Pure and Applied Mathematics Quarterly
Volume 5, Number 1(Special Issue: In honor ofJean-Pierre Serre, Part 2 of 2 )375—434, 2009

On the p-adic Beilinson Conjecture for Number Fields
A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot
Dedicated to Jean-Pierre Serre on the occasion of his eightieth birthday.
Abstract: We formulate a conjectural p-adic analogue of Borel’s theorem
relating regulators for higher K-groups of number fields to special valuesof the corresponding ζ-functions, using syntomic regulators and p-adic Lfunctions. We also formulate a corresponding conjecture for Artin motives,and state a conjecture about the precise relation between the p-adic andclassical situations. Parts of the conjectures are proved when the number
field (or Artin motive) is Abelian over the rationals, and all conjectures areverified numerically in various other cases.
Keywords: Beilinson conjecture, Borel’s theorem, number field, Artin motive, syntomic regulator, p-adic L-function
1. Introduction
The Beilinson conjectures about special values of L-functions [2] are a far reaching generalization of the class number formula for the Dedekind zeta function.
For every smooth, projective algebraic variety X over the rationals it predicts theleading term of the Taylor expansion of L(H i (X), s) at certain points, up to arational multiple, in terms of arithmetic information associated with X, namely,
its algebraic K-groups Kj (X) [42]. More generally, these conjectures can also be
formulated for motives.
Received July 25, 2007.
2000 Mathematics Subject Classification. Primary: 19F27; secondary: 11G55, 11R42, 11R70

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A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot

There have been several important steps taken towards verification of these
conjectures in various cases, although, strictly speaking they have only beenverified completely in the case where X is the spectrum of a number field, where
they follow from famous theorems of Borel [10, 11].
To motivate what follows, let us briefly recall the conjecture that interests us
the most (for an introduction see [23, 45]). One associates with X two cohomologyi (X , R(n)), which is angroups. The first one is the Deligne cohomology HD/R
i (X , Q(n)),
R-vector space. The second is “integral” motivic cohomology HM/Z
which may be defined as a certain subspace of K2n−i (X)⊗Z Q. There is a regulator
map defined by Beilinson,
(1.1)

i
i
HM
(X/Z , Q(n)) → HD
(X/R , R(n)) .

i (X , R(n)) has a rational structure coming from the reIf 2n > i + 1 then det HD/Rilations between HD (XR , R(n)) and the de Rham and singular cohomology groupsof X [45, p.30].

The first part of the conjecture is that the map in (1.1) induces an isomorphism between the left-hand side tensored with R and the right-hand side, andi (X , R(n)). The
consequently provides a second rational structure on det HD/Rsecond part of the conjecture states that, assuming a suitable functional equation for L(H i−1 (X), s), these two rational structures differ from each other by
the leading term in the Taylor expansion of L(H i−1 (X), s) at s = i − n. Becauseof the expected functional equation one can reformulate the conjecture in termsof L(H i−1 (X), n) (see [2, Corollary 3.6.2] or [31, 4.12]).
As mentioned before, this conjecture has only been verified in the case of
number fields, due to difficulties in the computation of motivic cohomology. What
has been verified in several other cases is a form of the conjecture in whichi (X , R(n)) elements
one assumes the first part. For this one finds dimR HD/R
i (X , Q(n)), checks that their images under (1.1) are independent, henceof HM/Zi (X , Q(n)) according to the first part of the conjecture,should form a basis of HM/Z
and verifies the second part using these elements.
The idea that there should be a p-adic analogue of Beilinson’s conjectures has
been around since the late 80’s. Such a conjecture was formulated and proved by
Gros in [29, 30] in the case of Artin motives associated with Dirichlet characters.
In the weak sense mentioned before, it was proved for certain CM elliptic curves

On the p-adic Beilinson Conjecture for Number Fields

377

in [17] (where the relation with the syntomic regulator is proved in [4] and further
elucidated in [7]), and for elliptic modular forms it follows from Kato’s work(see [46]).
The book [41] contains a very general conjecture about the existence and properties of p-adic L-functions, from which one can derive a p-adic Beilinson conjecture. Rather than explain this in full detail we shall give a sketch of thisconjecture similar to the sketch above of the Beilinson conjecture.
For the p-adic Beilinson conjectures one has to replace Deligne cohomology
with syntomic cohomology [30, 40, 3], the Beilinson regulator with the syntomic
regulator, and L-functions with p-adic L-functions.
i (Y, n) is defined for a smooth scheme Y of finite
Syntomic cohomology Hsyn
type over a complete discrete valuation ring of mixed characteristic (0, p) withperfect residue field. For the Q-variety X we obtain, for all but finitely many
primes, a map

(1.2)

i
i
HM
(X/Z , Q(n)) → Hsyn
(Y, n)

where Y is a smooth model for X over Zp . This cohomology group is a Qp -vector
space.
The theory of p-adic L-functions starts with Kubota and Leopoldt’s p-adic
ζ-function [35], ζp (s), which is defined by interpolating special values of complex
valued Dirichlet L-functions. This principle has been extended to ζ- and Lfunctions in various situations, resulting in corresponding p-adic functions for
totally real number fields [1, 15, 22], CM fields [32, 33] and modular forms [37].
(Given the occasion, let us note that the approach of Deligne and Ribet using
modular forms was initiated by Serre [47].)
The p-adic Beilinson conjecture therefore has many similarities with its complex counterpart. However, there is a very important difference. In general, when
2n > i + 1, there is no hope that (1.2) induces an isomorphism after tensoring theleft-hand side with Qp . To see this, consider a number field k with ring of algebraic integers Ok . By Borel’s theorem (see Theorem 2.2) we have, in accordancewith Beilinson’s conjectures,(
r2 when n ≥ 2 is even1
dimQ HM (Spec(k)/Z , Q(n)) =r1 + r2 when n ≥ 2 is odd

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A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot

with r1 (resp. 2r2 ) the number of real (resp. complex) embeddings of k. Thus,
motivic cohomology “knows” about the number of real and complex embeddingsof k, as does, by its definition, Deligne cohomology, which in this case becomes
r1 +2r2 such that x = x }. But
1 (Spec(k) , R(n)) ∼ {(x )HD=σ σ:k→C in R(n − 1)σσ/Rsyntomic cohomology, which depends only on the completion of k at p, does not.
1 (Spec(O ⊗ Z ), n) ∼ Qr1 +2r2 .
In this case, we obtain Hsyn
= p
k Z p
The solution to this problem suggested in [41] is to make the p-adic L-function
depend on a subspace of syntomic cohomology which is complementary to theimage of the regulator. While this might seem artificial, there are other reasonsfor choosing this solution. In most cases one chooses a particular subspace andobtains a special case of the conjecture.
In the important special case of a totally real number field, or, more generally,an Artin motive over Q associated with a Galois representation where the conjugacy class of complex conjugation acts trivially (let us call these totally real
Artin motives), no such subspace is required when n ≥ 2 is odd (see Proposition 3.12). The same holds for Artin motives where this conjugacy class acts as
multiplication by −1 and n ≥ 2 is even (we may think of those as the “negativepart” of CM Artin motives).
If χ denotes the character associated with either representation, then the
Beilinson conjecture relates the regulator of K2n−1 with the Artin L-functionof χ at n. For the p-adic L-functions one has to consider χ ⊗ ωp1−n with ωp the
Teichmüller character for the prime number p. Then the fixed field of the kernelof the representation underlying χ ⊗ ωp1−n is totally real, and it is perhaps nocoincidence that in precisely this case the existence of a p-adic L-function that is
not identicaly zero has been established, by [1, 15, 22] for the case of fields andby [28] for Artin motives.
The goal of the present work is to describe in detail the conjectures for thecases of totally real fields, totally real Artin motives, as well as the “negativepart” of CM Artin motives, and describe the (conjectural) relation between theclassical and p-adic conjectures. We test everything numerically in several cases,
and also deduce most of the conjectures for Abelian Artin motives from work by
Coleman [16].
There have been several developments that allow us to carry out this numerical verification. In [19] de Jeu proved part of Zagier’s conjecture concerning a

On the p-adic Beilinson Conjecture for Number Fields

379

more explicit description of the K-theory (tensored with Q) of number fields (seeSection 4). While this conjecture is not known to give the K-theory of such fields
in all cases, it does in practice. Thus it provides a way of computing them, andPaul Buckingham wrote a computer implementation for this. In [5] Besser and de
Jeu computed the syntomic regulator for (essentially) the part of the K-theoryof a number field described by Zagier’s conjecture and showed that it is given by
applying the p-adic polylogarithm. Those p-adic polylogarithms were invented
by Coleman [16] using his theory of p-adic integration but are not so easy tocompute. In [6] Besser and de Jeu devised an algorithm for this computation.
Taken together, these developments allow us to compute (1.1) and (1.2) for number fields. Finally, building on earlier work in [44], Roblot has dealt with thecomputational aspects of computing p-adic L-functions for Abelian charactersover Q or a real quadratic field [43].
This paper is organized as follows. In Section 2 we recall Borel’s theorem aswell as various facts about L-functions and p-adic L-functions, and formulate aconjectural p-adic analogue of Borel’s theorem. In Section 3 we introduce Artin
motives with coefficients in a number field E in terms of representations of the
Galois group, determine when the Q-dimension of the left-hand side of (1.2)
equals the Qp -dimension of the right-hand side (corresponding to equality in
Proposition 3.12), define both classical and p-adic L-functions with coefficientsin E, and formulate the motivic Beilinson conjecture with coefficients in E, Conjecture 3.18, a small part of which is the same as a conjecture by Gross. In
Section 4 we describe the set-up for finding elements in the K-groups of number fields using Zagier’s conjecture, and the classical and p-adic regulators on
them. We also prove most of Conjecture 3.18 for Abelian Artin motives over Q(Proposition 4.17 and Remark 4.18). In Section 5 we discuss a few computational
aspects of implementing Zagier’s conjecture and describe the Artin motives that
we consider later for the numerical examples, and in the process prove Gross’s
conjecture for Artin motives obtained from S3 and D8 -extensions of Q. Then in
Section 6 we sketch how the p-adic L-functions can be computed in certain cases,
and make the required Brauer induction explicit for the Artin motives we wantto consider. Finally, the last section is devoted to the results of the numericalcalculations for examples.
Amnon Besser, Paul Buckingham and Rob de Jeu would like to thank theEC network Arithmetic Algebraic Geometry for travel support. Rob de Jeu

380

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot

would like to thank the Tata Institute of Fundamental Research for a productive
stay during which this paper was worked on, and the Nuffield Foundation fora grant under the Undergraduate Research Bursary programme (NUF-URB03)
that enabled Paul Buckingham to develop a computer program for finding thenecessary elements in K-theory. Amnon Besser would like to thank the Institutefor Advanced Study in Princeton for a stay during which part of the paper wasworked on, and the Bell companies Fellowship and the James D. Wolfensohn
fund for financial support while at the institute. Finally, the authors would liketo thank Alfred Weiss for very useful explanations about p-adic L-functions, and,in particular, for bringing to their attention the paper [28].
Notation
Throughout the paper, for an Abelian group A we let AQ denote A⊗Z Q. If B is a
subgroup of C and n an integer then we let B(n) = (2πi)n B ⊆ C. We normalize
the absolute value | · |p on the field of p-adic numbers Qp by |p|p = p−1 , and use
the same notation for its respective extensions to an algebraic closure Qp of Qp
ˆ of Q .
and a completion C = Qppp

2. The p-adic Beilinson conjecture for totally real fields
Let k be a number field with r1 real embeddings, 2r2 complex embeddings,
ring of algebraic integers Ok , and discriminant Dk . As is well-known, Ok∗ is a
finitely generated Abelian group of rank r = r1 + r2 − 1, and its regulator Rpsatisfies w |Dk | Ress=1 ζk (s) = 2r1 (2π)r2 |Cl(Ok )| R, with Cl(Ok ) the class groupof Ok , and w the number of roots of unity in k. Because K0 (Ok ) ∼= Cl(Ok ) ⊕ Z∗∼
and K1 (Ok ) = Ok , so |Cl(Ok )| = |K0 (Ok )tor | and w = |K1 (Ok )tor |, this can beinterpreted as a statement about the K-theory of Ok , and it is from this pointof view that it can be generalized to ζk (n) for n ≥ 2. Namely, in [42], Quillen
proved that Km (Ok ) is a finitely generated Abelian group for all m. Borel in[10] computed its rank when m ≥ 2. For m even this rank is zero, but for odd
m it is r1 + r2 or r2 , and in [11] he showed that a suitably defined regulator of∼
K2n−1 (Ok ) is related to ζk (n). Since K2n−1 (Ok ) → K2n−1 (k) when n ≥ 2 we
can rephrase his results for K2n−1 (Ok ) in terms of K2n−1 (k). Also, we replace
Borel’s regulator map regB : K2n−1 (C) → R(n − 1) (n ≥ 2) with Beilinson’s
regulator map reg∞ (see [45, § 4]), which is half the Borel map regulator by [12,Theorem 10.9]. Because k ⊗Q C ∼= ⊕σ:k→C C, and n > dim Spec(k ⊗ C), we obtain

On the p-adic Beilinson Conjecture for Number Fields

381

by [45, p.9]
(2.1)

1
HD
(Spec(k ⊗ C)/R , R(n)) ∼= H 0 (Spec(k ⊗ C)/R , R(n − 1))³ M´+∼
R(n − 1) ,=σ:k→C

which consists of those (xσ )σ with xσ = xσ . Finally, for any embedding σ : k → C
we let σ∗ : K2n−1 (k) → K2n−1 (C) be the induced map.
Theorem 2.2. (Borel) Let k be a number field of degree d, with r1 real embeddings and 2r2 complex embeddings, and let n ≥ 2. Then the rank mn of K2n−1 (k)equals r2 if n is even and r1 + r2 if n is odd. Moreover, the map
M
K2n−1 (k) →R(n − 1)
σ:k→C
(2.3)
α 7→ (reg∞ ◦ σ∗ (α))σ
embeds K2n−1 (k)/torsion as a lattice in (⊕σ R(n − 1))+ ∼= Rmn , and the volumeVn (k) of a fundamental domain of this lattice satisfiesp(2.4)
ζk (n) |Dk | = qπ n(d−mn ) Vn (k)for some q in Q∗ .
Remark 2.5. In Theorem 2.2 mn = 0 precisely when k is totally real and n ≥ 2 isodd. In this case the given relation holds (with Vn (k) = 1) by the Siegel-Klingen
theorem [39, Chapter VII, Corollary 9.9].
This theorem is equivalent with Beilinson’s conjecture for k. In order to deal
with this in detail (see Remarks 2.20 and 3.24) and in order to introduce p-adic
L-functions we recall some facts about Artin L-functions [39, Chapter VII, § 1012]. Let k be a number field, d = [k : Q], and let χ be a C-valued Artin characterof Gal(k/k). For a prime number ` we defineY
Eul` (s, χ, k) =Eull (s, χ, k)l|`where Eull (s, χ, k) is the reciprocal of the Euler factor for l and the product isover primes l of k lying above `. Then for s ∈ C with Re(s) > 1 we can write theArtin L-function of χ asY
L(s, χ, k) =Eul` (s, χ, k)−1 .
`382

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot

For an infinite place v of k we let(
LC (s)χ(1)
Lv (s, χ, k) =+−LR (s)n LR (s + 1)nwhen v is complex,
when v is real,P
where LR (s) = π −s/2 Γ(s/2), LC (s) = 2(2π)−s Γ(s) and n± = d2 χ(1)± v 12 χ(Frw ),with the sum taken over the real places v of k and Frw the generator of the imageof Gal(C/R) in Gal(k/k) corresponding to any extension w : k → C of v. ThenY
Λ(s, χ, k) = L(s, χ, k)
Lv (s, χ, k)v|∞

extends to a meromorphic function that satisfies the functional equation (cf. [39,p. 541])Λ(s, χ, k) = W (χ)C(χ, k)1/2 C(χ, k)−s Λ(1 − s, χ, k)with W (χ) a constant of absolute value 1 and C(χ, k) = |Dk |χ(1) Nmk/Q (f(χ, k))for f(χ, k) the Artin conductor of χ. Therefore
1

(2.6)

L(1 − s, χ, k) = W (χ)C(χ, k)s− 2 (2(2π)−s Γ(s))dχ(1)+−× (cos(πs/2))n (sin(πs/2))n L(s, χ, k) .

