1995), no. 3, 443–551.
(Matthew Emerton) Mathematics Department, Northwestern University, Evanston, IL(Robert Pollack) Department of Mathematics, Boston University, Boston, MA(Tom Weston) Department of Mathematics and Statistics, University of Massachusetts,
Amherst, MA
E-mail address, Matthew Emerton: emerton@math.northwestern.edu
E-mail address, Robert Pollack: rpollack@math.bu.edu
E-mail address, Tom Weston: weston@math.umass.edu