Gauss sums, Jacobi sums and cyclotomic units related to torsion Galois modules
Let be a finite group and let be a tamely ramified -Galois extension of number fields. We show how Stickelberger’s factorization of Gauss sums can be used to determine the stable isomorphism class of various arithmetic -modules attached to . If and denote the rings of integers of and respectively, we get in particular that defines the trivial class in the class group and, if is also assumed to be locally abelian, that the square root of the inverse different (whenever it exists) defines the same class as . These results are obtained through the study of the Fröhlich representatives of the classes of some torsion modules, which are independently introduced in the setting of cyclotomic number fields. Gauss and Jacobi sums, together with the Hasse-Davenport formula, are involved in this study. These techniques are also applied to recover the stable self-duality of (as a -module). Finally, when is the binary tetrahedral group, we use our results in conjunction with Taylor’s theorem to find a tame -Galois extension whose square root of the inverse different has nontrivial class in .
Let be a finite group and let be a Galois extension of number fields with Galois group . Let and denote the rings of integers of and , respectively; then is an -module, and in particular a -module, whose structure has long been studied. In the case where is a tame extension, is known to be locally free by Noether’s theorem and the investigation of its Galois module structure culminated with M. Taylor’s theorem [tay2] expressing the class defined by in the class group of locally free -modules in terms of Artin root numbers (see Theorem LABEL:pearl).
Other -modules which appear naturally in our context have also been studied, among which the inverse different of the extension and, when it exists, its square root . The existence of is of course equivalent to being a square, a condition which can be tested using Hilbert’s valuation formula [Serre, IV, Proposition 4]. In particular, when is tame, exists if and only if the inertia group of every prime of in has odd order (which is the case for example if the degree is odd). In the tame case, both these modules are locally free -modules by a result of S. Ullom in [Ullom-cohomology] (applying to any -stable fractional ideal of ).
The modules and are related by duality. For a fractional ideal of , the dual of with respect to the trace from to is the fractional ideal
Then is -isomorphic to the -dual of , namely , and by definition one has , namely and are dual of each other. It can be shown that, for any fractional ideal of , one has , which implies that , when it exists, is the only self-dual fractional ideal of .
The duality relation between and accounts for comparing their -module structures. In the tame case, it essentially amounts to comparing their classes and in , and A. Fröhlich conjectured that is stably self-dual, namely that
equality which was later proved by M. Taylor [tay1] under slightly stronger hypotheses and by S. Chase [ChaseTors] in full generality. Taylor’s proof uses Fröhlich’s Hom-description of , while Chase examines the torsion module
It is worth noting that both proofs make crucial use of the stable freeness of the Swan modules of cyclic groups, a result due to R. Swan [SwanPRFG].
The study of the Galois module structure of was initiated by B. Erez, who proved in particular that, when is tamely ramified and of odd degree, the class it defines in is trivial, see [Erez2]. His proof follows the same strategy as that of Taylor in [tay2], using in particular Fröhlich’s Hom-description of . Since Taylor’s theorem implies the triviality of when is of odd degree, we get:
and both classes are in fact trivial.
In this paper, we consider a tame -Galois extension such that the square root of the inverse different exists. We introduce the torsion module
check it to be -cohomologically trivial, hence to define a class in , and show that this class is trivial, yielding a new proof of Equality (2) without any assumption on the degree of the extension. Nevertheless we need to assume that is locally abelian, namely that the decomposition group of every prime ideal of is abelian (note that this condition is automatically satisfied at unramified primes). The proofs of Erez [Erez2] and Taylor [tay2] involve the study of Galois Gauss sums, which may be regarded as objects of analytic nature since they come from the functional equation of Artin -functions. In our work instead only classical Gauss and Jacobi sums are involved, which might seem more satisfactory if one considers that (2) is a purely algebraic statement.
It turns out that the strategy of our proof also applies, without any restriction on the decomposition groups, to the torsion module defined above as well as to another torsion module, denoted . The module , whose definition will be given in Section 2, was introduced and studied by Chase in [ChaseTors]. We will prove that both and define the trivial class in . This allows us on the one hand to recover Equality (1), and on the other hand to deduce that the -module defines the trivial class in . This result looked new to us at first sight, but we show in Proposition LABEL:ONn that in , hence it is easily deduced from Taylor’s theorem.
These results are stated in Theorem 2 below. After a reduction to the case of a local totally ramified tame Galois extension with Galois group , we introduce new torsion Galois modules, which define classes and in . The study of these classes is the core of our work. In Theorem 1 we give explicit expressions of representative morphisms of and in terms of Gauss and Jacobi sums, which through Fröhlich’s Hom-description of yield the triviality of and . Theorem 3 will show a surprising consequence of the generalisation of Equality (2) to even degree extensions.