Following [28, pp. 80–81] we shall now describe a p-adic L-function Lp (s, χ, k)when k is a totally real number field, p a prime and χ : Gal(k/k) → Qp a suitable
Artin character. We begin with the case of 1-dimensional Artin characters.
If σ : Qp → C is any isomorphism then σ ◦ χ is a complex Artin character, so
we have the Artin L-function L(s, σ ◦ χ, k). We may also view χ as a character ofa suitable ray class group, so by [39, Corollary 9.9 and page 509] all L(m, σ ◦ χ, k)
for m ∈ Z≤0 are in Q(σ ◦ χ) and the values(2.7)

L∗ (m, χ, k) = σ −1 (L(m, σ ◦ χ, k))

are independent of the choice of σ.
In the same way, we define(2.8)

Eul∗` (m, χ, k) = σ −1 (Eul` (m, σ ◦ χ, k)) ,which is clearly independent of the choice of σ. To construct the p-adic L-function
one finds a p-adic analytic or meromorphic function on an open ball around 0 thatinterpolates the values L∗ (m, χ, k). For 1-dimensional χ with the fixed field kχ ofits kernel totally real this was achieved independently by Deligne and Ribet [22],Barsky [1], and Cassou-Noguès [14, 15]. We shall sketch a proof of the followingOn the p-adic Beilinson Conjecture for Number Fields

383

theorem and Remark 2.13 below in Section 6. (When kχ is not totally real thep-adic L-function is identically zero since the values interpolated are all zero by
the functional equation of the L-function.)
Theorem 2.9. For p prime, let B in Cp be the open ball with centre 0 and radius
qp−1/(p−1) where q = p if p > 2 and q = 4 if p = 2. If k is a totally real number
field and χ : Gal(k/k) → Qp a 1-dimensional Artin character, then there exists a
unique Cp -valued function Lp (s, χ, k) on B satisfying the following properties:(1) Lp (s, χ, k) is analytic if χ is non-trivial and meromorphic with at most asimple pole at s = 1 if χ is trival;(2) if m is a negative integer such that m ≡ 1 modulo ϕ(q) thenLp (m, χ, k) = Eul∗p (m, χ, k)L∗ (m, χ, k) .
If χ : Gal(k/k) → Qp is any Artin character then by Brauer’s induction theorem [39, (10.3)] there exist 1-dimensional Artin characters χ1 , . . . , χt on subgroups
G1 , . . . , Gt of Gal(k/k) of finite index, and integers a1 , . . . , at , such that
(2.10)χ=t
X

Gal(k/k)ai IndGi(χi ) .

i=1

If kχ is totally real then we can assume that the same holds for the fixed fields
ki of the Gi , and we define the p-adic L-function of χ by
(2.11)Lp (s, χ, k) =tY

Lp (s, χi , ki )ai ,i=1

which is a meromorphic function on B (see Section 6). If m is a negative integer
satisfying m ≡ 1 modulo ϕ(q) then the value Lp (m, χ, k) is defined and equals
Eul∗p (m, χ, k)L∗ (m, χ, k) by well-known properties of Artin L-functions (see [39,Prop. 10.4(iv)]), showing the function is independent of how we express χ as asum of induced 1-dimensional characters.
Remark 2.12. In [28] Greenberg proves that the Main Conjecture of Iwasawa
theory implies the p-adic Artin conjecture, that is, that the p-adic L-function of
an Artin character χ is analytic on the open ball B if it does not contain the trivial
character, and has at most a simple pole at s = 1 otherwise. It therefore follows
from the proof of the Main Conjecture for p odd by Wiles [53] that Lp (n, χ, k)384

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot

for p 6= 2 is defined for all integers n 6= 1. In particular, the values of the padic L-functions in Conjecture 2.17 below exist by Theorem 2.9, and those inConjecture 3.18 exist when p 6= 2 but have to be assumed to exist when p = 2.
Remark 2.13. Let Wp ⊂ Z∗p be the subgroup of (p − 1)-th roots of unity if p
is odd and let W2 = {±1}. The Teichmüller character on Gal(k/k) is defined as
the composition(2.14)∼∼

ωp : Gal(k/k) → Gal(Q/Q) → Gal(Q(µq )/Q) → (Z/qZ)∗ → Wp ,where the last map sends an element of (Z/qZ)∗ to the unique element of Wp towhich it is congruent modulo q. For an Artin character χ : Gal(k/k) → Qp andan integer l, χωpl is also an Artin character. If m ≤ 0 satisfies l + m ≡ 1 modulo
ϕ(q) and either χ is 1-dimensional or kχ is totally real then Lp (m, χωpl , k) =Eul∗p (m, χ, k)L∗ (m, χ, k).
The p-adic Beilinson conjecture is going to predict the special values of p-adic
L-functions in terms of a p-adic regulator. The required regulator is the syntomic
regulator [3, Theorem 7.5]. Let F be a complete discretely valued field of characteristic 0 with perfect residue field of characteristic p and let X be a scheme that issmooth and of finite type over the valuation ring OF . Then the above mentioned
i (X, n), as well as synpaper associates to X its rigid syntomic cohomologies Hsyni (X, n). In thistomic regulators (i.e., Chern characters) regp : K2n−i (X) → Hsynwork, unlike in [3], we shall need to change the base field F . We therefore preferi (X/O , n). For the formulation of
to denote the syntomic cohomology by HsynF
the conjecture the following basic fact is required.
1 (Spec(O )/O , n) ∼ F , for all n > 0 and conseLemma 2.15. We have Hsyn=
F
F
quently we have a syntomic regulator regp : K2n−1 (OF ) → F . Furthermore, themap regp commutes in the obvious way with finite extensions of fields and with
automorphisms of such fields, provided that their residue fields are algebraic overthe prime field.

Proof. The first claim follows from part 3 of [3, Proposition 8.6]. For the second
claim we note that by part 4 of the same proposition we have in this case an isomorphism between syntomic and modified syntomic cohomology (the latter only
exists under the additional assumption on the residue field). The compatibility
with finite base changes now follows from the same result for modified syntomic

On the p-adic Beilinson Conjecture for Number Fields

385

cohomology in [3, Proposition 8.8]. The same holds for automorphisms, although
not explicitly stated in the above reference, since the relevant base change results,
e.g., Proposition 8.6.4, hold for this type of base change as well.
¤
As a consequence of the lemma we can also, by abuse of notation, define for any
complete discretely valued subfield F ⊂ Qp the regulator regp : K2n−1 (OF ) → F .
In [5], two of the authors of the present work showed how one can sometimes
compute the map regp by using p-adic polylogarithms.
We now restrict our attention to a totally real number field k of degree d. Our
goal will be to formulate a conjecture that is the p-adic analogue of Theorem 2.2
for K2n−1 (k) with n ≥ 2. Since this K-group is torsion when n is even but has
rank d when n is odd, we only consider odd n ≥ 2.
In preparation for the more general construction that will follow in Section 3 let
us first reformulate Theorem 2.2 in this special case. Let a1 , . . . , ad form a Z-basis
1/2,∞
of Ok and let σ1∞ , . . . , σd∞ be the embeddings of k into C. Then we define Dk=∞det(σi (aj )), a real root of the discriminant of k. Similarly, if α1 , . . . , αd form a∞ (α )). Then
Z-basis of K2n−1 (k)/torsion then we let Rn,∞ (k) = det(reg∞ ◦ σi∗
j
1/2,∞
Dk
and Rn,∞ (k) are well-defined up to sign, and the relation in Theorem 2.2
in this case is equivalent with
1/2,∞

(2.16)

ζk (n)Dk= q(n, k)Rn,∞ (k)with q(n, k) in Q∗ .
Now let F ⊂ Qp be the topological closure of the Galois closure of k embedded
1/2,p=in Qp in any way. If σ1p , . . . , σdp are the embeddings of k into F then Dkpdet(σi (aj )) is a root in F of the discriminant of k. For σ : k → F an embedding
we denote the induced map K2n−1 (k) ∼= K2n−1 (Ok ) → K2n−1 (OF ) by σ∗ . Then
1/2,pp
we define a p-adic regulator in F by Rn,p (k) = det(regp ◦ σi∗(αj )). Both Dk
and Rn,p (k) are well-defined up to sign.
Remark 2.13 suggests that the role of ζk (n) in a p-adic analogue of Theorem 2.2
Qshould be played by Lp (n, ωp1−n , k)/Eulp (n, k) where ζk (s) = l Eull (s, k)−1 forRe(s) > 1, so we can hope that
1/2,p

Lp (n, ωp1−n , k)Dk
for some qp (n, k) in Q∗ .

= qp (n, k) Eulp (n, k)Rn,p (k)386

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot
1/2,∞

1/2,p

and Rn,p (k)/Dk
are invariant unMore precisely, because Rn,∞ (k)/Dk
der reordering the σi∞ or σip , and transform in the same way if we change thebases of Ok and K2n−1 (k)/torsion, we can make the following conjecture.
Conjecture 2.17. For k a totally real number field, p prime, and n ≥ 2 odd, wehave, with notation as above:(1) in F the equality
1/2,p

Lp (n, ωp1−n , k)Dk

= qp (n, k) Eulp (n, k)Rn,p (k)holds for some qp (n, k) in Q∗ ;(2) in fact, qp (n, k) = q(n, k);(3) Lp (n, ωp1−n , k) and Rn,p (k) are non-zero.
As mentioned in the introduction, this, and the corresponding parts of Conjecture 3.18 below, can be deduced (with some effort) from a much more general
conjecture of Perrin-Riou [41, 4.2.2].
Remark 2.18. The conjecture is similar to the result that the residue of ζp (s, k)
at s = 1 is related to the Leopoldt regulator of Ok∗ through exactly the same
formula as for the residue of ζ(s, k) and the Dirichlet regulator [18], with part(4)corresponding to the Leopoldt conjecture. However, we have not tried to determine if Colmez’s normalization of the sign for the regulator is the same as
here, especially given the sign error in the proof of Lemma 4.3 in loc. cit. (seeSection 6).
Remark 2.19. In the definition of Rn,∞ (k) and Rn,p (k) we can use a basis ofa subgroup of finite index of K2n−1 (k)/torsion or even a Q-basis of K2n−1 (k)Q ,without affecting the rationality of q(n, k), qp (n, k) or their equality. Similarly
1/2,∞
we can replace the Z-basis of Ok with a Q-basis of k in the definitions of Dk
1/2,pand Dk .
Remark 2.20. As is well-known, for k and n as in Conjecture 2.17, by (2.6)ζk (s) at s = 1 − n has a zero of order d and the first non-zero coefficient in its
n−1/2
Taylor expansion, ζk] (1 − n), equals (2πi)d(1−n) ((n − 1)!/2)d Dkζk (n). If we
1/2,∞
1/2
take Dk= Dk in (2.16) then we obtainen,∞ (k)ζk] (1 − n) = ((n − 1)!/2)d q(n, k)Dkn−1 R

On the p-adic Beilinson Conjecture for Number Fields

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en,∞ (k) = (2πi)d(1−n) Rn,∞ (k). In comfor Beilinson’s renormalized regulator R
puter calculations ((n − 1)!/2)d q(n, k)Dkn−1 often has only relatively small prime
factors, so the larger prime factors in q(n, k) correspond to Dk1−n . This phenomenon also occurs in the calculations for Conjecture 3.18 below (see Remark 7.8).
Remark 2.21. (1) The thought that Lp (n, ωp1−n , k) is non-zero for n ≥ 2 andodd when k is a totally real Abelian extension of Q is mentioned by C. Soulé in[49, 3.4].
(2) F. Calegari [13] (see also [8]) proved that, for p = 2 and 3, ζp (3), which inthose cases equals Lp (3, ωp−2 , Q), is irrational. (More results along these lines aredescribed in Remark 3.20(3) below.)(3) Parts (1) and (2) of Conjecture 2.17 hold when k is a totally real Abelian
number field, and in fact the corresponding parts of a much stronger conjecturethat we shall describe in Section 3 hold for cyclotomic fields (see Proposition 4.17).
(4) We numerically verified part (3) of Conjecture 2.17, and its more refinedversion Conjecture 3.18(4) below, in certain cases; see Remark 4.19 and Section 7.
3. A motivic version of the conjecture
If E is any extension of Q, and k/Q is finite and Galois with Galois group G,
then we let M E = E ⊗Q k and K2n−1 (k)E = E ⊗Q K2n−1 (k)Q , which are E[G]modules. The goal of this section is to refine Conjecture 2.17 to a conjecture forArtin motives with coefficients in E, or equivalently, idempotents in the groupring E[G], when E is a number field.
Definition 3.1. If π is an idempotent in E[G] then we write MπE for πM E andK2n−1 (MπE ) for πK2n−1 (k)E .
Now fix an embedding φ∞ : k → C. The pairing G × k → C mapping (σ, a) toφ∞ (σ(a)) leads to an E-bilinear pairing(3.2)

E[G] × M E → E ⊗Q C
(eσ, e0 ⊗ a) 7→ ee0 ⊗ φ∞ (σ(a)) .

and we consider its restriction( · , · )∞ : E[G]π × MπE → E ⊗Q C .

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Similarly, replacing φ∞ with a fixed embedding φp : k → Qp we obtain( · , · )p : E[G]π × MπE → E ⊗Q F .
where F ⊂ Qp is the topological closure of φp (k), which is independent of φpsince k/Q is Galois.
Lemma 3.3. Let π in E[G] be an idempotent. Then dimE (MπE ) = dimE (E[G]π).
E and E[G](1−π)×M E .
Proof. Since π 2 = π, (3.2) is identically 0 on E[G]π×M1−ππ∗
But the determinant of this pairing is, up to multiplication by E , equal to
1/2,∞
Dk, hence non-zero. This implies the lemma.
¤

We now introduce pairings similar to (·, ·)∞ and (·, ·)p but replacing MπE with
K2n−1 (MπE ). If we denote the map K2n−1 (k)Q → K2n−1 (C)Q induced by φ∞ by
φ∞∗ and let Φ∞ be the composition with the Beilinson regulator map reg∞ forC,Φ∞ = reg∞ ◦ φ∞∗ : K2n−1 (k)Q → K2n−1 (C)Q → R(n − 1) ⊂ C ,then the pairing G × K2n−1 (k)Q → C given by mapping (σ, α) to Φ∞ (σ(α)) gives
rise to an E-bilinear pairing(3.4)

E[G] × K2n−1 (M E ) → E ⊗Q C
(eσ, e0 ⊗ α) 7→ ee0 ⊗ Φ∞ (σ(α))

and we consider its restriction[ · , · ]∞ : E[G]π × K2n−1 (MπE ) → E ⊗Q C .
By Lemma 2.15 and the definition of F the syntomic regulator gives us
regp : K2n−1 (OF )Q → F .
If we write φp∗ for the composition K2n−1 (k)Q ∼= K2n−1 (Ok )Q → K2n−1 (OF )Q ,with the second map induced by φp : k → F , and let Φp be the composition
regp ◦ φp∗ : K2n−1 (k)Q → K2n−1 (F )Q → F ,then we can similarly obtain a pairing[ · , · ]p : E[G]π × K2n−1 (MπE ) → E ⊗Q F .
We now fix ordered E-bases of E[G]π, MπE , and K2n−1 (MπE ).

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Definition 3.5. For ∗ = ∞ or p we let D(MπE )1/2,∗ be the determinant of thepairing (·, ·)∗ , computed with respect to our fixed bases of E[G]π and MπE .
Note that D(MπE )1/2,∞ is non-zero by the proof of Lemma 3.3, and the same
argument works for D(MπE )1/2,p .
Definition 3.6. If dimE (E[G]π) equals dimE (K2n−1 (MπE )) then for ∗ = ∞ or p
we let Rn,∗ (MπE ) be the determinant of the pairing [·, ·]∗ , computed with respect
to our fixed bases of E[G]π and K2n−1 (MπE ).
For future use we prove the following.
Lemma 3.7. If dimE (E[G]π) = dimE (K2n−1 (MπE )) then
(1) Rn,∞ (MπE )/D(MπE )1/2,∞ is independent of the basis of E[G]π, of φ∞ , andlies in E ⊗ R;
(2) Rn,p (MπE )/D(MπE )1/2,p is independent of the basis of E[G]π, of φp , andlies in E ⊗ Qp .
Proof. We prove the second statement, the proof of the first being entirely similar. Choosing a different E-basis of E[G]π corresponds to letting an E-linear
transformation act on E[G]π, and in the given quotient the resulting determinant cancels. Since k/Q is Galois we get all possible embeddings of k into Qp by
replacing φp with φp ◦ σ for σ in G. For both Rn,p (MπE ) and D(MπE )1/2,p thiscorresponds to letting σ act on E[G]π, and the resulting determinant cancels as
before. That the quotient lies in E ⊗ Qp follows because it lies in E ⊗ F and wehave just proved that it is invariant under Gal(F/Qp ) by Lemma 2.15.
¤
We now investigate when the two dimensions in Definition 3.6 are equal. The
answer is given by Proposition 3.12 below, but we need some preliminary results.
Proposition 3.8. If k/Q is a finite Galois extension with Galois group G and
π is an idempotent in E[G] then dimE (K2n−1 (MπE )) ≤ dimE (E[G]π). Equality
holds precisely when any (hence every) τ in the image in G of the conjugacy classof complex conjugation in Gal(Q/Q) acts by (−1)n−1 on E[G]π.
Proof. For the statement we can first replace E by a finitely generated subfield
since π contains only finitely many elements of E, next embed E into C and use
this embedding to enlarge E to C. So we may assume that E = C.

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According to Theorem 2.2 the pairing (3.4) gives an injection
R ⊗Q K2n−1 (k)Q → R[G]∨ = HomR (R[G], R)(3.9)

α 7→ fα with fα (σ) =1
Φ∞ (σ(α))(2πi)n−1

and by extending the coefficients we get an injection C ⊗Q K2n−1 (k)Q → C[G]∨ .
The image of π(K2n−1 (k)Q ⊗Q C) under the last map vanishes on C[G](1 − π)since (1 − π)π = 0 so that πK2n−1 (k)Q ⊗Q C injects into (C[G]/C[G](1 − π))∨ ∼=(C[G]π)∨ , which proves our first inequality.
As for equality, we know by Theorem 2.2 that (3.9) has as its image the subspace of R[G]∨ where, under the action of Gal(Q/Q) on R[G]∨ via (σ̃f )(σ) =f (σ̃ −1 σ), the conjucagy class of complex conjugation in Gal(Q/Q) acts as multiplication by (−1)n−1 . The same will therefore hold with complex coefficients.
Since any element τ in the conjugacy class of complex conjugation has order 1
or 2, C[G]π decomposes into eigenspaces for the eigenvalues ±1 and the desired
equality can only hold if τ acts as multiplication by (−1)n−1 on all of C[G]π. ¤We now determine precisely when the equality of dimensions as in Proposition 3.8 can occur, and for this we need a preliminary result.
Proposition 3.10. Let ψ be a representation of Gal(Q/Q) over Q that factorizes
through the Galois group of a finite Galois extension of Q and for which ψ(τ )acts as multiplication by (−1)n−1 for any τ in the conjugacy class of complex conjugation in Gal(Q/Q). Then the fixed field of Ker(ψ) is a finite Galois extensionof Q that is totally real if n is odd, and CM if n is even.
Proof. That the fixed field k of Ker(ψ) is a finite Galois extension of Q is clear.
When n is odd ψ(τ ) = 1 for any τ in the conjugacy class of complex conjugation
so that k is totally real. For even n we let ω be the composition Gal(Q/Q) →∗∼Gal(Q(µ4 )/Q) → {±1} ⊂ Q . Then Ker(ψω) contains τ and Ker(ψ) ∩ Ker(ω),
hence its fixed field is a totally real Galois extension k + of Q, contained in k(µ4 ).
Similarly k is contained in the CM field k + (µ4 ) so, since k is not totally real, it
must be CM.
¤
Remark 3.11. For a CM field k with k/Q Galois, its maximal totally real subfield
is Galois over Q and is the fixed field of an element of order two in the centre ofGal(k/Q), which we shall refer to as the complex conjugation of k.

