Verifying a p Adic Abelian Stark Conjecture at s

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Niveau: Supérieur, Doctorat, Bac+8
Verifying a p-Adic Abelian Stark Conjecture at s = 1 X.-F. Roblot IGD, Universite Lyon I D. Solomon ? King's College London November 3, 2003 Abstract In a previous paper [13], the second author developed a new approach to the abelian p-adic Stark conjecture at s = 1 and stated related conjectures. The aim of the present paper is to develop and apply techniques to numerically investigate one of these – the ‘Weak Refined Combined Conjecture' – in fifteen cases. 1 Introduction In the 1970's and 80's Harold Stark [14] made a series of conjectures concerning the values at s = 1 and s = 0 of complex Artin L-series attached to a Galois extension of number fields K/k. Subsequently, much theoretical and computational work has been done, extend- ing and testing these conjectures, with particular attention paid recently to certain refined conjectures in the case where K/k is abelian ([7], [5]). In [13], a new approach to the abelian case of the p-adic conjecture at s = 1 was developed and several related conjectures were stated. The main aim of the present paper is to develop and apply techniques to numerically investigate one of these – the ‘Weak Refined Combined Conjecture' (Conjecture 3.6 of [13], here Conjecture 2.2) – in a number of cases.

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Verifying a
p-Adic
Abelian
Stark Conjecture at
X.-F. Roblot D. SolomonIGD,Universit´eLyonIKingsCollegeLondon November 3, 2003
Abstract
s
=
In a previous paper [13], the second author developed a new approach to the abelian p-adic Stark conjecture ats The aim of the present= 1 and stated related conjectures. paper is to develop and apply techniques to numerically investigate one of these – the ‘Weak Refined Combined Conjecture’ – in fifteen cases.
1 Introduction
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In the 1970’s and 80’s Harold Stark [14] made a series of conjectures concerning the values ats= 1 ands= 0 of complex ArtinL-series attached to a Galois extension of number fieldsK/k much theoretical and computational work has been done, extend-. Subsequently, ing and testing these conjectures, with particular attention paid recently to certain refined conjectures in the case whereK/kis abelian ([7], [5]). In [13], a new approach to the abelian case of thep-adic conjecture ats= 1 was developed and several related conjectures were stated. The main aim of the present paper is to develop and apply techniques to numerically investigate one of these – the ‘Weak Refined Combined Conjecture’ (Conjecture 3.6 of [13], here Conjecture 2.2) – in a number of cases. In Section 2, we shall recall the definitions of the complex andp-adic ‘twisted zeta functions’. They depend on two parameters: a proper idealfofOk, and a setTof primes ideals ofOk(which, for the purpose ofp-adic interpolation, must contain the primes above p). Then the statements of the two ‘combined conjectures’ of [13] are given. (The term ‘combined’ refers to the fact that each conjecture predicts both a complex and ap-adic equality.) The main reference for this section is, of course, [13], but also [12] which contains a reformulation developed by the second author of a refined complex abelian conjecture at s= 0 originally made by Rubin in [7]. Briefly, the ‘Weak Refined Combined Conjecture’ takes the following form: we assemble all the complex (resp.p-adic) twisted zeta-functions for givenfandTinto a single group-ring-valued function Φf,T(s) (resp.Φf,T ,p(s), assuming
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thatTcontains the primes abovep assuming that the primes in). Then,Tdo not dividef, the value of the latter ats= 1 is conjectured to be equal to the complex (resp.p-adic) group-ring-valued regulator of a certain elementηf,Tmultiplied by an explicit algebraic constant. The elementηf,Tis constructed from certainS-units of the fieldKwhich in this case is simply the ray-class fieldk(f). Section 3 develops a new formula to compute the element Φf,T ,p(1). We concentrate on the case wherekis real quadratic although our technique should extend to other totally real fields. Relying as it does on Shintani’s method and the theory ofp-adic measures, this technique is very different in nature from that used to evaluate complexL-functions. (For the latter we use [1].) We stress that it passes most naturally not by the analogousp-adic L-functions but by thep Indeed,-adic twisted zeta-functions themselves. this was one of the major reasons for introducing these functions and, in preparation, their complex analogues. Finally, Section 4 is devoted to the numerical investigation of the ‘Weak Refined Com-bined Conjecture’ over a real quadratic field. We first explain some procedures (for example a continued fraction method based on ideas of Zagier) that greatly shorten the calculation of Φf,T ,p we explain the basis of our method Then(1) using the formula of the previous section. for verifying the conjecture. Since [k:Q] = 2 andKis totally real, our conjectures are ‘second order’ in the sense that the relevant complexL-functions have at least a double zero ats The= 0.side’ of the conjectures is that both the com- corresponding fact on the ‘other plex andp One-adic regulators must be of rank 2. consequence is that, unlike verifications of the (complex) first order abelian Stark Conjectures (see for example [6]), the regulators themselves do not determineS-units ofK . Wetherefore need different methods for finding Kandηf,Tand new criteria for affirming that the latter satisfies the combined conjecture to the precision of our computations. In fact, we use the methods of [6] (which actually rely on the first-order complex conjecture!) to independently and verifiably construct the ray-class fieldKillustrate our methodology by numerically confirming the conjecture in then . We fifteen different examples, using a number of different primespin each example. The result-ing data are displayed in tables at the end of the paper. We hope that they will serve to stimulate further interest in these conjectures, their possible refinements and extensions.
