On the -adic Stark conjecture at
Let be a finite Galois extension of totally real number fields and let be a prime. The ‘-adic Stark conjecture at ’ relates the leading terms at of -adic Artin -functions to those of the complex Artin -functions attached to . We prove this conjecture unconditionally when is abelian. Moreover, we also show that for certain non-abelian extensions the -adic Stark conjecture at is implied by Leopoldt’s conjecture for at . As an application, we provide strong new evidence for special cases of the ‘equivariant Tamagawa number conjecture’ for Tate motives and the closely related ‘leading term conjectures’ at and .
2010 Mathematics Subject Classification:11R23, 11R42
Let be a finite Galois extension of totally real number fields and let . Let be a prime. In the case that is abelian, the fundamental work of Deligne and Ribet [MR579702] and of Pierette Cassou-Noguès [MR524276] shows that one can attach to each irreducible character of a -adic Artin -function that interpolates values of the corresponding complex Artin -function at negative integers. This construction can be generalised to the case that is non-abelian by using Brauer induction.
The ‘-adic Stark conjecture at ’ originated with Serre [MR0506177], was discussed by Tate [MR782485], and was clarified by Burns and Venjakob [MR2290587, MR2749572]. Roughly speaking, this conjecture relates the leading terms at of the complex Artin -function and the -adic Artin -function attached to a character of . For the trivial character it is equivalent to the ‘-adic class number formula’ of Colmez [MR922806] together with Leopoldt’s conjecture for at . More generally, the leading terms of the two -functions attached to a character of are related by a certain ‘comparison period’, which is non-zero if Leopoldt’s conjecture holds for at .
In the present article, we prove the -adic Stark conjecture at unconditionally when is abelian by building on work of Ritter and Weiss [MR1423032] and using standard results on Dirichlet -functions and Kubota-Leopoldt -adic -functions. Moreover, we show that it is implied by Leopoldt’s conjecture for at when every character of is a virtual permutation character (in particular, the symmetric groups have this property). By combining the proofs of these two results, we also show that if and where is a finite field with elements and the semidirect product is defined by the natural action, then Leopoldt’s conjecture for at again implies the -adic Stark conjecture at for .
These results are motivated by their applications to the equivariant Tamagawa number conjecture (ETNC) for Tate motives and the closely related leading term conjectures, both at and at . Building on work of Bloch and Kato [MR1086888], Fontaine and Perrin-Riou [MR1265546], and Kato [MR1338860], Burns and Flach [MR1884523] formulated the ETNC for any motive over with the action of a semisimple -algebra, describing the leading term at of an equivariant motivic -function in terms of certain cohomological Euler characteristics. This is a powerful and unifying formulation which, in particular, recovers the Birch and Swinnerton-Dyer conjecture. We refer the reader to the survey article [MR2088713] for a more detailed overview.
Let be a finite Galois extension of number fields (not necessarily totally real) and let . In the case of Tate motives, the ETNC relates certain arithmetic complexes to the leading terms at integers of the equivariant complex Artin -function attached to . Burns [MR1863302] formulated the leading term conjecture (LTC) at and he showed that this conjecture for is equivalent to the ETNC for the pair . The advantage of this new formulation is that it is more explicit. Moreover, the LTC at recovers Stark’s conjecture at (as interpreted by Tate in [MR782485]), the ‘strong Stark conjecture’ of Chinburg [MR724009] and Chinburg’s ‘-conjecture’ [MR724009, MR786352]. It is also equivalent to the ‘lifted root number conjecture’ of Gruenberg, Ritter and Weiss [MR1687551] and implies numerous other conjectures involving leading terms of Artin -functions at (see [MR2882689, Lecture 3] for a partial list of such conjectures). Breuning and Burns [MR2371375] formulated the LTC at , which simultaneously refines both Stark’s conjecture at (as formulated by Tate in [MR782485]) and Chinburg’s ‘-conjecture’ [MR786352]. In [MR2804251], Breuning and Burns also showed that, under the assumption of Leopoldt’s conjecture (at all primes), the LTC at for is equivalent to the ETNC for the pair . If the ‘global epsilon constant conjecture’ of Bley and Burns [MR2005875] holds then the LTCs at and are equivalent.
Again let be a finite Galois extension of totally real number fields. Let and let be a prime. Burns [MR3294653] showed that under the assumption that certain -invariants attached to and vanish for all odd prime divisors of , the -adic Stark conjecture for all characters of and for all odd primes implies the ETNC for the pair . The proof relies crucially on the descent machinery of Burns and Venjakob [MR2749572], and on the equivariant Iwasawa main conjecture, which has been proven by Ritter and Weiss [MR2813337] and by Kakde [MR3091976] independently, under the assumption that the relevant -invariant vanishes.