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Proposition 3.12. Let k/Q be a finite Galois extension with Galois group G,
E any extension of Q, and π an idempotent of E[G]. Let k 0 be the fixed fieldof the kernel of the representation of G on E[G]π. Then for n ≥ 2 the equality
dimE (E[G]π) = dimE (K2n−1 (MπE )) holds precisely in the following cases:(1) k 0 is totally real and n is odd;(2) k 0 is a CM field, n is even, and the complex conjugation of k 0 acts on
E[G]π as multiplication by −1.
Proof. From Propositions 3.8 and 3.10 we see that there cannot be any othercases. Conversely, by Proposition 3.8 equality holds in both.
¤
We now recall and introduce some terminology for later use.
Definition 3.13. Let G be a finite group and E any extension of Q.
(1) If π is a central idempotent of E[G] such that E[G]π is a minimal (nonzero) 2-sided ideal of E[G] then π is a primitive central idempotent.
(2) If π is a primitive central idempotent of E[G] such that E[G]π ∼= Mm (E)as E-algebras for some m, then we call a primitive idempotent corresponding to π any element in E[G]π ⊆ E[G] that maps to a matrix in
Mm (E) that is conjugate to a matrix with 1 in the upper left corner and
0’s elsewhere. (Since all automorphisms of Mm (E) as E-algebra are inner
this is independent of the isomorphism E[G]π ∼= Mm (E).)
Remark 3.14. (1) If E[G]π ∼= Mm (E) then any idempotent in E[G]π can be
written as a sum of orthogonal primitive idempotents corresponding to π as one
sees immediately by diagonalizing the matrix A that π maps to, which satisfies
A2 = A.
(2) If G = Gal(k/Q) and π is a primitive central idempotent of E[G] then thedimensions of E[G]π̃ and π̃K2n−1 (k)E are equal for some non-zero idempotent π̃
in E[G]π if and only if the same holds for one (hence any) primitive idempotent
corresponding to π. Indeed, the dimensions for π̃ do not change if we replace itwith a conjugate in E[G], and they add up for sums of orthogonal idempotents.
We now introduce L-functions, both classical and p-adic, in the following context. Let E be a finite extension of Q, k a number field, and consider a characterψ of a representation of G = Gal(k/k) on a finite dimensional E-vector space V

392

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblotthat factorizes through Gal(k/k 0 ) for some finite extension k 0 of k. Then for every
embedding σ : k → C we have the Artin L-function L(s, σ(ψ), k), with for every
prime P of k the reciprocal EulP (s, σ(ψ), k) of the Euler factor corresponding∼
to P. Under the natural isomorphism E ⊗Q C → ⊕σ Cσ the L(s, σ(ψ), k) correspond to a canonical E ⊗Q C-valued L-function that we denote by L(s, ψ ⊗ id, k).
Similarly, for every prime P of k we have an E ⊗Q C-valued EulP (s, ψ ⊗ id, k)corresponding to the EulP (s, σ(ψ), k).
We now move on to the p-adic L-functions, and assume that k is totally real, pa prime number, a any integer and ωp the Teichmüller character Gal(Q/k) → Q∗p(see (2.14)). If τ : E → Qp is an embedding and kτ (ψ)ωpa is totally real then from
Section 2 we have the p-adic L-function Lp (s, τ (ψ)ωpa , k), which is not identically∼zero. In this case, using the natural isomorphism E ⊗Q Qp → ⊕τ Qp,τ , they give
us an E ⊗Q Qp -valued p-adic L-function on Zp or Zp \ {1} that we denote by
Lp (s, ψ ⊗ ωpa , k).
Lemma 3.15. The values of Lp (s, ψ ⊗ ωpa , k) are in E ⊗Q Qp .
Proof. Using Brauer induction for ψ (cf. (2.10)) it suffices to prove this when ψ
is 1-dimensional. But then for each τ : E → Qp the function Lp (s, τ (ψ)ωpa , k) canbe described as in (6.3) with l = −1, from which the result is clear.
¤
Remark 3.16. It follows from Remark 2.13 that Lp (s, ψ ⊗ ωpa , k) satisfies
Lp (m, ψ ⊗ ωpa , k) = Eulp (m, ψ ⊗ id, k)L(m, ψ ⊗ id, k)
for integers m ≤ 0 congruent to 1 − a modulo φ(q), where both sides lie in
E = E ⊗Q Q inside E ⊗Q Qp and E ⊗Q C respectively, andY
Eulp (s, ψ ⊗ id, k) =EulP (s, ψ ⊗ id, k) ,P|p

the product being over all primes of k dividing p.
Remark 3.17. If E = Q then we shall identify E ⊗Q C with C, and write
L(s, ψ, k), etc., instead of L(s, ψ ⊗ id, k), etc. Similarly we identify E ⊗Q Qp with
Qp and write Lp (s, ψωpa , k) instead of Lp (s, ψ ⊗ ωpa , k).
We now have all the ingredients for the generalization and refinement of Conjecture 2.17. Starting with a finite Galois extension k/Q with Galois group G,

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E a finite extension of Q, and π an idempotent in E[G], we let ψπ be the natural representation of Gal(Q/Q) on E[G]π and χπ its associated character. If
dimE (E[G]π) = dimE (πK2n−1 (k)E ) for some n ≥ 2 then, for any prime p and any
embedding τ : E → Qp , τ (ψπ )ωp1−n is trivial on the conjugacy class of complex
conjugation in Gal(Q/Q) by Proposition 3.8. In particular, Qτ (ψπ )ωp1−n is totallyreal and therefore Lp (s, χπ ⊗ ωp1−n , Q) is not identically zero. With F ⊂ Qp the
topological closure of φp (k) as before, using ordered bases for E[G]π, MπE andK2n−1 (MπE ), we have D(MπE )1/2,∞ and the regulator Rn,∞ (MπE ) in E ⊗ C as wellas D(MπE )1/2,p and Rn,p (MπE ) in E ⊗ F .

Conjecture 3.18. Let notation be as above. If n ≥ 2 and dimE (E[G]π) is equal
to dimE (K2n−1 (MπE )), then(1) in E ⊗Q C we haveL(n, χπ ⊗ id, Q)D(MπE )1/2,∞ = e(n, MπE )Rn,∞ (MπE )for some e(n, MπE ) in (E ⊗Q Q)∗ ;(2) in E ⊗Q F we have
Lp (n, χπ ⊗ ωp1−n , Q)D(MπE )1/2,p = ep (n, MπE ) Eulp (n, χπ ⊗ id, Q)Rn,p (MπE )for some ep (n, MπE ) in (E ⊗Q Q)∗ ;(3) in fact, ep (n, MπE ) = e(n, MπE );(4) Lp (n, χπ ⊗ ωp1−n , Q) and Rn,p (MπE ) are units in E ⊗Q Qp and E ⊗Q F
respectively.
Remark 3.19. (1) One sees as in the proof of Lemma 3.7 that the validity ofeach part of the conjecture is independent of the chosen bases of E[G]π, MπE andK2n−1 (MπE ).
∼(2) Since E ⊗Q C → ⊕σ Eσ , where the sum is over all embeddings σ : E → C,an identity in E ⊗Q C is equivalent to the corresponding identities for all such
embeddings. The same holds if we replace C with Qp .
In Remark 3.24 we shall make explicit how part (1) of this conjecture is equivalent with Beilinson’s conjecture [2, Conjecture 3.4] for an Artin motive associatedwith π. First, we make various remarks about its dependence on E, etc., and on
its relation with Conjecture 2.17.

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Remark 3.20. (1) Conjecture 3.18(1) can be formulated for any idempotent π
if k/Q is any finite Galois extension (see the end of Section 5), in which case it isa conjecture by Gross (see [38, p. 210]). In that case it was proved by Beilinson
if the action of G = Gal(k/Q) on E[G]π is Abelian (see loc. cit.). One can also
prove that it holds for any π in E[G] if this action factors through S3 or D8 (seeProposition 5.9).
(2) If the action of G on E[G] is Abelian then parts (1)-(3) of Conjecture 3.18
hold for any π to which the conjecture applies (see Remark 4.18).
(3) Extending and simplifying earlier work by F. Calegari [13], F. Beukers in [8]proves that ζp (2) is irrational when p = 2 or 3. This value equals Lp (2, χωp−1 , Q)
with χ the primitive character on (Z/4Z)∗ for p = 2, and the primitive characteron (Z/3Z)∗ for p = 3. Moreover, if χ is the odd primitive character on (Z/8Z)∗
then he also shows that L2 (2, χω2−1 , Q) is irrational. It follows that the conjectureholds in full for π corresponding to the non-trivial representation of Gal(k/Q) and√√√n = 2 when (k, p) is one of (Q( −1), 2), (Q( −2), 2) and (Q( −3), 3).
(4) We have verified numerically that part (4) of the conjecture holds in certain
cases; see Remark 4.19 as well as Section 7.
Remark 3.21. (1) If π = π1 + · · · + πm with πi2 = πi and πi πj = 0 wheni 6= j, then the conjecture for π is implied by the conjecture for all πi becauseQD(⊕i MπEi )1/2,∞ = i D(MπEi )1/2,∞ , etc., as one easily sees by using bases.
(2) If π is in E[G] and E 0 is an extension of E, then we may view π as an0element of E 0 [G] as well, and the conjectures for MiE and MiE are equivalent: we0
can use the same bases over E 0 as over E, so that D(MiE )1/2,∞ = D(MiE )1/2,∞
in E ⊗ C ⊆ E 0 ⊗ C, and the same holds for all the other ingredients (including
the L-functions).
(3) By comparing bases one sees immediately that if π and π 0 are conjugate
under the action of E[G]∗ then the conjectures for π and for π 0 are equivalent.
(4) If, for a primitive central idempotent πi of E[G], E[G]πi ∼= Mm (E) for somem, and π is a primitive idempotent corresponding to πi , then the conjecture forπi is implied by the conjecture for π. Indeed, we can decompose πi into a sumof orthogonal primitive idempotents as in Remark 3.14(1), and the truth of theconjecture for πi is implied by its truth for each such primitive idempotent. But
all such primitive idempotents are conjugate to π hence (3) above applies.

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Remark 3.22. If k/Q is a Galois extension with Galois group G and H a subHgroup of G with fixed field k H , then Km (k)H
Q = Km (k )Q , a result known asP
Galois descent. With πH = |H|−1 h∈H h, an idempotent in E[G], this implies
that πH Km (k) = Km (k H ).
Remark 3.23. (1) Let k/Q be a Galois extension with Galois group G. If N isPa normal subgroup of G corresponding to k 0 = k N and πN = |N |−1 h∈N h, then2 = π and the natural map φ : E[G] → E[G/N ] has kernel E[G](1 − π ) andπN
NN
induces an isomorphism E[G]πN → E[G/N ]. Indeed, it is clear that πN is central
in E[G] since N is normal in G and that 1−πN is in the kernel of φ. Also, since N
acts trivially on E[G]πN this is an E[G/N ]-module generated by one element, so
its dimension over E cannot be bigger than |G/N |. Since φ is obviously surjective
our claims follow. Therefore, in this situation, for any idempotent π in E[G/N ]there is a canonical idempotent π̃ lifting π to E[G]πN , and the natural map
E[G]π̃ → E[G/N ]π is an isomorphism of E[G]- and E[G/N ]-modules.
Then the statements of Conjecture 3.18 for π in E[G/N ] or for π̃ in E[G] areequivalent. Namely,π̃K2n−1 (k)E = π̃πN K2n−1 (k)E = π̃K2n−1 (k 0 )E = πK2n−1 (k 0 )E
inside K2n−1 (k)E so that we can use the same bases for either side. The same
holds for π̃(E⊗k) and π(E⊗k 0 ) inside E⊗k. Moreover, E[G]π̃ is the pullback to Gof the G/N -representation E[G/N ]π, so that L(s, E[G]π̃, Q) = L(s, E[G/N ]π, Q)
and similarly for the p-adic L-functions.
(2) If k is a totally real number field k let k̃ be its (totally real) Galois closurePover Q. Then k = k̃ H for some subgroup H of G = Gal(k̃/Q), π = |H|−1 h∈H his an idempotent in Q[G], and for n ≥ 2 odd Conjecture 3.18 for π is equivalent
to Conjecture 2.17 for k. Namely, K2n−1 (k)Q = πK2n−1 (k̃)Q ⊆ K2n−1 (k̃)Q by
Remark 3.22 and π k̃ = k so that we can use the same Q-bases in each case.
Moreover, Q[G]π ∼= IndG
H (1H ) = Q[G] ⊗Q[H] 1H with 1H the trivial 1-dimensional
representation of H, as one easily sees by mapping β in Q[G]π to β ⊗ v andPP
σ aσ σ ⊗ (λv) to λσ aσ σπ, {v} being a basis of 1H . By well-known propertiesof Artin L-functions [39, Prop. 10.4(iv)] this implies that ζk (s) = L(s, Q[G]π, Q),
and similarly for the p-adic L-functions.
Remark 3.24. We make the relation between Conjecture 3.18(1) and Beilinson’s conjecture [2, Conjecture 3.4] for (Artin) motives explicit since the relation

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396

between the two elements of E ∗ involved also shows up very explicitly in our computer calculations, suggesting that the element for the formulation at s = 1 − n
is simpler than for our formulation at s = n (see Remark 7.8). We provide somedetails since we could not find a detailed enough reference in the literature.
With notation as in Conjecture 3.18 and (2.6) we have1

(3.25)∨∨∨s− 2
L(1 − s, χ∨(2(2π)−s Γ(s))χ (1)π ⊗ id, Q) = W (χπ ⊗ id)C(χπ ⊗ id)+−× (cos(πs/2))n (sin(πs/2))n L(s, χπ ⊗ id, Q) ,∨∨with χ∨π the dual character of χπ , W (χπ ⊗id) in E⊗C, and C(χπ ⊗id) = C(χπ ⊗id)in Q∗ . If m = dimE (E[G]π), then n+ = m and n− = 0 for n odd, and n− = m
and n+ = 0 for n even. In either case L(s, χ∨π ⊗ id, Q) has a zero of order m at
s = 1 − n. Moreover(3.26)∨
1/2
W (χ∨= en im(n−1) D(MπE )1/2,∞
π ⊗ id)C(χπ ⊗ id)for some en in E ∗ = (E ⊗Q Q)∗ by [20, Propositions 5.5 and 6.5] since D1/2,∞ canbe taken to be the same for MπE and the associated determinant representation(cf. [31, p.360]). Hence the first non-vanishing coefficient in the Taylor expansion]∨of L(s, χ∨π ⊗ id, Q) around s = 1 − n, L (1 − n, χπ ⊗ id, Q), equalsn−1
δn en (2πi)m(1−n) ((n − 1)!/2)m C(χ∨D(MπE )1/2,∞ L(n, χπ ⊗ id, Q)
π ⊗ id)with δn = (−1)m(n−1)/2 when n is odd, and δn = (−1)m(n+2)/2 when n is even.
In particular, Conjecture 3.18(1) is equivalent with
(3.27)m∨n−1
en,∞ (M E )L] (1 − n, χ∨e(n, MπE )R
π ⊗ id, Q) = δn en ((n − 1)!/2) C(χπ ⊗ id)
πen,∞ (M E ) = (2πi)m(1−n) Rn,∞ (M E ).
with the renormalized regulator Rππ
Let us compare this with Beilinson’s conjecture for a motive associated with π.
We associate motives covariantly to smooth projective varieties over Q as in [2,§ 2.4]. The Galois group G acts on the left on k, hence on the right on Spec(k), andwe let Mπ be the motive corresponding to π under this action. Then G acts on
the left on the cohomology theories on Spec(k) as well as its K-theory, and after
tensoring with E the corresponding groups for Mπ are the images under π. Thus,
1 (ME
the relevant motivic cohomology of Mπ is HM
π/Z , Q(n)) = K2n−1 (Mπ ).

On the p-adic Beilinson Conjecture for Number Fields

397

We shall need the non-degenerate E-bilinear pairing
M
E[G] ×E→E
τ :k→CPP( σ aσ σ, (bτ )τ ) 7→ σ aσ bφ∞ ◦σ−1 .
It factors through the left E[G]-action on ⊕τ E (given by σ((bτ )τ ) = (bτ ◦σ−1 )τ ),hence is trivial on E[G](1−π)×π(⊕τ E) and E[G]π ×(1−π)(⊕τ E). We therefore
obtain a non-degenerate E-bilinear pairing(3.28)

h · , · i : E[G]π × π(⊕τ E) → E

that identifies E[G]π and π(⊕τ E) as dual E[G]-modules.
Tensoring (2.1) with E and applying π we get
¡ M
¢
1
HD
(Mπ/R , R(n)) ∼E ⊗ R(n − 1) .
=π
τ :k→C

Note that the left-hand side is a subspace of the right-hand side by (2.1), but
because dimE (K2n−1 (MπE )) = dimE (E[G]π) = dimE (π(⊕τ E)) by our assumption on π and (3.28), and the regulator map (2.3) tensored with R is injective,
equality must hold. For the Beilinson regulator we therefore have to compare
1 (Mtwo E-structures on det HD
π/R , R(n)), the first one coming from Betti cohomology,
M
¡ M
¢
0
HB
(Mπ/R , Q(n − 1)) ∼E ⊗ Q(n − 1)) ⊆ π
E ⊗ R(n − 1) ,= π(
τ :k→Cτ :k→Cand the second one being induced by the Beilinson regulator map (2.3),
¡ M
¢
1
1
HM
(Mπ/R , Q(n)) → HD
(Mπ/R , R(n)) = π
E ⊗ R(n − 1) .
τ :k→C

Choosing an E-basis of π(⊕τ E), and multiplying it by (2πi)n−1 to obtain an
E-basis of π(⊕τ E ⊗ Q(n − 1)), it is easy to see that Beilinson’s regulator RBei forMπ satisfies
en,∞ (M E )/ deth · , · i
RBei = (2πi)m(1−n) det[ · , · ]∞ / deth · , · i = R
πwhere all determinants are computed using the chosen E-bases.
Finally, we compare L-functions. We have
µ M ¶
M
0
∼
Q
Q
⊗
Q
Het
(Spec(k) ⊗ Q, Ql ) ∼=l = l Q
τ :k→Qτ :k→Q

398

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot

0 (M , Q ) ∼ Q ⊗ π(⊕ E) as Q ⊗ E[G]-modules. The Q plays no role,so that Het
π
τ
l = l
l
l
and as in (3.28) we see that π(⊕τ :k→Q E) is dual to E[G]π as E[G]-module. As
the motivic L-function uses the geometric rather than the arithmetic Frobenius(cf. [34, p.26]), we obtain that L(s, Mπ ) of [2, § 3] is equal to L(s, χπ ⊗ id, Q).
If θ : E[G] → E[G] is the E-linear involution obtained by replacing each σ in G
with σ −1 , then we need to consider M0π = Mθ(π) instead of Mπ . But the map
E[G]π ×E[G]θ(π) → E mapping (α, β) to the coefficient of the neutral element of
G in θ(β)α is easily seen to identify E[G]θ(π) and E[G]π as dual E[G]-modules,
so that L(s, M0π ) = L(s, χ∨π ⊗ id, Q).

4. More explicit K-groups and regulators
In this section we first describe an inductive procedure that conjecturally gives
K2n−1 (k)Q (n ≥ 2) for any number field k. It is originally due to Zagier [54],but we essentially give a reformulation by Deligne [21]. We also describe results
concerning Conjecture 3.18 when the action of G on E[G]π is Abelian.
In order to describe Zagier’s conjecture we need the functions
X zl
(4.1)
Lin (z) =(n ≥ 0)ln
l≥1

for z in C with |z| < 1 if n = 0 or 1, and |z| ≤ 1 if n ≥ 2. In particular,
Li1 (z) is the main branch of − log(1 − z). Using that d Lin+1 (z) = Lin (z) d log(z)they extend to multi-valued analytic functions on C \ {0, 1}. By simultaneously
continuing all Lin along the same path one obtains single-valued functions on
C \ {0, 1} (see [54, § 7] or [19, Remark 5.2]) by putting
µn−1
¶
X bj
j
(4.2)
Pn (z) = πn−1(2 log |z|) Lin−j (z)(n ≥ 1) ,j!
j=0with bj the j-th Bernoulli number and πn−1 the projection of C = R(n−1)⊕R(n)onto R(n − 1). These functions satisfy Pn (z̄) = Pn (z) as well as(4.3)

Pn (z) + (−1)n Pn (1/z) = 0

and
(4.4)

Pn (z m ) = mn−1X
ζ m =1

when m ≥ 1 and z m 6= 1.