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Thep-adic Stark conjectures ats= 1
The main reference for this section are [12] (for the complex twisted functions) and [13].
2.1 Complex twisted zeta functions ¯ LetkQCbe any number field of finite degree overQand letOits ring of integers. LetIbe any fractional ideal ofkandξany character on (the additive group of)Iwith values inµ(C The), the complex roots of unity.annihilatorofξis the idealfOgiven by f={a∈ O:ξ(ax) = 1xI}. Suppose thatzis the formal product of some subset of the real places ofkand writemfor thecyclethat is the formal productfz. We denote by
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Emthe subgroup of finite index inE(K) :=O×of the units that are congruentconsisting to 1 modulom For any finite setin the usual sense.Tof finite places (prime ideals) ofO, the groupEmacts by multiplication on the following subset ofI S(I,z, T) :={aI:akz×and (aI1, T) = 1} wherekz×denotes the elements ofk×which are positive at all places dividingzand the notation (J, T) = 1 indicates that an idealJofOhas support disjoint fromT. ForsC,<(s)>1 we consider the absolutely convergent Dirichlet series, called the ‘twisted zeta-function’ for these data, defined by ξ(a) ZT(s;ξ, I,m) :=a∈S(I,zX,T)/Em|Iξ:((aa))|s=a∈S(I,zX,T)/EmN(aI1)s =N IsXξ(a)|Nk/Q(a)|s(1) a∈S(I,z,T)/Em LetWfbe the set of all pairs (ψ, J), whereψis a character of annihilatorfon a fractional idealJ [13] a natural equivalence relation (depending on. Inz) was defined onWfin such 0 a way thatZT(s;ξ, I,m) equalsZT(s;ξ0 ,, Im) if (ξ, I) and (ξ0, I0) are equivalent. LetWm denote the quotient set ofWf Thenby this equivalence relation. for any equivalence class wWmwe can unambiguously defineZT(s;w) :=ZT(s;ξ, I,m) for any (ξ, I)w. Let Clm(k) denote the ray-class group ofkmodulom. Thus Clm(k) :=If(k)/Pm(k) whereIf(k) denotes the group of fractional ideals prime tofandPm(k) the subgroup consisting of those of the form (a) for someak×, a1 (modm). For anycin Clm(k) andwinWmwe letcwdenote the element ofWmgiven by the equivalence class of the pair (ξ|aI,aI)Wf where (ξ, I) is any pair in the classwanda∈ If(k) any integral ideal in the classc. This map is well-defined and determines an action of Clm(k) onWm can check that this. One action is free and transitive. LetDOdenote the absolute different ofkand writeξf0for the character onf1D1 which sendsato exp(2πiTrk/Q(a)). Thus the pair (ξf0,f1D1) lies inWfand we writew0m for its class inWm. Letk(m)Cbe theray-class fieldoverkmodulom Galois group. The Gm:= Gal(k(m)/k) is isomorphic to Clm(k)viathe Artin map which sendscClm(k) to σc,m=σcGm. We letCGmdenote the complex group-ring ofGmand make the Definition 2.1For any cyclem=fzforkand any finite setTof prime ideals ofO, we writeΦm,Tfor the function
Φm,T:{sC:<(s)>1}CGm s7XZT(s;cw0m)σc1 cClm(k) The basic properties of Φm,T(s) are given in [12,§3] and [13,§ particular Theorem 2.22]. In of [13] gives a relation between Φm,Tand the primitiveL-functions of the characters ofGm (or Clm(k)).
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