We prove a refinement of the above result that allows us to work prime-by-prime. In other words, we show that for a fixed odd prime , the vanishing of the relevant -invariant attached to and together with the -adic Stark conjecture for all characters of imply the ETNC for the pair , where is the localisation of at . A key ingredient is a direct proof that for any fixed prime , the -adic Stark conjecture at for all characters of implies Stark’s conjecture at for all such characters. By combining this prime-by-prime descent result with our new results on the -adic Stark conjecture at , we obtain new evidence for the relevant case of the ETNC. Thus by tweaking the aforementioned result of Breuning and Burns [MR2804251] to work prime-by-prime, we also obtain results for the LTC at . Then by using known cases (and by proving a new case) of the global epsilon constant conjecture, we obtain results on the LTC at and thus the ETNC for the pair .
We now give three concrete examples of the new results obtained. For the first example, let be an odd prime and let be a positive integer. If is any totally real Galois extension with (see definition above) then under the assumption that Leopoldt’s conjecture for at holds, the LTC for at holds. We note that Leopoldt’s conjecture for a given number field and prime can be verified computationally (see Remark LABEL:rmk:Leopoldt-compute). For the second example, let denote the cyclic group of order and let where is a positive integer and acts on by inversion (in the case we have ). If is any totally real Galois extension with then under the assumption that Leopoldt’s conjecture for at holds, the LTCs for at and both hold. For the final example, let be any finite group. Then there exist infinitely many Galois extensions of totally real number fields with such that, if for all odd prime divisors of both Leopoldt’s conjecture holds for at and a certain -invariant attached to and vanishes, then the LTCs for at and both hold outside their -primary parts.
The first named author acknowledges financial support provided by EPSRC First Grant EP/N005716/1 ‘Equivariant Conjectures in Arithmetic’. The second named author acknowledges financial support provided by the DFG within the Collaborative Research Center 701 ‘Spectral Structures and Topological Methods in Mathematics’. The authors are indebted to Werner Bley for his help in running the Magma code bundled in Debeerst’s PhD thesis [debeerst-thesis] (this is used in the proof of Theorem LABEL:thm:GEGG-for-C3m:C2-extensions). Moreover, the authors are grateful to Alex Bartel, Tim Dokchitser and Otmar Venjakob for helpful conversations and correspondence.
Notation and conventions
All rings are assumed to have an identity element and all modules are assumed to be left modules unless otherwise stated. We fix the following notation:
|the symmetric group of degree|
|the alternating group of degree|
|the cyclic group of order|
|the dihedral group of order|
|the subgroup of generated by double transpositions|
|the finite field with elements, where is a prime power|
|the affine group isomorphic to defined in §LABEL:subsec:affine|
|a primitive -th root of unity|
|the trace map for the field extension|
|the field of fractions of an integral domain|
|the ring of matrices over a ring|
|the set of infinite places of a number field|
|the set of places of a number field above a rational prime|
A finite Galois extension of totally real number fields will usually be denoted by . By contrast, will usually denote a finite Galois extension of number fields, neither of which is necessarily totally real.
2. Algebraic Preliminaries
2.1. Representations and characters of finite groups
Let be a finite group and let be a field of characteristic . We write for the set of characters attached to finite-dimensional -valued representations of , and for the ring of virtual characters generated by . Moreover, we let denote the subset of irreducible characters in and let denote the ring of -valued virtual characters of . Thus we have containments
We let denote the trivial character of and for we write for the usual inner product of virtual characters. For a subgroup of and we write for the induced character; for a normal subgroup of and we write for the inflated character. For and we set and note that this defines a group action from the left even though we write exponents on the right of .
We write for the ring of characters of virtual permutation representations of , that is, -linear combinations of characters of the form where ranges over subgroups of . It is important to note that each of the inclusions
may be strict.
2.2. Endomorphisms of modules over group algebras
Let be a finite group and let be a field of characteristic . For any we fix a -module with character . For any -module and any we write for the -vector space
and for the induced map . We note that is independent of the choice of . The following is similar to [MR782485, Chapitre I, 6.4].
Let be a -module and let . Let be a subgroup of and let denote considered as a -module. Let be a normal subgroup of and let denote the -module of -invariants of .
If then .
If then and .
If then and .
In the appropriate bases, the matrix for is a block matrix whose blocks are the matrices for and , and this gives claim (i). Claim (ii) follows from Frobenius reciprocity, i.e., the natural isomorphism . Similarly, the natural isomorphism gives claim (iii). ∎
3. Stark’s conjecture at
3.1. Artin -functions
Let be a finite Galois extension of number fields and let . Let be a finite set of places of containing the set of infinite places . For each character we write for the -truncated (complex) Artin -function attached to (see [MR1697859, Chapter VII, §10]). We recall that and that is invariant under induction and inflation of characters. Moreover, , the -truncated Dedekind zeta-function attached to , which has a simple pole at . In fact, writing for the leading term at of , it is well-known that
(see the discussion of [MR0218327, p. 225], for instance).
3.2. Stark’s conjecture at
For any place of we write for the completion of at . Let . Define by and denote the kernels of the trace maps by and . Then we have a commutative diagram of -modules with exact rows