Pn (ζz)On the p-adic Beilinson Conjecture for Number Fields

399

en (k) be a free Abelian
We can now describe the conjecture. For n ≥ 2 we let B∼
group on generators [x]n with x 6= 0, 1 in k. Defineen (k) → (⊕σ:k→C R(n − 1))+
Pen : B[x]∼n 7→ (Pn (σ(x)))σ ,with (σ : ⊕k→C R(n − 1))+ = {(aσ )σ such that aσ = aσ̄ }. Then we define inductively, for n ≥ 2,

2

 ^ ∗
k
if n = 2
en (k) →dn : BZ∗B
n−1 (k) ⊗Z k if n > 2
by
(
[x]∼
n 7→

(1 − x) ∧ x if n = 2[x]n−1 ⊗ x if n > 2 ,where [x]n−1 denotes the class of [x]∼n−1 in Bn−1 (k), which is defined asen (k)/ Ker(dn ) ∩ Ker(Pen ) .
Bn (k) = B
There are some universal relations, one of which is that [x]n + (−1)n [1/x]n = 0,
a consequence of (4.3).
Conjecture 4.5. If n ≥ 2 then
(1) there is an injection
Ker(dn )Ker(dn ) ∩ Ker(Pen )→ K2n−1 (k)Qwith image a finitely generated group of rank equal to dimQ (K2n−1 (k)Q );(2) Beilinson’s regulator map is essentially given by Pen : the diagram
Ker(dn )Ker(dn ) ∩ Ker(Pen )/ K2n−1 (k)Q

RRR
RRR
RRR
RR
(n−1)!Pen RR(

Q

σ:k→C reg∞ ◦σ∗²(⊕σ:k→C R(n − 1))+

commutes.
Remark 4.6. For n = 2 the corresponding results were already known before
Zagier made his conjecture (see [9, 50], as well as [26, § 2]).

400

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot

From the results in [19, § 5] one obtains the following.
Theorem 4.7. Let k be a number field and let n ≥ 2 be an integer. Then there
is an injection
Ker(dn )Ψn :→ K2n−1 (k)Q
Ker(dn ) ∩ Ker(Pen )with finitely generated image, such that the diagram in Conjecture 4.5(2) commutes.
Remark 4.8. The part of Zagier’s conjecture that remains open is the question
if the rank of the image of Ψn equals dimQ (K2n−1 (k)Q ). For a cyclotomic field kthis holds for any n ≥ 2 as we shall recall in Example 4.10 below, but it is also
known for arbitrary number fields for n = 2, as mentioned above, or n = 3 (see[25, § 3] and [27, Appendix]).
Remark 4.9. Although not stated explicitly in [19], it is clear from the construction there that the map Ψn in Theorem 4.7 is natural in k. In particular,
if k/Q is Galois then Gal(k/Q) acts on Ker(dn )/ Ker(dn ) ∩ Ker(Pen ) (throughthe obvious action on the generators [x]∼n ) as well as on K2n−1 (k)Q , and Ψn isGal(k/Q)-equivariant.
Example 4.10. If k is a cyclotomic field then Zagier’s conjecture is known infull. By Theorem 4.7 it suffices to see that the rank of the image of Ψn equals
dimQ K2n−1 (k)Q . For an N -th root of unity ζ 6= 1, N [ζ]n is in Ker(dn )/ Ker(dn )∩Ker(Pen ). If k = Q(ζ) for a primitive N -th root of unity ζ with N > 2 thendimQ K2n−1 (k)Q = [k : Q]/2 for n ≥ 2, and the N [ζ j ]n with 1 < j < N/2 andgcd(j, N ) = 1 have R-independent images under Pen [54, pp.420–422] so that they
are Z-independent in Ker(dn )/ Ker(dn ) ∩ Ker(Pen ), and the same holds for theirimages under the injective map Ψn . For k = Q one can easily check directly from
Theorem 4.7 and (4.2) that for odd n ≥ 2 the element 2Ψ([−1]n ) is non-zero.
For the p-adic regulator we need to describe the p-adic polylogarithms introduced in [16]. We first make a choice of a branch of the p-adic logarithm. Recall
that a homomorphism logp : OC∗ p → Cp is uniquely determined by the requirement that for |x|p < 1 it is given by the usual power series for log(1+x). To extend
it to a homomorphism logp : Cp ∗ → Cp it suffices to make an arbitrary choice oflogp (p). Any such extension will be called a branch of the p-adic polylogarithm.
In what follows we fix one such choice.

On the p-adic Beilinson Conjecture for Number Fields

401

Coleman first produced the p-adic polylogarithm out of a more extensive theoryof what is now called Coleman integration. For the p-adic polylogarithm it is,however, possible to give a more elementary and explicit theory. To do this,consider the class of functions f : Cp \ {1} → Cp that satisfy(1) for any a ∈ Cp with |a − 1|p = 1 we have a power series expansion forf (z) in z − a that converges for |z − a|p < 1;(2) for 0 < |z − 1|p < 1 (resp. |z|p > 1) f (z) is given by a polynomial in
logp (z − 1) (resp. logp (1/z)) with coefficients that are Laurent series inz − 1 (resp. 1/z), convergent for 0 < |z − 1|p < 1 (resp. |1/z|p < 1).
It is easy to see that differentiation sends this class surjectively onto itself. The padic polylogarithms are functions Lin,p (z) (n ≥ 0) in this class with the following
properties:z(1) Li0,p (z) = 1−z;(2) Lin,p (0) = 0 for n ≥ 0;(3) dLin+1,p (z) = Lin,p (z) dz
z for n ≥ 0;(4) for n ≥ 0 there are power series gn,p (v), convergent for |v|p < p1/(p−1) ,such that gn,p (1/(1−z)) = Lin,p (z)− p1n Lin,p (z p ) when |z−1|p > p−1/(p−1) .

Note that 1/z has a singularity at 0 but Lin,p (z)/z does not, because of theassumption that Lin,p (0) = 0. In fact, it is easy to see that Lin,p (z) is given by
the p-adically convergent power series (4.1) for |z|p < 1.
In [6] two of the authors of the present work described an efficient algorithm
for the computation of p-adic polylogarithms up to a prescribed precision. This
will be used in Section 7 to compute the p-adic regulator for n ≥ 2 that we now
proceed to describe in terms of suitable combinations of logn−m(z)Lim (z).
p
By Proposition 6.4 and the correct version of Proposition 6.1 of [16] the Lin,p (z)for n ≥ 0 and z in Cp \ {1} satisfy
Lin,p (z) + (−1)n Lin,p (1/z) = −1
lognp (z)n!

when z 6= 0, and(4.11)

Lin,p (z m ) = mn−1X
ζ m =1

when m ≥ 1 and z m 6= 1.

Lin,p (ζz)

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot

402

For z in Cp \ {0, 1} and fixed n ≥ 2, we define(4.12)

Pn,p (z) =n−1
X

cj logjp (z)Lin−j,p (z)(n ≥ 2)j=0

P
for any cj in Cp satisfying c0 = 1 and n−1
j=0 cj /(n − j)! = 0, so that Pn,p (z) +n(−1) Pn,p (1/z) = 0. This is the case for cj = bj with bj the j-th Bernoulli
numbers as before, although the resulting formula is different from (4.2). Anotherpossible natural candidate for the function Pn,p (z) is Ln,p (z)+Ln−1,p (z) logp (z)/n,
where
n−1
X (−1)m
Ln,p (z) =logm
p (z)Lin−m,p (z)m!
m=0(see [5, (1.3) and Remark 1.5]).
The relations corresponding to (4.3) and (4.11) for Pn,p (z) (n ≥ 2) are thenPn,p (z) + (−1)n Pn,p (1/z) = 0
when z 6= 0, 1, and(4.13)

Pn,p (z m ) = mn−1

XPn,p (ζz)ζ m =1

when m ≥ 1 and z m 6= 0, 1.
For any branch of the p-adic logarithm one then has the following result for thep-adic regulator map (see [5, Theorem 1.10 or page 909]). Since its formulation
depends on the embedding of k we deal with one such embedding at a time.
Theorem 4.14. Let k be a number field, and let F ⊂ Qp be the topological closureof the Galois closure of any embedding k → Qp . For σ : k → F let
Bnσ (k) = h[x]n | σ(x), 1 − σ(x) are in OF∗ i ⊆ Bn (k) =Thenen (k) → F
B[x]n 7→ Pn,p (σ(x))
induces a mapσ
Pn,p: Bnσ (k) → Fen (k)B
Ker(dn ) ∩ Ker(Pen ).

On the p-adic Beilinson Conjecture for Number Fields

403

and the solid arrows in
Bnσ (k) ∩Ker(dn )Ker(dn )ÂÄ/eKer(dn ) ∩ Ker(Pn ) YYY Ker(dn ) ∩ Ker(Pen )YYYΨn/ K2n−1 (k)QYYYYYYσ(n−1)!Pn,p
YYYYYYregp ◦σ∗YYYYYYYYYYYYσ(n−1)!Pn,p
YYYYY', ²F

form a commutative diagram.
Remark 4.15. It was conjectured in [5, Conjecture 1.14] that the dotted arrowexists and that the full diagram commutes. This is known to hold for N [ζ]n if ζ
is any N -th root of unity other than 1 (see Theorem 1.12 of loc. cit.).
Remark 4.16. The formulae for the complex and p-adic regulators involved in
Theorems 4.7 and 4.14 are stated up to sign in [19, Proposition 4.1] (with anormalizing factor) and [5, Theorem 1.10(2) or Proposition 7.14] respectively,
since they depend on the choice of Ψn , which is natural only up to sign. In
the p-adic case it is clear from [5, Proposition 7.10] that the sign is (−1)n if the“relativity isomorphism” in the p-adic case is normalized as in Proposition 5.7
of loc. cit. and the K-theoretic relativity isomorphism as in (3.1) of loc. cit.
corresponds to this under the naturality of the regulator map. In order to seethat the sign for the complex regulator is the same with the same choice of Ψn ,we observe that one can use the techniques of [5, Appendix A and Section 5] todescribe the target of the complex regulator map for [z]n with z in C\{0, 1} in [19]by means of a complex with the same formal structure as in [5, Definition 5.2], butusing R(n−1)-valued C ∞ -forms ψ with logarithmic poles on copies of (P1C\{1, z})j
indexed by increasing functions f : [1, . . . , j] → [1, . . . , n − 1]. Using the mapsR
ψ 7→ (P1 )j d arg(tf (1) ) ∧ · · · ∧ d arg(tf (j) ) ∧ ψ for a j-form ψ with index f :C[1, . . . , j] → [1, . . . , n − 1], and suitable normalizing factors, one writes downan R-linear map Π∞ on such forms, with values in R(n − 1), that vanishes on
exact forms and satisfies the analogous normalization condition as Π = Πp in [5,Proposition 5.7]. Moreover, as in [19, Section 2.5] one sees that the subcomplex
of forms on the component where j = n − 1 that vanish if some tl = 0 or ∞ gives
a quasi-isomorphism under inclusion, and that the image of the regulator of [z]nis given by εn (z) as in Section 4 of loc. cit. (up to sign since one has to use thecorrect formula for ωn , which can be obtained as in [5, Section 6]). Using thecomputations on pages 236 and 237 of [19] one then sees that Π∞ maps εn (z)404

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot
´

³P

n−1 1
l
l=0 l! (− log |z|) Lin−l (z) , which
induces the same map on Ker(dn )/ Ker(dn ) ∩ Ker(Pen ) as (−1)n (n − 1)! Pn (z).

to (−1)n (n − 1)!Pn[ (z), where Pn[ (z) = πn−1

Since the relativity isomorphism in K-theory only depends on a choice of tl = 0
or ∞ for l = 1, . . . , n − 1 this shows that the signs in the formulae for the complex
and p-adic regulators of [z]n match for the same choice of Ψn , and we adjust Ψn
so that those signs disappear.
We conclude this section by considering Conjecture 3.18 when the action of G
on E[G]π is Abelian.
Proposition 4.17. Let N ≥ 2, k = Q(µN ), G = Gal(k/Q) = (Z/N Z)∗ where
a in (Z/N Z)∗ corresponds to ψa in G satisfying ψa (ζ) = ζ a for all ζ in µN .
Assume that E contains a root of unity of order equal to the exponent of G and
let π in E[G] be the idempotent corresponding to an irreducible character χ of
G. Then parts (1), (2) and (3) of Conjecture 3.18 hold for π and n ≥ 2 ifχ(−1) = (−1)n−1 .
Proof. That Conjecture 3.18(1) holds in this case is well-known (cf. [38]) but wemake it explicit here for part (3) of the conjecture (cf. [30, Proposition 3.1]).
We deal with the case N = 2 later so by Remark 3.23(1) we may assume N > 2
P−1
and χ primitive. The idempotent π corresponding to χ is |G|−1 0N
a=1 χ (a)⊗ψa ,where the prime indicates that we sum over a that are relatively prime to N .
Note that by our assumption on χ(−1) and Proposition 3.8 the dimension of
K2n−1 (MπE ) equals 1. Fixing a generator ζ of µN ⊂ k ∗ we use π, π(1 ⊗ ζ) andπ(1 ⊗ [ζ]n ) as basis vectors of E[G]π, MπE and K2n−1 (MπE ) respectively, where
we simplify notation by writing [ζ]n instead of Ψn ([ζ]n ) as in Theorem 4.7.
In order to verify (1) in E ⊗Q C it suffices by Remark 3.19(2) to consider
all embeddings σ : E → C. If η = φ∞ (ζ) with φ∞ as in Section 3 then the
σ-component of D(MπE )1/2,∞ becomes
ÃN
!−1
N
X0
X
0 −1
a
−1
−a
−1
E 1/2,∞
χσ (a)η
D(Mπ )σ= |G|χσ (a)η = N |G|
a=1

a=1

by [36, (3) on p.84]. Similarly, by Theorem 4.7 the σ-component of Rn,∞ (MπE ) is
Rn,∞ (MπE )σ = (n−1)! |G|−1

N
X
0
a=1

a
−1
χ−1
σ (a)Pn (η ) = (n−1)! |G|

N
X
0
a=1

a
χ−1
σ (a)Lin (η )On the p-adic Beilinson Conjecture for Number Fields

405−a ))/2 by our assumption on χ(−1),
because Pn (η a ) = (Lin (η a ) + χ−1
σ (−1)Lin (ηwhere Lin is computed using the power series in (4.1). They relate to the σ1/2,∞
component of L(n, χ, Q) via L(n, χσ , Q) = (n − 1)!−1 Rn,∞ (MπE )σ /D(MπE )σaccording to (2) on page 172 of [16] with the correct sign in the exponent of theGauss sum as used here (cf. [54, p. 421]). Since L(n, χσ , Q) 6= 0 this also showsthat π(1 ⊗ [ζ]n ) gives a basis of K2n−1 (MπE ) as claimed.

Similarly, for (2) and (3) we consider all embeddings τ : E → Qp . If nowP−a −1
φp (ζ) = ηp then the τ -component of D(MπE )1/2,p is N |G|−1 ( 0N
a=1 χτ (a)ηp )P
0N
a
and the τ -component of Rn,p (MπE ) equals (n − 1)! |G|−1 a=1 χ−1τ (a)Pn,p (ηp ) by
Remark 4.15, independent of our assumption on χ(−1). By (3) on page 172 of[16] we have
Lp (n, χτ ωp1−n , Q) = (n − 1)!−1 Eulp (k, χτ , Q)Rn,p (MπE )τ /D(MπE )1/2,p.
τTherefore Conjecture 3.18(1)-(3) hold with e(n, MπE ) = ep (n, MπE ) in Q∗ ⊆ E ∗ .
Now assume that N = 2 so that χ is the trivial character 1, π = 1 and n ≥ 2
is odd. Taking 1, 1 ⊗ 1 and 1 ⊗ [−1]n as bases of E[G] = E, MπE = E ⊗ Q and
K2n−1 (Q)E we find from Theorem 4.7, (4.1), (4.2) and (4.4) that
L(n, 1, Q) = ζ(n) =

2n−1
2n−1
P
(−1)
=
Rn,∞ (M1E ) ,n
1 − 2n−1(n − 1)!(1 − 2n−1 )again proving that [−1]n gives a basis for K2n−1 (Q)Q . Coleman proves thatLp (n, ωp1−n , Q) = (1 − p−n ) lim 0 Lin,p (x)x→1

where the limit is taken in any subfield of Cp that is of finite ramification degree
over Qp . But by (4.12) and (4.13) we can rewrite this as
Lp (n, ωp1−n , Q) =(1 − p−n )2n−1(1 − p−n )2n−1
P(−1)=
Rp (M1 ) .
n,p
1 − 2n−1(n − 1)!(1 − 2n−1 ) n,∞¤

Remark 4.18. The π in the proof of Proposition 4.17 are primitive in the senseof Definition 3.13. So, by Remarks 3.21(4) and 3.23(1), if Conjecture 3.18 applies
to π in E[G], then parts (1), (2) and (3) of it hold for π for any E if the action
on E[G]π is Abelian. In particular, parts (1) and (2) of Conjecture 2.17 hold forany totally real Abelian number field by Remark 3.21.
Remark 4.19. Computer calculations show that Lp (n, ωp1−n , Q) is in Q∗p whenp = 2, . . . , 19 and n = 2, . . . , 20 is odd, verifying Conjecture 2.17(3) in those406

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot

cases. Similarly, with notation and assumptions as in Proposition 4.17, for the
470 primitive characters χ of Gal(Q(µN )/Q) = (Z/N Z)∗ with 2 ≤ N ≤ 50,
Lp (n, χ ⊗ ωp1−n , Q) lies in (E ⊗ Qp )∗ for those values of p and n = 2, . . . , 20
whenever χ(−1) = (−1)n−1 . Thus Conjecture 3.18(4) also holds in those cases.
5. Computing K-groups in practiceLet n ≥ 2 and let k be a number field. It follows from Theorem 4.7 thatZagier’s approach as described in Section 4 can be used to obtain at least somepart of K2n−1 (k)Q . Using the notation introduced in Section 4, in order to carry
this approach out in practice one starts with two sets S ⊆ S 0 of primes of Ok0and considers only [x]∼n where x is an S-unit and 1 − x an S -unit. When we∼∼
consider several hundred of such [x]∼n ’s (and avoid using both [x]n and [1/x]nsince [x]n + (−1)n [1/x]n = 0 as mentioned in Section 4) this method is well suited
to computer calculations. The only point in Zagier’s approach that cannot be
carried out algebraically is to determine which elements in Ker(dn ) are actuallyin Ker(dn ) ∩ Ker(Pen ). This is done using standard methods for finding linear
relations between the Pen (αj )’s for a basis {αj } of Ker(dn ) with small integral
coefficients. Typically, we used about the first 50 decimals after the decimal pointof all the polylogarithms of all elements of k involved in the αj embedded into C in
all possible ways. The relations that were found this way were then verified to holdup to at least 30 additional decimal places. As a final check, when we obtained
a subgroup of Ker(dn )/ Ker(dn ) ∩ Ker(Pen ) with the same rank as K2n−1 (k), thenumber q in (2.4) (but now using the regulator Ven (k) for the resulting subgroupof K2n−1 (k)Q rather than Vn (k); cf. Remark 2.19) was computed with a precisionof up to 120 decimals. Its reciprocal appeared to be rational rather convincingly.
For example, for the cubic field k = Q[x]/(x3 − x2 − 3x + 1) with discriminant148, one of the fields in Example 7.1, we find for ζk (n)|Dk |1/2 /Ven (k) the values− 1.7531044558071585098612125639152666179693206722 × 10−2(n = 3)− 4.1170685884062518549452525064367732455835167754 × 10−9(n = 5)

with reciprocals− 57.041666666666666666666666666666666666666666662(n = 3)− 242891265.59999999999999999999999999999999999998(n = 5) .

On the p-adic Beilinson Conjecture for Number Fields

407

As the calculation of a basis of K2n−1 (k)Q , using the method outlined above,
tends to take (substantially) longer when [k : Q], the absolute value of the discriminant of k, or n increase, we describe two straightforward methods to reduce
this. They, and most of our arguments below, will rely on Remark 3.22.
Method 5.1. If k is an Abelian extension of Q, hence is contained in a cyclotomic field k 0 , then we can find K2n−1 (k)Q by computing K2n−1 (k 0 )Q as in
Example 4.10 and applying Remark 3.22 (using Remark 4.9).
Since parts (1), (2) and (3) of Conjecture 3.18 are known for all idempotentsof E[G] to which it applies if G is Abelian by Remark 3.21 and Proposition 4.17,
this will not be used for verifying Conjecture 3.18. Instead, in the cases that weshall consider we rely on the following method in order to find the K-groups.
Method 5.2. Let k/Q be Galois with Galois group G, E/Q any extension, and M
an irreducible E[G]-module in K2n−1 (k)E . If H is a subgroup of G with M H 6= 0
then M ⊆ E[G] · K2n−1 (k H )Q . If for every irreducible M we can take H 6= {e},then we reduce to finding K2n−1 (l)Q together with the action of G on it inside
K2n−1 (k)Q for proper subfields l of k.
We now discuss in Examples 5.3, 5.5 and 5.7 how to use Method 5.2 for certain
Galois extensions k/Q. In the decomposition of K2n−1 (k)E according to central
idempotents πi of E[G] as in Section 3 we concentrate on those primitive central
πi for which the action on E[G]πi is not Abelian and describe corresponding primitive idempotents π (as in Definition 3.13(2)) for later use. Due to restrictions onwhen we can calculate the p-adic L-functions numerically (see Section 6) we only
consider k that are Abelian over a quadratic subfield. Of course Proposition 3.12
describes when Conjecture 3.18 applies to π, but we work out the structure of
the K-groups and the Galois action in more detail.
Example 5.3. Let k/Q be an S3 -extension with quadratic subextension q. With
E = Q and σ in G of order three, π1 = (e + σ + σ 2 )/3 and π2 = 1 − π1 areorthogonal central idempotents with π2 primitive (as in Definition 3.13(2)). In
Q[G] = Q[G]π1 ⊕ Q[G]π2 the first summand consists of Abelian representationsof G and the last of two copies of the irreducible 2-dimensional representation Vof G. A corresponding primitive idempotent for π2 is π = π2 πH for any subgroup
H = hτ i of G of order 2 where πH = (e + τ )/2.

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot

408

By Remark 3.22 we have π1 K2n−1 (k)Q = K2n−1 (q)Q so that
(5.4)

K2n−1 (k)Q = K2n−1 (q)Q ⊕ π2 K2n−1 (k)Q .

The last summand is a Q[G]-module isomorphic to V tn for some tn ≥ 0. As
V = πV ⊕ τ 0 πV for any τ 0 6= τ of order 2, we find, if c = k H ,π2 K2n−1 (k)Q = π2 K2n−1 (c)Q ⊕ τ 0 π2 K2n−1 (c)Q
again by Remark 3.22. So we can reduce to the calculation of K2n−1 (q)Q and
K2n−1 (c)Q together with the action of G on the latter.
A dimension count in (5.4) using Theorem 2.2 determines tn . We distinguish
two cases.
Case 1. If q is real then tn = 0 for n even and 2 for n odd.
Case 2. If q is imaginary then tn = 1.
Conjecture 3.18 applies to π when tn = 2.
Example 5.5. Let k/Q be a D8 -extension with D8 = hσ, τ |σ 4 = τ 2 = (στ )2 = ei.
We fix an isomorphism G = Gal(k/Q) ∼= D8 and use notation as in Figure 5.1.
{e} OOhτ ioooÄooo ÄÄÄ
o
o
Ä
ooo

?? OO
?? OOO
?
OO

??
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??
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hσ 2 τ i

hσ 2 ihτ, σ 2 ihσi hστ, σ 2 i

Ä
ÄÄ
ÄÄ

??
??
?

hστ i hσ 3 τ i

Ä
ÄÄ
ÄÄ

D8

Ä
ÄÄ
ÄÄ

k1

kO
ooÄoÄ ??O?OOOOoo?? OOOo Ä?
OOO
ooo ÄÄÄ
o
o
o
k

2
??
??
ÄÄ
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ÄÄ
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q1

k3
q

k

4
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ÄÄ
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k5

q

2
3
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Q

Figure 5.1. The subgroups of G = D8 and their fixed fields in k/Q.
Then Q[G] = Q[G]π1 ⊕ Q[G]π2 where π1 = (1 + σ 2 )/2 and π2 = 1 − π1 areorthogonal central idempotents. The first term is an Abelian representation of
G and the second as a Q[G]-module is isomorphic to V 2 for V the irreducible
2-dimensional representation of G. Then π2 is a primitive central idempotent
and π = π2 πH a corresponding primitive idempotent if we let H = hτ i andπH = (1 + τ )/2.

On the p-adic Beilinson Conjecture for Number Fields

409

Since π1 K2n−1 (k)Q = K2n−1 (k3 )Q by Remark 3.22, for K2n−1 (k)Q we find
(5.6)

K2n−1 (k)Q = K2n−1 (k3 )Q ⊕ π2 K2n−1 (k)Qwith the second term a Q[G]-module isomorphic with V tn for some tn ≥ 0. If τ 0
in G has order two and does not commute with τ then V = πV ⊕ τ 0 πV , so by
Remark 3.22 we obtain
K2n−1 (k)Q = K2n−1 (k3 )Q ⊕ π2 K2n−1 (k1 )Q ⊕ τ 0 π2 K2n−1 (k1 )Q
and we reduce to the calculation of K2n−1 (k3 )Q and K2n−1 (k1 )Q together withthe action of G on the latter.
We can find tn by counting dimensions in (5.6), and distinguish three cases.
Case 1. For k3 and k totally real tn = 0 for n even and tn = 2 for n odd.
Case 2. For k3 totally real but k not tn = 2 for n even and tn = 0 for n odd.
Case 3. For k3 not totally real tn = 1.
Again Conjecture 3.18 applies to π when tn = 2.
Example 5.7. Let k/Q be an S3 × Z/3Z-extension, identify G = Gal(k/Q) andS3 × Z/3Z via a fixed isomorphism, and choose a generator σ of A3 ⊂ S3 . Let q
be the quadratic subfield of k, s0 the fixed field of A3 × {0}, s00 the fixed field of{e} × Z/3Z and si for i = 1, 2 the fixed field of Hi = h(σ, i)i.
If E is an extension of Q that contains a primitive third root of unity ζ3 then
(5.8)

E[G] = E[G]π1 ⊕ E[G]π2 ⊕ E[G]π3 ⊕ E[G]π4∼= E[Z/6Z] ⊕ W12 ⊕ Wζ23 ⊕ Wζ22
3

where

π1 = ((e, 0) + (σ, 0) + (σ 2 , 0))/3
π2 = (1 − π1 )((e, 0) + (e, 1) + (e, 2))/3
π3 = (1 − π1 )((e, 0) + ζ32 (e, 1) + ζ3 (e, 2))/3
π4 = (1 − π1 )((e, 0) + ζ3 (e, 1) + ζ32 (e, 2))/3

are orthogonal central idempotents of E[G] satisfying π1 + π2 + π3 + π4 = (e, 0),and Wa is the irreducible representation of G given by (ρ, m)(v) = am ρ(v) forPv in the irreducible 2-dimensional representation of S3 . With πHi = 13 h∈Hi h,a primitive idempotent corresponding to the primitive central idempotent πj(j = 3, 4) is π = πj πHi .

410

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot

Using Remark 3.22 for A3 × {0} and {e} × Z/3Z we see that
K2n−1 (k)E = π1 K2n−1 (k)E ⊕ π2 K2n−1 (k)E ⊕ π3 K2n−1 (k)E ⊕ π4 K2n−1 (k)E
= K2n−1 (s0 )E ⊕(1 − π1 )K2n−1 (s00 )E ⊕ π3 K2n−1 (k)E ⊕ π4 K2n−1 (k)E .
Because Wa = WaH1 ⊕ WaH2 when a 6= 1 we haveπj K2n−1 (k)E = πj K2n−1 (s1 )E ⊕ πj K2n−1 (s2 )E(j = 3, 4)

so that we reduce to the calculation of K2n−1 (s)Q for the sextic subfields s of k,
together with the action of G when s = s00 , s1 or s2 .
We have π3 K2n−1 (k)E ∼= Wζt3n and π4 K2n−1 (k)E ∼= Wζt2n for the same tn as one
3sees by taking E = Q(ζ3 ) and considering the obvious action of Gal(E/Q) on
E[G]. Since the dimension of (1 − π1 )K2n−1 (s00 )E was obtained in Example 5.3
we get the following results for tn .
Case 1. For q real tn = 0 when n is even and tn = 2 when n is odd.
Case 2. For q imaginary tn = 1.
Conjecture 3.18 applies to π corresponding to πj (j = 3, 4) when tn = 2.
We finish this section with a result that was promised in Remark 3.20(1).
Note that the assumption dimE (E[G]π) = dimE (πK2n−1 (k)E ) is not needed
here: we consider the regulator map with values in π(⊕τ :k→C E ⊗ R(n − 1))+
obtained from (2.3), and use Beilinson’s regulator RBei comparing E-structures
as in Remark 3.24. Beilinson’s conjecture [2, Conjecture 3.4] states that forL(s, χ∨π ⊗ id, Q) the order of vanishing at s = 1 − n is dimE (πK2n−1 (k)E ) andthat the first non-vanishing coefficient of its Taylor expansion at s = 1 − n equals
eRBei for some e in E ∗ .
Proposition 5.9. Let k be a G = S3 -extension (resp. G = D8 -extension) of
Q, and let n ≥ 2. Then this conjecture holds for any primitive idempotent π
corresponding to π2 in Example 5.3 (resp. Example 5.5).
Proof. It suffices to prove this when E = Q. If G = S3 then by Remark 3.21(3)we may assume πH = π1 πH + π with π1 πH and π = π2 πH orthogonal idempotents, where we use notation as in Example 5.3. The conjecture for πH follows
from Theorem 2.2 for c = k H by the arguments of Remark 3.23 and the funcP
tional equation (2.6) of ζc (s). Similarly, it holds for π1 πH = 61 g∈G g, where itOn the p-adic Beilinson Conjecture for Number Fields

411

corresponds to that theorem for k G = Q, and it follows that it must hold for π as
well. If G = D8 the argument is identical, using Theorem 2.2 for k1 and q1 . ¤6. Computing p-adic L-functions
We briefly sketch the method used to compute values of p-adic L-functionsof 1-dimensional Artin characters. Values of p-adic L-functions of characters ofhigher dimension can then be deduced using (2.11). The method we employ
is somewhat similar to, but more technical than the construction in [18], andgeneralizes the one used in [44] to compute values of p-adic L-functions of real
quadratic fields at s = 1. Note that several modifications are necessary comparedto [18] since the construction there is for partial ζ-functions of the ray class groupmodulo p whereas we need to work with partial ζ-functions of arbitrary ray classgroups. Also, we need to replace p = 2 in a certain number of results by q = 4.
We leave it to the careful reader to check that these changes are indeed possible,
or refer to the forthcoming article [43] where the details of the actual method
used will be given.
Let k be a totally real number field and let d = [k : Q]. Let χ be a Qp -valued
1-dimensional Artin character of Gal(k/k). As in Section 2, we let q = p if p is
odd, and q = 4 if p = 2.
By class field theory, χ can be viewed as a character of a suitable ray classgroup of k. We denote by f the conductor of χ, which is an integral ideal of k, andwe let g be the modulus with infinite part g∞ consisting of all infinite places of k,and finite part g0 equal to the least common multiple of f and q. Let a1 , . . . , ahbe representatives of the classes of the ray class group of k modulo g, and let
ζai (s), i = 1, . . . , h, be the corresponding partial ζ-functions. According to [39,
Chapter VII, p.526] we have, for any isomorphism σ : Qp → C,hXσ ◦ χ(ai )ζai (s) = Eulp (s, σ ◦ χ, k)L(s, σ ◦ χ, k) .

i=1

Since all ζai (m) are rational for integers m ≤ 0 by loc. cit. Chapter VII, Corollary 9.9, we may identify them with σ −1 (ζai (m)) in Qp . Thus for such m we
have
(6.1)

h
X
i=1

χ(ai )ζai (m) = Eul∗p (m, χ, k)L∗ (m, χ, k)

412

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot

where L∗ (m, χ, k) and Eul∗p (m, χ, k) are as in (2.7) and (2.8).
Now let β 6= 1 in Ok be such that(1) β ≡ 1 (mod g0 ) and v(β) > 0 for every infinite place v of k;(2) Ok /(β) ' Z/bZ, where b = N (β) is the norm of the principal ideal (β).
Scaling the measures obtained in [18, Lemme 4.4] for the ai we obtain Qp -valuedei (i = 1, . . . , d) on Zd , depending on β and ai , such that for all integers
measures λpm ≤ 0,
Z
m−1
dm
−m
ei .
(b
− 1)ζa (m) = (−1) N (ai )(x1 · · · xd )−m dλi

The proof of Lemma 4.3 of loc. cit. shows, contrary to the statement of thatei are supported on (−1 + qZp )d . Therefore pullinglemma, that the measures λback over multiplication by −1 on Zdp we obtain measures λi on Zdp , supportedon (1 + qZp )d , such that, for the same m,Z(6.2)(bm−1 − 1)ζai (m) = N (ai )−m (x1 · · · xd )−m dλi .
For a fixed x in 1 + qZp the function s 7→ xs is defined and analytic on Zp .
Since, in general, N (ai ) will not be congruent to 1 modulo q, we do the following.
For x in Z∗p let hxi = xωp (x)−1 where ωp is the Teichmüller character on Z∗p(see (2.14)), so that hxi is in 1 + qZp . Then for any integer l and any integral
ideal a of k that is coprime to p, the function s 7→ ωp (a)l hN (a)i−s is an analytic
Zp -valued function on Zp , whose value equals N (a)−m at any integer m withm + l ≡ 0 modulo ϕ(q). We define the p-adic partial ζ-function of the class of ai
by mapping s in Zp \ {1} to
Z 0
¡ s−1
¢−1
−s
ζp,ai (s) = b−1hN (ai )i(x1 · · · xd )−s dλi ,where the prime indicates that we restrict the domain of integration to (1+qZp )d .
Let Ψi be the function defined by the above integral, that is, for s in Zp ,Z 0
Ψi (s) =(x1 · · · xd )−s dλi .
For a in Zp with |a−1|p ≤ q −1 , and s in Zp , one can write a−s = expp (−s logp (a)),
where expp and logp are the p-adic exponential and logarithm functions respectively; and thus the function s 7→ a−s can be expressed as a power series with coefficients in Zp , where the coefficient of sm has absolute value at most q −m pm/(p−1) .

On the p-adic Beilinson Conjecture for Number Fields

413

Developing (x1 · · · xd )−s in this way as a power series of s and using the fact thatthe measures λi have bounded norm by [18, Lemme 4.2 bis], we see that Ψi (s)can also be expressed as a power series in Qp [[s]] where the absolute value of thecoefficient of sm is at most Ci q −m pm/(p−1) for some Ci > 0. In the same way,
the function s 7→ hN (ai )i−s can be expressed as a power series in Zp [[s]] whose
coefficients satisfy the same bounds. Similarly, since the only solution for s ofbs−1 = 1 with |s| < qp−1/(p−1) is s = 1, it follows from the p-adic Weierstrass
preparation theorem that the function s 7→ bs−1 − 1 can be expressed as the
product of a power of p, the polynomial s − 1, and an invertible power series in
Zp [[s]] with the absolute value of the coefficient of sm bounded by q −m pm/(p−1) .
Therefore ζp,ai (s) can be expressed as the quotient of a power series in Qp [[s]]with the absolute value of the coefficient of sm bounded by Ci0 q −m pm/(p−1) with
Ci0 > 0, and s − 1.
If we define p-adic functions of s,
(6.3)

L(l)
p (s, χ, k) =hXωp (ai )l χ(ai )ζp,ai (s)(l modulo φ(q)) ,i=1

then it follows from (6.1), (6.2), and the equality ζp,ai (m) = ωp (ai )m ζai (m) for
any integer m ≤ 0, that
(6.4)

∗
∗
L(l)
p (m, χ, k) = Eulp (m, χ, k)L (m, χ, k)

for all non-positive m such that l + m ≡ 0 modulo ϕ(q). In particular, the p-adic(−1)
L-function of χ is given by Lp (s, χ, k) = Lp (s, χ, k). Moreover, its residue at
s = 1 is zero if χ is non-trivial because the functions ωp (ai )−1 ζp,ai (s) all have thesame residue there by [18, Corollaire on page 388].
The estimates on the coefficients of the power series expansions above imply theclaims in Theorem 2.9(1). Through the p-adic Weierstrass preparation theorem
they also imply that the function in (2.11) can be written as the quotient of apower series in Qp (χ)[[s]] that converges for s in Cp with |s| < qp−1/(p−1) , and apolynomial in Qp (χ)[s].
We note in passing that if l is any integer thenLp (s, χωpl , k) =

h
X
i=1

ωp (ai )l−1 χ(ai )ζp,ai (s) = L(l−1)(s, χ, k) ,p

414

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot

so that we obtain from (6.4) that
Lp (m, χωpl , k) = Eul∗p (m, χ, k)L∗ (m, χ, k)
for all m ≤ 0 with l + m ≡ 1 modulo ϕ(q), as stated in Remark 2.13.
In order to compute L(s, χ, k) for a fixed s in Zp , it suffices to compute ζp,ai (s),for which one mainly has to compute the Ψi (s). The measures λi can be represented as power series in d variables,X
Fi (w1 , . . . , wd ) =ai,n1 ,...,nd w1n1 · · · wdnd ,n1 ,...,nd ≥0with (bounded) coefficients
ai,n1 ,...,nd =

Z µ

¶
µ ¶
x1
xd
···
dλi
n1
nd

that can be computed using Shintani’s method [48], which is practical for calculations if the base field k is Q or real quadratic. On the other hand, the function
h on Zp defined byx−s if x is in 1 + qZ ,
p
x 7→
0
otherwise,¡ ¢P
is continuous and thus admits a Mahler expansion n≥0 cs,n nx , where the coefficients satisfy cs,n → 0 when n → ∞ and can be easily computed recursively.
Then we haveZX
Ψi (s) = h(x1 ) · · · h(xd ) dλi =ai,n1 ,...,nd cs,n1 · · · cs,nd .
n1 ,...,nd ≥0

Thus, by computing enough terms in the above sum, we can get a good approximation of the integral, and therefore, by putting everything together, of thep-adic L-function.
In order to apply this we make (2.10) explicit in the cases we are interested in,writing L-functions of representations for the L-functions of the corresponding
characters. In particular, we write the classical L-functions of the irreducible
2-dimensional representations of S3 and D8 in Examples 5.3 and 5.5, and theirreducible 2-dimensional representations Wa with a 6= 1 of S3 × Z/3Z in Example 5.7, as the L-functions of Abelian characters over quadratic fields. For later
use we also note how to write some of them in terms of ζ-functions of certain
fields.

On the p-adic Beilinson Conjecture for Number Fields

415

We take the field of coefficients E to be C in the remainder of this section, anddenote the trivial 1-dimensional representation of any group G by 1G .
Example 6.5. Let AltS3 be the 1-dimensional representation of S3 through thesign character, and V the irreducible 2-dimensional representation. Since
S3
2
3
∼
∼
IndSA33 ◦ IndA{e} (1{e} ) = Ind{e} (1{e} ) = 1S3 ⊕ AltS3 ⊕ V∼ 1S ⊕ AltS it follows that IndS3 (V 0 ) ∼and IndSA33 (1A3 ) == V for either non3
3
A3
0
trivial 1-dimensional representation V of A3 . Applying this to the situation of
Example 5.3 we see that L(s, V, Q) = L(s, V 0 , q). Similarly, if H is any subgroupof S3 of order 2, then IndSH3 (1H ) ∼= 1S3 ⊕ V . In the situation of Example 5.3 thismeans that ζc (s) = ζQ (s)L(s, V, Q).
Example 6.6. If D8 is as in Example 5.5, Va,b for a, b = ±1 is the 1-dimensional
representation of D8 where σ acts as multiplication by a and τ by b, and V is theirreducible 2-dimensional representation of D8 , then for any subgroup H of D8
H
2
8
∼
IndD
H ◦ Ind{e} (1{e} ) = (⊕a,b=±1 Va,b ) ⊕ V .

∼If H = hτ, σ 2 i then IndH{e} (1{e} ) = ⊕a,b=±1 Wa,b with Wa,b the 1-dimensional
representation of H where σ 2 acts as multiplication by a and τ by b. Because
D8
D8
8
∼
∼
∼
IndD
H (W1,b ) = ⊕a=±1 Va,b we have IndH (W−1,1 ) = IndH (W−1,−1 ) = V , so withnotation as in Figure 5.1 we get L(s, V, Q) = L(s, W−1,1 , q1 ) = L(s, W−1,−1 , q1 ).
8∼
Similarly IndDhτ i (1hτ i ) = (⊕a=±1 Va,1 ) ⊕ V so that ζk1 (s) = ζq1 (s)L(s, V, Q) as well.
∼ 4
Remark 6.7. If H = hσi in Example 6.6 then IndH{e} (1{e} ) = ⊕m=1 Uim with Ua
the 1-dimensional representation of hσi where σ acts as multiplication by a. Now
D8
8
∼
∼
IndD
H (Ua ) = Va,1 ⊕ Va,−1 when a = ±1 so that IndH (Ua ) = V when a 6= ±1,
and in Example 5.5 we also have L(s, V, Q) = L(s, Ui , q2 ) = L(s, U−i , q2 ).
Example 6.8. If G = S3 × Z/3Z and H = A3 × Z/3Z then
H
2
2
2
∼
IndG
H ◦ Ind{(e,0)} (1{(e,0)} ) = W ⊕ W1 ⊕ Wζ3 ⊕ Wζ 2
3

with W the direct sum of the 1-dimensional representations of G and the Wa∼
as in Example 5.7. Write IndH{(e,0)} (1{(e,0)} ) = ⊕a,b Va,b where, for a and b cube
roots of unity, Va,b is the 1-dimensional representation of H on which (σ m , n) acts∼
as multiplication by am bn . Since IndG
H (⊕b V1,b ) = W we find by considering the∼action of (e, 1) that IndG
H (Va,b ) = Wb when a 6= 1. In particular, in the notationof Example 5.7, we have L(s, Wb , Q) = L(s, Va,b , q) since q = k H .

416

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot

It is easy to see that the coefficients of p−s in the Euler factors of L(s, Wζ3 , Q)
and L(s, Wζ32 , Q) are conjugate, but since not all of them can be real it is notpossible to express those functions in terms of ζ-functions of subfields of k.

7. Examples
In this section we describe the evidence for Conjecture 3.18 in the case ofprimitive idempotents π in E[Gal(k/Q)] as in Examples 5.3, 5.5 and 5.7, with
E = Q in the first two and E = Q(ζ3 ) in the last. In the first two cases part (1)of the conjecture is known by Proposition 5.9, but in the last case our evidence
for this part is numerical; the same holds for parts (2) through (4) in all cases.
We would like to stress that the statement about the value of the p-adic Lfunction in part (4) of the conjecture is actually proved by its numerical verification in all cases that we considered since we can check that an element of E ⊗ Qp
is a unit of that ring by computing a close enough approximation of it. The same
holds for the statement about the p-adic regulator when n = 2 or 3 since then,in the cases we considered, the calculation in Zagier’s conjecture is done purely
algebraically so we know we find a subgroup of the K-group. (For higher valuesof n the calculation depends on high-precision calculations of regulators, which
doesn’t allow us to conclude this.)
For the numerical calculations of the p-adic regulator, Rn,p (MπE ), we assumed
that the syntomic regulator map on the subgroup of K2n−1 (k)Q described inσ (as in Remark 4.15). Out of theZagier’s Conjecture 4.5 is given by the Pen,pmany possible choices for Pn,p (z) as described in (4.12) we used
Pn,p (z) = Lin,p (z) − logn−1(z)Li1,p (z)/n! = Lin,p (z) + logpn−1 (z) logp (1 − z)/n!
psince it is relatively simple from a computational point of view. The calculationsof the Pn,p (z) for the branch with logp (p) = 0 were done in versions 2.11-7, 2.12-19
and 2.12-21 of [51], using an implementation of the algorithm described in [6].
Note that the primitive idempotents in Examples 5.3 and 5.5 are unique up
to conjugation in Q[G] since they all correspond to π2 . But in Example 5.7
replacing ζ3 with ζ32 changes a primitive idempotent corresponding to π3 into one
corresponding to π4 , and conversely. Because the same holds for the classical andp-adic L-functions (with values in E ⊗Q C and E ⊗Q Qp respectively), as well asOn the p-adic Beilinson Conjecture for Number Fields

417

Table 1. Splitting field of x3 − x2 − 3x + 1, where C(χπ ) = 22 · 37.
e(3, Mπ )−1 = 2−2 · 3−2 · 372

n=3p

R3,p (Mπ )/D(Mπ)1/2,p2
3
5(1.00010101001000000010000101111100100010101110100110010110110011000000)2 × 20
(1.010012121202211021200001102121121100201201100121200211021212)3 × 34
(4.2320143330214023104113344110103131003140)5 × 56

7
11

(6.354304301412412415450326016336635)7 × 76
(2.62161235A928A3423563A7888A)11 × 116p

Lp (3, χπ ωp−2 , Q)

2
3
5
7
11

(1.00000010100111000100001100100011111010110000101001101110011000101000)2 × 22
(1.1202122002221002110112100012002001211111011101010220)3 × 30
(4.11040024440232442233024131040014140)5 × 50
(5.23516363226501261362543533110)7 × 70
(A.9542728A692401225487A278)11 × 110
e(5, Mπ )−1 = 24 · 32 · 5−2 · 374

n=5p

R5,p (Mπ )/D(Mπ)1/2,p2(1.00001010110110011101000011101001000001100000100010000011011110011101)2 × 26

3

(1.200111022020102011202110020222210102001020101122110122022)3 × 312

57
11(3.214210342400224303122204142313310423002)5 × 58
(1.2103004405355450123124426604231)7 × 710(1.2069AA607609242834655465)11 × 1110p

Lp (5, χπ ωp−4 , Q)

2
3
5
7
11

(1.00000101111001110001111000010111110100101010001010001001100111100011)2 × 22
(1.2022122002112102001110211120022002211212022201001210)3 × 30
(3.34404422113121141420412040113130342)5 × 50
(4.66001565513622316645262615350)7 × 70
(1.1A26918431A09200099A81A7)11 × 110

the Euler factor for p, the conjectures for primitive idempotents corresponding
to π3 and π4 are actually equivalent.
In Examples 7.1, 7.4 and 7.5 we have E = Q so that we identify E ⊗Q C with
C, etc., and use notation as in Remark 3.17.
Example 7.1. Let k be a totally real S3 -extension of Q. Use notation as inCase 1 of Example 5.3, and let π be the primitive idempotent corresponding to
π2 of that example. Since E = Q we shall write Mπ instead of MπE in all notationreferring to Conjecture 3.18.
Since πK2n−1 (k)Q = π2 K2n−1 (c)Q by Remark 3.22 we computed the K-theoryof c using the (numerical) methods of Section 5 and tried to recognize the number(7.2)

e(n, Mπ ) = L(n, χπ , Q) · D(Mπ )1/2,∞ /Rn,∞ (Mπ )418

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot
Table 2. Splitting field of x3 − 6x − 2, where C(χπ ) = 22 · 33 · 7.
e(3, Mπ )−1 = −2−2 · 34 · 72

n=3p

R3,p (Mπ )/D(Mπ)1/2,p235(1.10000001110000110101010101111101011001000101111010100001000101111010)2 × 20
(2.1122020011220001210112202100011222122000102121211020022012)3 × 34
(3.4232302100220101433222440213432224202143)5 × 56

7
11

(4.2544363600510010013523314315122)7 × 77
(5.481569541633A6875525A00911)11 × 116p

Lp (3, χπ ωp−2 , Q)

2
3
5
7
11

(1.01111101110011001100110111100111100010100000100110111100101110110010)2 × 22
(1.1211000101210110112021221022022211101101102021011220)3 × 30
(2.13000020122302011221412322140400441)5 × 50
(4.55510152416245405156545556000)7 × 72
(1.14A21577033A19227344806A)11 × 110
e(5, Mπ )−1 = −24 · 314 · 5−2 · 74

n=5p

R5,p (Mπ )/D(Mπ)1/2,p2(1.10101101100000101100101100100110101101111111000110011100110000011111)2 × 26

3

(2.120122222011210012110212011212210211022212221020022201)3 × 314

57
11(4.040210142044334334412003042243311012202)5 × 58
(3.2403415331415026502116243110353)7 × 79
(7.160964096503643537518492)11 × 1110p

Lp (5, χπ ωp−4 , Q)

2
3
5
7
11

(1.01010111000010111011111101010101111011001011010111110100101100010111)2 × 22
(1.2002000121100001212101001020022211022201122100111000)3 × 30
(4.44311102333442121212402410333144343)5 × 50
(3.41526030363465662002262442000)7 × 70
(2.687349321A27130485753530)11 × 110

for n = 3 and 5 as an element of Q∗ by employing the same methods as describedin Section 5 for number fields, where L(n, χπ , Q) was computed as ζc (n)/ζQ (n)(see Example 6.5) using Pari-GP [52]. We succeeded in all cases, and then verified
numerically for p = 2, 3, 5, 7 and 11 if Lp (n, χπ ωp1−n , Q) 6= 0 and(7.3)

e(n, Mπ ) · Eulp (n, χπ , Q) ·Rn,p (Mπ )· Lp (n, χπ ωp1−n , Q)−1
D(Mπ )1/2,pin Q∗p was equal to 1.
We put our results for the four totally real S3 -extensions k of Q with smallest
discriminants in Tables 1 through 4. Note that Rn,p (Mπ )/D(Mπ )1/2,p is in Qp
by Lemma 3.7 and the same holds for Lp (n, χπ ωp1−n , Q) by Lemma 3.15, so that(approximations of) those numbers can be easily represented in our tables. We

On the p-adic Beilinson Conjecture for Number Fields

419

Table 3. Splitting field of x3 − 4x − 1, where C(χπ ) = 229.
e(3, Mπ )−1 = −2−9 · 3−2 · 2292

n=3p

R3,p (Mπ )/D(Mπ)1/2,p2
35(1.01110100101010000001111100101001000110110101000001001011100111100010)2 × 20
(2.012012211010121011021101220021012212100110002220200211021221)3 × 34
(4.3433241433301132322003123241441021322342)5 × 56

7
11

(6.262401634562505353633061004510116)7 × 76
(3.6717A20401481085265342054)11 × 117

p

Lp (3, χπ ωp−2 , Q)

2
3
5
7
11

(1.01100001011001101110100010111011001000000110000010000100110111111101)2 × 23
(2.2120211202111122100220011211121110002020101200021220)3 × 30
(3.03414332241433441404241012442104412)5 × 50
(3.24033044132523034433054506150)7 × 70
(9.21A61A480A10801760893710)11 × 111
e(5, Mπ )−1 = −2−8 · 36 · 11 · 2294

n=5p

R5,p (Mπ )/D(Mπ)1/2,p2(1.11110110101010011011010001011100111001101011001101110111000000010001)2 × 25

3

(2.222112021012010001201220011021022211112022112001020100)3 × 316

57
11(4.3414223144333221124000410221113420102)5 × 510(2.5612036605646266635264251053110)7 × 710(8.5A9220278400758A0193AA6)11 × 1111

p

Lp (5, χπ ωp−4 , Q)

2
3
5
7
11

(1.01010000111010100010011010011100110010001101000101101100101100100011)2 × 23
(2.1121000022222000210222202101202022011021200021010220)3 × 30
(4.42011132342110332341130324344440134)5 × 50
(1.25032245164656402236525435500)7 × 70
(5.375705346AA0810959728302)11 × 110

have denoted an element (a0 +a1 p+a2 p2 +· · · )×ps in Q∗p with all aj in {0, . . . , p−1}
and a0 6= 0 by (a0 .a1 a2 · · · )p × ps , writing A to represent 10 when p = 11.
For the p-adic regulators we computed each Pn,p (z) up to O(pL(p) ) with pL(p)approximately equal to 1030 . The relevant part of Rn,p (Mπ )/D(Mπ )1/2,p has
been given in the tables whenever it fitted. For the values of the p-adic Lfunction we can only prove that the relative error is at most 1 + O(pM (p) ), where
M (2) = 72, M (3) = 47, M (5) = 32, M (7) = 26 and M (11) = 22. This provesthat Lp (n, χπ ωp1−n , Q) does not vanish; together with the verification that (7.2)
equals 1 numerically this proves part (4) of Conjecture 3.18 when n = 3 and makes
it very plausible when n = 5. The value that we found for (7.3) was 1 + O(pN (p) )with N (2) ≥ 82, N (3) ≥ 52, N (5) ≥ 36, N (7) ≥ 27 and N (11) ≥ 24, giving
numerical evidence for parts (2) and (3) of the conjecture, but also suggesting420

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot
Table 4. Splitting field of x3 − 6x2 + 2, where C(χπ ) = 22 · 34 · 5.
e(3, Mπ )−1 = 2−2 · 36 · 5

n=3p

R3,p (Mπ )/D(Mπ)1/2,p2
35(1.11110010110000001011011110010111010011100101111011011000111111110001)2 × 20
(2.120020100000012102010221000201210120200202222221111002100)3 × 36
(3.3432222440014423344440441124111231400414)5 × 54

7
11

(5.611651266622656555064166304513241)7 × 76
(2.805A8344265760A69A2490828A)11 × 116p

Lp (3, χπ ωp−2 , Q)

2
3
5
7
11

(1.10100000100110111110111110101101001001110101011111100111101100001111)2 × 22
(1.1120211121012011012202202010100211210110122222201210)3 × 30
(2.13242313233011201133333143412022134)5 × 50
(3.00123442142613541412624063510)7 × 70
(2.2725A17844929880A8412281)11 × 110
e(5, Mπ )−1 = −25 · 318 · 52 · 324762301

n=5p

R5,p (Mπ )/D(Mπ)1/2,p2(1.00010100011011001110000011110010000100001101100111101000101001001110)2 × 27

3

(1.0200220011011111211112101212120212201200221122102001)3 × 318

57
11(1.321100221113142104412032003401031323110)5 × 57
(3.5553063665305405650105042513546)7 × 710(A.568607725325302A5663A746)11 × 1110p

Lp (5, χπ ωp−4 , Q)

2
3
5
7
11

(1.10111101010011001111001000001111111000110110101100010100000000001010)2 × 22
(1.0002101112212121121001002201100122111211210111102110)3 × 30
(2.30334130213120322410240113023441000)5 × 50
(4.62042101122332630656130062420)7 × 70
(5.491153185997A48326862A46)11 × 110

that the relative precision of the value of the p-adic L-function is slightly higher
than we can prove, justifying the higher precision given in the tables.
Similar calculations were carried out for the next four such S3 extensions andthe same primes, with very similar results.
The other examples proceed mostly along the same lines as Example 7.1. The
values of L(n, χπ , Q) in Examples 7.4 and 7.5 below were computed using theformula at the end of Example 6.6, but in Example 7.6 the values of L(n, χE
π , Q),
with E = Q(ζ3 ) for ζ3 a primitive cube root of unity, were computed using thealgorithms described in [24] and its associated program. In Examples 7.4 and 7.5
we determined if e(n, χπ ) (as in (7.2)) was in Q∗ using the methods of Section 5,
and for Example 7.6 we wrote e(n, MπE )−1 = 1 ⊗ a + ζ3 ⊗ b in E ⊗Q R and thenrecognized a and b as integers divided by products of some relatively small primes.

On the p-adic Beilinson Conjecture for Number Fields

421

Table 5. Splitting field of x4 − x3 − 3x2 + x + 1, where C(χπ ) = 5 · 29.
e(3, Mπ )−1 = 2−6 · 3−1 · 292

n=3p

R3,p (Mπ )/D(Mπ)1/2,p2
35(1.11111010111100011100011000110000111000001111111101101000111001001001)2 × 22
(1.201220211120010110010200120221012101101121011201212202210212)3 × 35
(2.21221421224100432231134432324224211113143)5 × 53

7
11

(2.260524146626624605001621634435032)7 × 76
(2.74697855444977862531A25026)11 × 116p

Lp (3, χπ ωp−2 , Q)

2
3
5
7
11

(1.01010010100111000101001000111001101011100000001100110101111010001100)2 × 22
(1.1011210202202201201020222100121100002100020220220021)3 × 30
(4.30210324423422014142420103340302330)5 × 50
(1.16355063143251352506540201034)7 × 70
(9.9A7720A981020A9A32071790)11 × 110
e(5, Mπ )−1 = 2−4 · 32 · 52 · 294

n=5p

R5,p (Mπ )/D(Mπ)1/2,p2(1.01101101001101100011000000000011101110110010001010001100111111101001)2 × 28

3(1.0002011102111201211120012012011221112210212111112002202122)3 × 312

57
11(1.030204001130134123130241412102302410321)5 × 57
(5.6213331545344431613600254300115)7 × 710(9.98414459608789A37989AA86)11 × 1110p

Lp (5, χπ ωp−4 , Q)

2
3
5
7
11

(1.01111100101100101110101001110010110011011010001011101111000000111001)2 × 22
(1.1111012010001010110000200000021011010010220212100211)3 × 30
(4.01341421411140002024312030010110013)5 × 50
(4.10032134632553222245313526015)7 × 70
(8.6637671480756A9916284377)11 × 110

In each case we then verified for p = 2, . . . , 11 if (7.3) (or its equivalent) was equal
to 1 in Qp or, for Example 7.6, in Q(ζ3 ) ⊗Q Qp . The precision for Pn,p (z) and thep-adic L-functions in Examples 7.4 and 7.5 was as in Example 7.1. In Example 7.6
all Pn,p (z) were computed up to O(pL(p) ) with pL(p) approximately 1016 since inthis case the check if the equivalent of (7.3) is 1 is done in Q(ζ3 ) ⊗Q Qp so twocoefficients are checked; the values of the p-adic L-function in this case, which
also lie in Q(ζ3 )⊗Q Qp , were determined up to multiplication by 1⊗(1+O(pM (p) ))with M (p) as in Example 7.1.
Example 7.4. Let k be a totally real D8 -extension of Q as in Case 1 of Example 5.5, and use notation as in that example. We again write Mπ instead of MπE
since E = Q and consider odd n ≥ 2. Because πK2n−1 (k)Q = π2 K2n−1 (k1 )Q we
computed K2n−1 (k1 )Q numerically using the methods of Section 4. Our resultswith n = 3 and 5 for the first four such extensions when ordered according to422

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot
Table 6. Splitting field of x4 − 2x3 − 3x2 + 4x + 1, where C(χπ ) = 22 · 32 · 11.
e(3, Mπ )−1 = 2−2 · 32 · 112

n=3p

R3,p (Mπ )/D(Mπ)1/2,p2
35(1.11001111111100110011011000011011110100000010001100010000111000001110)2 × 25
(1.101222112200111120002122200210022002021121200101112022212121)3 × 32
(1.3213042203212141302102134014142431100344)5 × 56

7
11

(1.444434205533111532005253505615656)7 × 76
(9.61613555149153993505670160)11 × 115

p

Lp (3, χπ ωp−2 , Q)

2
3
5
7
11

(1.11110101101001001100111001001100001010010111011100011010101000010001)2 × 24
(1.1221211120021010011021120022021022220202012220120020)3 × 30
(1.03123432421241311210024333243134200)5 × 50
(1.20515104501535164355503412253)7 × 70
(4.A19905A97667183A61256312)11 × 110

n=5e(5, Mπ )−1 = 27 · 310 · 5−2 · 117 · 151 · 1389251

p

R5,p (Mπ )/D(Mπ )1/2,p2(1.10101000100011001001110000000110011010000011010001100100101100100011)2 × 216

3(2.00020011020011210121200021121000100102202022102012021120)3 × 310

57
11(3.043124340101142041302443131432222411101)5 × 58
(3.1016042121151553442503432550426)7 × 710(4.A244AA713986547114383)11 × 1113

p

Lp (5, χπ ωp−4 , Q)

2
3
5
7
11

(1.01011001111000100100100110110111001110010101101110101100000001001101)2 × 24
(1.2200111010220200001122120000000211211201112212022102)3 × 30
(4.31111224412403210010123414333414223)5 × 50
(2.63463651641424006321031014001)7 × 70
(2.054143779893294A023A2760)11 × 111

the discriminant are in Tables 5 through 8. The value that we found for (7.3) inthis case was 1 + O(pN (p) ) where N (2) ≥ 76, N (3) ≥ 52, N (5) ≥ 34, N (7) ≥ 29
and N (11) ≥ 22. Such calculations were also carried out for the next four suchextensions, with similar results.
Example 7.5. Let k be a CM Galois extension of Q with Gal(k/Q) ∼= D8 asin Case 2 of Example 5.5, and use notation as in that example. We again write
Mπ instead of MπE since E = Q, and consider even n ≥ 2. Again πK2n−1 (k)Q =π2 K2n−1 (k1 )Q , so we computed K2n−1 (k1 )Q numerically using the methods ofSection 4. We put our results at n = 2 and 4 for the first four such extensions
when ordered by the value of the (positive) discriminant in Tables 9 through 12.
The value that we found for (7.3) in this case was 1 + O(pN (p) ) where N (2) ≥ 79,
N (3) ≥ 53, N (5) ≥ 35, N (7) ≥ 27 and N (11) ≥ 25. Such calculations were also
carried out for the next four such extensions, with similar results.

On the p-adic Beilinson Conjecture for Number Fields

423

Table 7. Splitting field of x4 − 6x2 + 6, where C(χπ ) = 27 · 32 .
e(3, Mπ )−1 = −28 · 32

n=3p

R3,p (Mπ )/D(Mπ)1/2,p2
35(1.11111110110101110110001001001000100111010011011111000101011110110110)2 × 210(2.022112012020121000012010222021210101120000211220112222110100)3 × 32
(2.041234240331110012313440303434111221114)5 × 57

7
11

(4.126126412601443066506634321456330)7 × 76
(1.508753A40028580A83265A6397)11 × 116p

Lp (3, χπ ωp−2 , Q)

2
3
5
7
11

(1.00111001111010100110000110000111111000001101110001010000100110100110)2 × 22
(1.1102020120122110111012211100112220010211120222001022)3 × 30
(3.24102423421030322330312322041041320)5 × 51
(3.53421553111552403452502135351)7 × 70
(9.993168A73395501045A926A6)11 × 110

n=5e(5, Mπ )−1 = −225 · 310 · 5−1 · 11 · 37 · 180097

p

R5,p (Mπ )/D(Mπ )1/2,p2(1.01000011110101000111110110010101100000101100100001110011010100110111)2 × 227

3(1.20110210022121202120201021000120210110011111202222221202)3 × 310

57
11(4.13321010222213321143100244321412404344)5 × 59
(1.6542260045104311315015635541014)7 × 710(4.35786968692892127976707)11 × 1111

p

Lp (5, χπ ωp−4 , Q)

2
3
5
7
11

(1.00100100101111111001101001011101011111110110100110100000111111010001)2 × 22
(1.0101222010121001211100001210022220120122121200102122)3 × 30
(2.44334240213144033340330341233100231)5 × 50
(2.43613231343555620664002413300)7 × 70
(A.957125591551731675713855)11 × 110

Example 7.6. Let k be a totally real S3 × Z/3Z extension of Q, as in Case 1of Example 5.7. We use notation as in that example, taking E = Q(ζ3 ) for aprimitive cube root of unity ζ3 . A different identification of Gal(k/Q) with S3 ×
Z/3Z might interchange π3 and π4 as well as corresponding primitive idempotents,
but, as we noted before, the validity of the conjecture for either is equivalent.
Since πK2n−1 (k)E = π3 K2n−1 (s1 )E we computed K2n−1 (s1 )Q as before. Also,in order to speed up the calculations of the regulator we note that the action of(e, 1) and (σ 2 , 0) on s1 = k h(σ,1)i is the same, so that π = π3 πH1 and(1 − π1 )((e, 0) + ζ3 (σ, 0) + ζ32 (σ 2 , 0))/3 = ((e, 0) + ζ3 (σ, 0) + ζ32 (σ 2 , 0))/3
give the same result when applied to K2n−1 (s1 )E . We put our results for the
first four s1 , when ordered according to their discriminant, for n = 3 and 5 in

424

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot
Table 8. Splitting field of x4 − 6x2 − 4x + 2, where C(χπ ) = 25 · 17.
e(3, Mπ )−1 = −22 · 3−2 · 172

n=3p

R3,p (Mπ )/D(Mπ)1/2,p235(1.10111000110111011100110001011111110001011110100001100011100001101001)2 × 27
(1.1001122110011102222210022111010102000101002122001210120111212)3 × 34
(2.1330133431400143111041343243044132341000)5 × 56

7
11

(1.251366021640656215554304534633666)7 × 76
(6.8781153571182A7A3A52578590)11 × 116p

Lp (3, χπ ωp−2 , Q)

2
3
5
7
11

(1.01100101000011011110000111011100100110110110111011000100000101100110)2 × 25
(2.2111111121010122120102220010120102202210101102021112)3 × 30
(2.03010131243321342244201240202313243)5 × 50
(2.45653546525345623002012601006)7 × 70
(1.5054539309942A8686113521)11 × 110
e(5, Mπ )−1 = −214 · 32 · 5−2 · 174

n=5p

R5,p (Mπ )/D(Mπ)1/2,p2(1.01101001100111100011100000011101100011011111111011011010000000100111)2 × 219

3(1.0022021200111200200110022100110101220100021201020020021001)3 × 312

57
11(4.41122221322441142131444023001440141102)5 × 59
(4.6425656621000232634542266362653)7 × 710(5.30669526A542413A15559106)11 × 1110p

Lp (5, χπ ωp−4 , Q)

2
3
5
7
11

(1.10000111001100100001010000010001000011100111101011000000110111100101)2 × 25
(2.1100001122200222122120000002011201000121010120122102)3 × 30
(1.00313434131411434403030110103232430)5 × 51
(4.51404541504052312152404613255)7 × 70
(2.A43315061300727A39739622)11 × 110

Tables 13 through 16. In Table 16 the number e(5, MπE )−1 is given by−25 · 310 · 5−2 · 134 · (7 + 3ζ3 ) · (68 − 43ζ3 ) · (1547202 − 1603487ζ3 ) · (a + bζ3 )with
a = 8482255892311139091186217686741233
andb = 3527440500058817421018094757947617 ,presumably because we experimentally found a subgroup of K2n−1 (s1 )Q that is
not close to actually being K2n−1 (s1 ). In all cases Lp (n, χπ ⊗ ωp1−n , Q) is a unit in
E ⊗Q Qp , and when we write the equivalent of (7.3) in the form 1⊗α+ζ3 ⊗β thenwe find that α − 1 and β are O(pN (p) ) with N (2) ≥ 50, N (3) ≥ 24, N (5) ≥ 20,

On the p-adic Beilinson Conjecture for Number Fields

425

Table 9. Splitting field of x4 − 2x3 + 5x2 − 4x + 2, where C(χπ ) = 23 · 17.
e(2, Mπ )−1 = −17n=2p

R2,p (Mπ )/D(Mπ)1/2,p2
35(1.11110000011111000100011010100111010001000101111111000010111001000001)2 × 28
(1.2001010102220112101111101220000000012020211022222012112010122)3 × 34
(2.01203020434342403202140220421203111323421)5 × 54

711

(4.1140305454665411023106103225364300)7 × 74
(9.074667649A54466861880828A41)11 × 114

p

Lp (2, χπ ωp−1 , Q)

2
3
5
7
11

(1.10011100010001001110011010011101110110111100101101100011101101010011)2 × 26
(1.2211021011200210000101010021221011201210202220121211)3 × 30
(4.33003403101112402103034043124123020)5 × 50
(6.55655254400302602216266140200)7 × 70
(4.84968AAA0466783629843316)11 × 110
e(4, Mπ )−1 = 24 · 32 · 173

n=4p

R4,p (Mπ )/D(Mπ)1/2,p2(1.01111010101011110100100110010011000001110001101011101000010011010010)2 × 215

3(2.02020202222222212011121000222200020002221002201220201002211)3 × 310

57
11(3.121141434441031100040041101320123332342)5 × 58
(4.30103246365105165401451263010636)7 × 78
(9.8864321414815928A83132426)11 × 118

p

Lp (4, χπ ωp−3 , Q)

2
3
5
7
11

(1.10101000111010101100110000110001001011001001001001010011001010011101)2 × 27
(1.0220110000101201002201220101112220112012211122211111)3 × 30
(4.24124003322330023340322023204112410)5 × 50
(1.34631136124161206563156313232)7 × 70
(6.484018160860423737043861)11 × 110

N (7) ≥ 14 and N (11) ≥ 12, with in this case the precision bounded by that of
the Rn,p (MπE ). Since e(n, MπE ), D(MπE )1/2,p and Eulp (n, χπ ⊗ id, Q) are units in
E ⊗Q F this also shows that Rn,p (MπE ) is a unit in E ⊗Q F (as conjectured in
Conjecture 3.18(4)) for n = 3 and makes it very plausible for n = 5.
Such calculations were also carried out for the next four such s1 with similar
results.
Remark 7.7. In Example 7.1, the conjecture for π and the non-vanishing ofLp (n, ωp1−n , Q) are equivalent to Conjecture 2.17 for the field c in the example by
Proposition 4.17 and Remark 3.21(1). Also, part (1) of the conjecture holds forπ by Proposition 5.9. The same statements are true for π and k1 in Example 7.4.
Remark 7.8. In the tables we have also included the value of the constantC(χπ ⊗ id) = C(χ∨π ⊗ id) in the functional equation (3.25) of L(s, χπ ⊗ id, Q)

426

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot
Table 10. Splitting field of x4 − x3 + 3x2 − 2x + 4, where C(χπ ) = 5 · 41.
e(2, Mπ )−1 = 2−3 · 5 · 41

n=2p

R2,p (Mπ )/D(Mπ)1/2,p2
3
5(1.00100101101010001111110110001011010111110011100101011001110011011000)2 × 26
(2.0021220111220210202122211101102111121010022202210012111112202)3 × 34
(4.2413230212134320403404223440242144143010)5 × 54

7
11

(5.6622665100264660251241453165220553)7 × 74
(2.0A380682536A50856633A5963AA)11 × 114

p

Lp (2, χπ ωp−1 , Q)

2
3
5
7
11

(1.10001100100010010110110100011001000110001101100111111011111110000000)2 × 25
(1.0000112110000221101100101100220112122121100200111220)3 × 30
(3.31322324200110323444242424141204000)5 × 51
(6.20642211231410105634103112315)7 × 70
(4.8488929AA248277415458863)11 × 110
e(4, Mπ )−1 = 2−5 · 32 · 53 · 413

n=4p

R4,p (Mπ )/D(Mπ)1/2,p2(1.01101011011011001001110001011011001001100001000100001111001011100010)2 × 29

3

(2.12201221002111122111110002102122122100002111212000210102220)3 × 310

57
11(3.03433213241214232102131324212010224400)5 × 58
(3.21341424531605465104510422214051)7 × 78
(A.180567AA0545A59652909A250)11 × 118

p

Lp (4, χπ ωp−3 , Q)

2
3
5
7
11

(1.11100001111100101110011000101000001100001100100001000011000001000111)2 × 26
(1.2110210100202111000001112120121020120200000202111102)3 × 30
(1.22414201312142130131112334430123200)5 × 51
(6.66200605216643655434600606423)7 × 70
(3.5059939560A2AA91106844A7)11 × 110

to make possible a comparison between its prime factors and those of e(n, Mπ ).
Namely, for the D(MπE )1/2,∞ used in the calculations, the en in (3.26) (which inour case are independent of n) are as follows.
Table en
1
3
2
3
3
3
4
3

Table en
5
4
6
4
7
4
8
4

Table en
9
4
10
4
11 −4
12
4

Table
en
13
1 ⊗ 9 + ζ3 ⊗ 9
14
1 ⊗ 9 + ζ3 ⊗ 915
1 ⊗ 0 + ζ3 ⊗ 9
16 1 ⊗ 0 + ζ3 ⊗ (−9)

As can be seen from these values and Tables 1-16, in the numbern−1
δn en ((n − 1)!/2)m C(χ∨e(n, MπE )π ⊗ id)On the p-adic Beilinson Conjecture for Number Fields

427

Table 11. Splitting field of x4 − 2x3 + 6x2 − 5x + 2, where C(χπ ) = 13 · 17.
e(2, Mπ )−1 = 2−3 · 13 · 17

n=2
p

R2,p (Mπ )/D(Mπ)1/2,p2
35(1.11100111011011110110000110110011010010001100010000011011000000100011)2 × 27
(1.001110110101022221010202101001220021110111221112100200010011)3 × 35
(4.12002342224442011223100142021013114003331)5 × 54

7
11

(6.0065051533250322312551556340446112)7 × 74
(9.58602382412A681650341722097)11 × 114

p

Lp (2, χπ ωp−1 , Q)

2
3
5
7
11

(1.01010001000000000001000000101011111111001100001000001100111111100100)2 × 26
(2.0212122200002021202200211122100200001001012201202000)3 × 31
(2.12300004023110442141103112001044200)5 × 50
(5.65035644055466446205106662353)7 × 70
(6.38803A213738A731776470A4)11 × 110
e(4, Mπ )−1 = −2−5 · 32 · 133 · 173

n=4p

R4,p (Mπ )/D(Mπ)1/2,p2(1.01010110010100100110100011110011010001011010001010001001110111101010)2 × 210

3(1.1111010110110112222112002001211122102011200022210022220110)3 × 311

57
11(2.300041000440134413002331411231121430312)5 × 58
(2.63246241360463115315231424221502)7 × 78
(A.49299953329A1125214A939A7)11 × 118

p

Lp (4, χπ ωp−3 , Q)

2
3
5
7
11

(1.00100111010100001110011000011111110110110001001100110010111000011110)2 × 27
(1.2001110201210120022112022021221001220010010202222220)3 × 31
(4.31111034231223303224440300102101300)5 × 50
(3.23250645010664211233114156026)7 × 70
(6.722A82921527466612856704)11 × 110

of (3.27) the larger prime factors occurring in e(n, MπE ) are often cancelled byn−1 . This suggests that the formulation of Beilinson’s conjecthose of C(χ∨π ⊗ id)ture at s = 1 − n normally involves simpler prime factors in E ∗ than the formulation at s = n in Conjecture 3.18(1), with the remaining complicated factors in
our examples quite possibly due to an awkward choice of basis for K2n−1 (MπE ).
References[1] D. Barsky. Fonctions zêta p-adiques d’une classe de rayon des corps de nombres totalement
réels. In Groupe d’Etude d’Analyse Ultramétrique (5e année: 1977/78), pages Exp. No. 16,
23. Secrétariat Math., Paris, 1978.
[2] A. A. Beilinson. Higher regulators and values of L-functions. J. Sov. Math., 30:2036–2070,
1985.
[3] A. Besser. Syntomic regulators and p-adic integration I: rigid syntomic regulators. Israel
Journal of Math., 120:291–334, 2000.

428

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot
Table 12. Splitting field of x4 + 3x2 − 6x + 6, where C(χπ ) = 22 · 32 · 11.
e(2, Mπ )−1 = −2−1 · 32 · 11n=2
pR2,p (Mπ )/D(Mπ)1/2,p2
3
5(1.00011111110011010111101100110001100000010001000010010011001000110100)2 × 26
(2.001121000222211121002102201101101100211202000001120001201101)3 × 32
(4.32211042044110431112434230011130322040334)5 × 54

711

(6.460003406606016165433631101363300)7 × 75
(4.355421A54525023347874A49229)11 × 113

p

Lp (2, χπ ωp−1 , Q)

2
3
5
7
11

(1.11101000100101000011100111111110011101011000001101111010111100101100)2 × 25
(1.2022010222110011100120101011011011020010020011221012)3 × 30
(3.22012020201133014223242323022213400)5 × 50
(2.43314511556206561022451602200)7 × 71
(7.514814537963976728A46001)11 × 110
e(4, Mπ )−1 = −2 · 38 · 113

n=4p

R4,p (Mπ )/D(Mπ)1/2,p2(1.00011101001000000111110111000101001111110011011101001101001001110101)2 × 211

3(2.0222010120001012000101112011222120000210000112021000101001)3 × 38

57
11

(4.200024100322411004323111441041141020122)5 × 58
(5.12536440256103300364331512021150)7 × 78
(6.6966231827671A91984A434A7)11 × 117

p

Lp (4, χπ ωp−3 , Q)

2
3
5
7
11

(1.10111000101101010100110110100100101011000111100000010010100110001101)2 × 26
(1.0221110122220100011212020012222021100000010022110011)3 × 30
(3.44200104443132341021214030231112200)5 × 50
(4.44003540500226200364534021346)7 × 70
(5.766584290652809A28501199)11 × 110[4] A. Besser. Syntomic regulators and p-adic integration II: K2 of curves. Israel Journal of
Math., 120:335–360, 2000.
[5] A. Besser and R. de Jeu. The syntomic regulator for the K-theory of fields. Annales Scientifiques de l’École Normale Supérieure, 36(6):867–924, 2003.
[6] A. Besser and R. de Jeu. Li(p) -service? An algorithm for computing p-adic polylogarithms.
Math. Comp., 77:1105–1134, 2008.
[7] A. Besser and C. Deninger. p-adic Mahler measures. J. Reine Angew. Math., 517:19–50,
1999.
[8] F. Beukers. Irrationality of some p-adic L-values. Acta Math. Sin. (Engl. Ser.), 24:663–686,
2008.
[9] S. Bloch. Higher regulators, algebraic K-theory, and zeta functions of elliptic curves. Manuscript from 1978 (“Irvine notes”). Published as volume 11 of CRM Monographs Series by
American Mathematical Society.
[10] A. Borel. Stable real cohomology of arithmetic groups. Ann. Sci. ENS, 4:235–272, 1974.

On the p-adic Beilinson Conjecture for Number Fields

429

Table 13. Splitting field of x6 − x5 − 6x4 + 7x3 + 4x2 − 5x + 1, where C(χπ ⊗ id) = 22 · 5 · 19.
n=3
p
2
3
5
7
11
p
2
3
5
7
11
n=5
p
2
3
5
7
11
p
2
3
5
7
11

e(3, MπE )−1 = 3−2 · 192
E
E
1/2,p
R3,p (Mπ )/D(Mπ )1 ⊗ ((1.010101011100101010)2 × 24 ) + ζ3 ⊗ ((1.111100111000011101)2 × 22 )
1 ⊗ ((2.121122121210110121)3 × 34 ) + ζ3 ⊗ ((1.100202121120121202)3 × 34 )
1 ⊗ ((4.402020001241122012)5 × 53 ) + ζ3 ⊗ ((1.402102204113114001)5 × 54 )
1 ⊗ ((2.1404154326422162)7 × 76 ) + ζ3 ⊗ ((5.1444362630564332)7 × 76 )
1 ⊗ ((6.9026829596822)11 × 116 ) + ζ3 ⊗ ((8.048680429402A)11 × 116 )Lp (3, χπ ⊗ ωp−2 , Q)
1 ⊗ ((1.010111010101101000)2 × 24 ) + ζ3 ⊗ ((1.111111110001001011)2 × 22 )
1 ⊗ ((1.120012101022112222)3 × 30 ) + ζ3 ⊗ ((2.212202101110022110)3 × 30 )
1 ⊗ ((1.222321004420021231)5 × 50 ) + ζ3 ⊗ ((1.123432442320044212)5 × 50 )
1 ⊗ ((6.215644553551135401)7 × 70 ) + ζ3 ⊗ ((1.642236344041330514)7 × 70 )
1 ⊗ ((6.6A0036761338492361)11 × 110 ) + ζ3 ⊗ ((8.749011AA3694540145)11 × 110 )e(5, MπE )−1 = 27 · 32 · 52 · 194 · (37754 + 43673ζ3 )E
R5,p (Mπ )/D(MπE )1/2,p
1 ⊗ ((1.000011100001001101)2 × 29 ) + ζ3 ⊗ ((1.101010111001110110)2 × 29 )
1 ⊗ ((1.110110120010021222)3 × 312 ) + ζ3 ⊗ ((2.020220021122220210)3 × 312 )
1 ⊗ ((3.020013014242432200)5 × 57 ) + ζ3 ⊗ ((1.220134043203242432)5 × 57 )
1 ⊗ ((4.15141551064450)7 × 710 ) + ζ3 ⊗ ((6.53120060244502)7 × 710 )
1 ⊗ ((4.285500134)11 × 1112 ) + ζ3 ⊗ ((2.A3934821906)11 × 1110 )Lp (5, χπ ⊗ ωp−4 , Q)
1 ⊗ ((1.110010010001000101)2 × 24 ) + ζ3 ⊗ ((1.111111100101010110)2 × 22 )
1 ⊗ ((1.211021010101222221)3 × 30 ) + ζ3 ⊗ ((2.122212200211101011)3 × 30 )
1 ⊗ ((1.401222401123304240)5 × 50 ) + ζ3 ⊗ ((1.432224122043203014)5 × 50 )
1 ⊗ ((2.441442356342000541)7 × 70 ) + ζ3 ⊗ ((3.664531413660053644)7 × 70 )
1 ⊗ ((6.3A4A10A60052302518)11 × 110 ) + ζ3 ⊗ ((4.7AA8843572652457A4)11 × 110 )[11] A. Borel. Cohomologie de SLn et valeurs de fonctions zêta aux points entiers. Ann. Scuola
Norm. Sup. Pisa Cl. Sci. (4), 4(4):613–636, 1977. Errata at vol. 7, p. 373 (1980).
[12] J. I. Burgos Gil. The regulators of Beilinson and Borel, volume 15 of CRM Monograph
Series. Amer. Math. Soc., Providence, RI, 2002.
[13] F. Calegari. Irrationality of certain p-adic periods for small p. Int. Math. Res. Not.,(20):1235–1249, 2005.
[14] P. Cassou-Noguès. p-adic L-functions for totally real number field. In Proceedings of theConference on p-adic Analysis (Nijmegen, 1978), volume 7806 of Report, pages 24–37.
Katholieke Univ. Nijmegen, 1978.
[15] P. Cassou-Noguès. Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta p-adiques.
Invent. Math., 51(1):29–59, 1979.
[16] R. Coleman. Dilogarithms, regulators, and p-adic L-functions. Invent. Math., 69:171–208,
1982.
[17] R. Coleman and B. Gross. p-adic heights on curves. In Algebraic number theory, pages
73–81. Academic Press, Boston, MA, 1989.

430

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot

Table 14. Splitting field of x6 − x5 − 8x4 − x3 + 12x2 + 7x + 1, where C(χπ ⊗ id) = 22 · 5 · 31.
n=3
p
2
3
5
7
11
p
2
3
5
7
11
n=5

e(3, MπE )−1 = 2−2 · 3−2 · 312 · ζ3
E
E
1/2,p
R3,p (Mπ )/D(Mπ )1 ⊗ ((1.100100010111101000)2 × 22 ) + ζ3 ⊗ ((1.111010100101111010)2 × 20 )
1 ⊗ ((1.221220200011201020)3 × 34 ) + ζ3 ⊗ ((2.020111011200022220)3 × 34 )
1 ⊗ ((1.141130342002111122)5 × 53 ) + ζ3 ⊗ ((4.342032011212400441)5 × 53 )
1 ⊗ ((4.4505556355566640)7 × 76 ) + ζ3 ⊗ ((3.0610023664215014)7 × 76 )
1 ⊗ ((7.3700473650A19)11 × 116 ) + ζ3 ⊗ ((1.3659A52396880)11 × 116 )Lp (3, χπ ⊗ ωp−2 , Q)
1 ⊗ ((1.101110001100101000)2 × 22 ) + ζ3 ⊗ ((1.010100111010111101)2 × 24 )
1 ⊗ ((2.111112222201000112)3 × 30 ) + ζ3 ⊗ ((1.120112022212020122)3 × 30 )
1 ⊗ ((4.321142113321234330)5 × 50 ) + ζ3 ⊗ ((1.130134403242024422)5 × 50 )
1 ⊗ ((5.530006300632330553)7 × 70 ) + ζ3 ⊗ ((2.200601212403230506)7 × 70 )
1 ⊗ ((2.00762927347972431A)11 × 110 ) + ζ3 ⊗ ((A.191A1780051997A650)11 × 110 )e(5, MπE )−1 = −26 · 33 · 52 · 314 · (2 + ζ3 ) · (2 − ζ3 ) · (7 − 15ζ3 ) · (54 + 31ζ3 )p

R5,p (MπE )/D(MπE )1/2,p2

1 ⊗ ((1.011100011000010010)2 × 28 ) + ζ3 ⊗ ((1.011100000000101011)2 × 28 )3

1 ⊗ ((2.202202022111202222)3 × 314 ) + ζ3 ⊗ ((1.112002110200111001)3 × 315 )57111 ⊗ ((3.141433024141141322)5 × 58 ) + ζ3 ⊗ ((2.002014030014311244)5 × 57 )
1 ⊗ ((6.50135302661002)7 × 710 ) + ζ3 ⊗ ((4.04356114536113)7 × 710 )
1 ⊗ ((5.A3306A68921)11 × 1110 ) + ζ3 ⊗ ((3.85A6650344A)11 × 1110 )p

Lp (5, χπ ⊗ ωp−4 , Q)

2
3
5
7
11

1 ⊗ ((1.100110110011011000)2 × 22 ) + ζ3 ⊗ ((1.110110100001011101)2 × 24 )
1 ⊗ ((2.201101211210020202)3 × 30 ) + ζ3 ⊗ ((1.212021202011010201)3 × 30 )
1 ⊗ ((4.011013012410044000)5 × 50 ) + ζ3 ⊗ ((1.431333301302323324)5 × 50 )
1 ⊗ ((2.130552365243450401)7 × 70 ) + ζ3 ⊗ ((2.601221155363314062)7 × 70 )
1 ⊗ ((9.53274A487A14818064)11 × 110 ) + ζ3 ⊗ ((9.89A625650825167861)11 × 110 )[18] P. Colmez. Résidu en s = 1 des fonctions zêta p-adiques. Invent. Math., 91(2):371–389,
1988.
[19] R. de Jeu. Zagier’s conjecture and wedge complexes in algebraic K-theory. Compositio
Mathematica, 96:197–247, 1995.
[20] P. Deligne. Valeurs de fonctions L et périodes d’intégrales. In Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis,
Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, pages 313–346. Amer. Math. Soc.,
Providence, R.I., 1979. With an appendix by N. Koblitz and A. Ogus.
[21] P. Deligne. Interprétation motivique de la conjecture de Zagier reliant polylogarithms etrégulateurs. Preprint, 1991.
[22] P. Deligne and K. Ribet. Values of abelian L-functions at negative integers over totally real
fields. Invent. Math., 59(3):227–286, 1980.
[23] C. Deninger and A. J. Scholl. The Beı̆linson conjectures. In L-functions and arithmetic(Durham, 1989), volume 153 of London Math. Soc. Lecture Note Ser., pages 173–209.
Cambridge Univ. Press, Cambridge, 1991.

On the p-adic Beilinson Conjecture for Number Fields

431

Table 15. Splitting field of x6 − x5 − 8x4 + 5x3 + 19x2 − 4x − 11, where C(χπ ⊗ id) = 5 · 139.
n=3
p
2
3
5
7
11
p
2
3
5
7
11
n=5
p
2
3
5
7
11
p
2
3
5
7
11

e(3, MπE )−1 = 2−5 · 3−2 · 1392
E
E
1/2,p
R3,p (Mπ )/D(Mπ )1 ⊗ ((1.000110100111100100)2 × 24 ) + ζ3 ⊗ ((1.000011011101110010)2 × 25 )
1 ⊗ ((2.102212010212021001)3 × 34 ) + ζ3 ⊗ ((2.122111112211000220)3 × 34 )
1 ⊗ ((1.400211310114320013)5 × 53 ) + ζ3 ⊗ ((1.434230430334022423)5 × 53 )
1 ⊗ ((2.2523154003106061)7 × 76 ) + ζ3 ⊗ ((2.3200021260361110)7 × 76 )
1 ⊗ ((9.4472801A61A44)11 × 116 ) + ζ3 ⊗ ((1.7232221727046)11 × 116 )Lp (3, χπ ⊗ ωp−2 , Q)
1 ⊗ ((1.000100101010000101)2 × 24 ) + ζ3 ⊗ ((1.000010100000000110)2 × 23 )
1 ⊗ ((2.111012101221111220)3 × 32 ) + ζ3 ⊗ ((1.010011111201220212)3 × 30 )
1 ⊗ ((3.240243311024013404)5 × 50 ) + ζ3 ⊗ ((4.142342004244233331)5 × 52 )
1 ⊗ ((6.543343430631523116)7 × 71 ) + ζ3 ⊗ ((2.342260265363120032)7 × 70 )
1 ⊗ ((4.8521241A647A883282)11 × 110 ) + ζ3 ⊗ ((1.46220220356163464A)11 × 110 )e(5, MπE )−1 = 2−4 · 32 · 52 · 1394 · (4 − 5ζ3 )E
ER5,p (Mπ )/D(Mπ )1/2,p
1 ⊗ ((1.000100111100110101)2 × 210 ) + ζ3 ⊗ ((1.011111001011110010)2 × 29 )
1 ⊗ ((1.202001002112012221)3 × 313 ) + ζ3 ⊗ ((1.021221211222021020)3 × 312 )
1 ⊗ ((3.404332041431011004)5 × 57 ) + ζ3 ⊗ ((3.213210304101144004)5 × 57 )
1 ⊗ ((4.11031052300040)7 × 710 ) + ζ3 ⊗ ((4.03411453000015)7 × 710 )
1 ⊗ ((6.8523188534)11 × 1111 ) + ζ3 ⊗ ((5.2A39A447198)11 × 1110 )Lp (5, χπ ⊗ ωp−4 , Q)
1 ⊗ ((1.101001011001000000)2 × 24 ) + ζ3 ⊗ ((1.000100011110101010)2 × 23 )
1 ⊗ ((1.200212010102010102)3 × 31 ) + ζ3 ⊗ ((1.122011112111021002)3 × 30 )
1 ⊗ ((3.124113213204101323)5 × 50 ) + ζ3 ⊗ ((2.114244212313224030)5 × 53 )
1 ⊗ ((6.430024313466045624)7 × 70 ) + ζ3 ⊗ ((5.065261416212661354)7 × 70 )
1 ⊗ ((5.A88931A0106A0172A3)11 × 110 ) + ζ3 ⊗ ((3.A4151A485571789978)11 × 110 )[24] T. Dokchitser. Computing special values of motivic L-functions. Experiment. Math.,
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432

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot

Table 16. Splitting field of x6 − x5 − 8x4 + 11x3 + 2x2 − 5x + 1, where C(χπ ⊗ id) = 22 · 32 · 13.
n=3
p
2
3
5
7
11
p
2
3
5
7
11
n=5
p
2
3
5
7
11
p
2
3
5
7
11

e(3, MπE )−1 = −2−2 · 32 · 132 · ζ3
E
E
1/2,p
R3,p (Mπ )/D(Mπ )1 ⊗ ((1.101100110101111100)2 × 22 ) + ζ3 ⊗ ((1.001000001010110111)2 × 20 )
1 ⊗ ((2.021221201100120120)3 × 35 ) + ζ3 ⊗ ((2.020012210111121212)3 × 37 )
1 ⊗ ((3.000431031211204434)5 × 56 ) + ζ3 ⊗ ((1.012314421212444244)5 × 56 )
1 ⊗ ((3.0426445363200511)7 × 76 ) + ζ3 ⊗ ((1.1035504615611621)7 × 76 )
1 ⊗ ((8.9799A36158285)11 × 116 ) + ζ3 ⊗ ((9.229923AAA0734)11 × 116 )Lp (3, χπ ⊗ ωp−2 , Q)
1 ⊗ ((1.100100111001100011)2 × 22 ) + ζ3 ⊗ ((1.101001000011101110)2 × 24 )
1 ⊗ ((1.022110110201121122)3 × 32 ) + ζ3 ⊗ ((2.102110200120211001)3 × 30 )
1 ⊗ ((2.424401443414003124)5 × 50 ) + ζ3 ⊗ ((4.440233034214103134)5 × 50 )
1 ⊗ ((5.421026022055251515)7 × 70 ) + ζ3 ⊗ ((4.510213122044500212)7 × 70 )
1 ⊗ ((A.187435006878255608)11 × 110 ) + ζ3 ⊗ ((6.47020389393AA07A54)11 × 110 )e(5, MπE )−1 = see Example 7.6
E
E
1/2,p
R5,p (Mπ )/D(Mπ )1 ⊗ ((1.001111100010111111)2 × 27 ) + ζ3 ⊗ ((1.100110001101001011)2 × 28 )
1 ⊗ ((1.100222220001012001)3 × 315 ) + ζ3 ⊗ ((1.201100102012122102)3 × 315 )
1 ⊗ ((1.302021433120040233)5 × 58 ) + ζ3 ⊗ ((1.414010322010320220)5 × 58 )
1 ⊗ ((6.54022035336014)7 × 710 ) + ζ3 ⊗ ((1.31434160250212)7 × 710 )
1 ⊗ ((9.5A062420171)11 × 1110 ) + ζ3 ⊗ ((A.873665790A8)11 × 1110 )Lp (5, χπ ⊗ ωp−4 , Q)
1 ⊗ ((1.101111101101011100)2 × 22 ) + ζ3 ⊗ ((1.001110000011000011)2 × 24 )
1 ⊗ ((2.001012111110022002)3 × 31 ) + ζ3 ⊗ ((2.011221210002201001)3 × 30 )
1 ⊗ ((3.241120123211444141)5 × 51 ) + ζ3 ⊗ ((2.103430110032401010)5 × 50 )
1 ⊗ ((5.454116060262152231)7 × 70 ) + ζ3 ⊗ ((5.160242506230053303)7 × 70 )
1 ⊗ ((9.9950088486A6479846)11 × 110 ) + ζ3 ⊗ ((A.178492061888054088)11 × 110 )[31] U. Jannsen. Deligne homology, Hodge-D-conjecture, and motives. In Beı̆linson’s conjectures
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434

A. Besser, P. Buckingham, R. de Jeu and X.-F. Roblot

Amnon Besser
Department of Mathematics
Ben-Gurion University of the Negev
P.O.B. 653
Be’er-Sheva 84105
IsraelPaul Buckingham
Department of Pure Mathematics
University of Sheffield
Hicks Building
Hounsfield Road
Sheffield S3 7RH
United Kingdom
Rob de Jeu
Faculteit der Exacte Wetenschappen
Afdeling Wiskunde
Vrije Universiteit
De Boelelaan 1081a
1081 HV Amsterdam
The Netherlands
E-mail: jeu@few.vu.nl
Xavier-François Roblot
Institut Camille Jordan
Université de Lyon, Université Lyon 1, CNRS – UMR 5208
43 blvd du 11 Novembre 1918, 69622 Villeurbanne Cedex
France