As suggested by various experts, it seems reasonable to expect that our methods also work in the context of sums of -th roots of codifferents developed by Burns and Chinburg [Burns-Chinburg].
We now explain our strategy more in detail. In Section 2 we follow Chase’s approach: we consider the torsion modules , and , then we reduce to the study, for every prime of , of their -module structure, where is the inertia group at (note that is cyclic of order coprime to the residual characteristic of , since is tame). In other words we show that it is sufficient to prove the triviality of the classes in of the local analogues , and , where is a cyclic totally ramified tame -adic extension with Galois group of order (prime to ). Up to this point, the only difference with Chase’s study of is that we need to be locally abelian when dealing with .
It is interesting to remark that and are particular instances of torsion modules arising from ideals, i.e. modules of the form or , where is a -stable ideal of . In fact we will perform the reduction step of Section 2 for general torsion modules arising from ideals (under the assumption that is locally abelian). Working in this more general situation requires no additional effort and allows us to easily recover the cases of and (while needs a somehow separate treatment). Moreover we will get almost for free a new proof of a result of Burns [BurnsARC, Theorem 1.1] on arithmetically realisable classes in tame locally abelian Galois extensions of number fields (see Subsection LABEL:section:arcburns). Nonetheless, for the sake of simplicity, in this introduction we shall stick with the modules , and .
The key ingredient in Chase’s proof of the triviality of is the link he establishes between the local torsion module and the Swan module (here ). Consider the torsion module associated to the Swan module, then
as -modules, where is the inertia degree of .
To deal with and , we need to enlarge the coefficient ring of the group algebra of . Let denote the group of th roots of unity in a fixed algebraic closure of and let denote the ring of integers of the cyclotomic field . We introduce new torsion -modules and , which depend on a character and a prime of not dividing . They may be considered as analogues of in the sense that, for an appropriate choice of and (in particular , the residual characteristic of ), we have
as -modules, where is the inertia degree of in . These constructions are described in Section LABEL:section:back-to-global. Furthermore it is easy to see that both and are cohomologically trivial -modules, hence they define classes in (see Section LABEL:ctsec).
Now let be any injective character and be a prime of not dividing . Set and . The core of this paper is the study of the classes and in . In Section LABEL:section:hom-des, we find equivariant morphisms and from the group of virtual characters of to the idèles of , representing and respectively in Fröhlich’s Hom-description of . In Section LABEL:section:stickelberger, we note that the contents of the values of and are principal ideals: this follows from Stickelberger’s theorem, which also gives explicit generators of these ideals in terms of Gauss and Jacobi sums respectively. Dividing and by suitably modified generators and of their contents yields morphisms with unit idelic values. With the help of the Hasse-Davenport formula we express the resulting morphisms and as Fröhlich’s generalized Determinants of some unit idèles of , showing that they lie in the denominator of the Hom-description, namely that and are trivial.
It is worth noting that applying these techniques to yields a proof of the triviality of in . In this proof, as in the original proof by Swan, cyclotomic units play a central role, analogous to that of the Gauss and Jacobi sums above, as it will soon be apparent.
To state our main results, we let and denote the Gauss and Jacobi sums defined respectively by
where denotes the residue field trace homomorphism, is a fixed th root of unity in and is the th power residue symbol. Then both and belong to . Let be a fixed generator of . In Section LABEL:section:stickelberger we define for such that
The corresponding expression for the cyclotomic unit is:
We can now formulate the main result of this paper, in terms of Fröhlich’s Hom-description of the class group (see Section LABEL:hdcg for more details).
The classes , and are trivial in . More precisely they are represented in by the morphisms with -components equal to at places of such that , and to
respectively, at prime ideals of such that , where are defined by
and satisfy , for any rational prime such that .
The triviality of is essentially Swan’s theorem [SwanPRFG, Corollary 6.1]. Our proof using Fröhlich’s Hom-description of the class group, given in Section LABEL:hom-rep-TZ, might be folklore, nevertheless the representative morphism we use to describe is slightly different from the one given in [Frohlich-Alg_numb, (I.2.23)] (see Remark LABEL:t-rep). The proof for and will follow from the results of Sections LABEL:section:hom-des and LABEL:section:stickelberger and will enlight the analogy between the torsion modules and we introduce and (see Theorem LABEL:desiderio and its proof).
In his early work [Erez] on the square root of the inverse different, Erez already establishes a link between this module and a Jacobi sum, in the case of a cyclic extension of of odd prime degree . Using a result of Ullom [Ullom-representations, Theorem 1], he shows that the class of in corresponds, through Rim’s isomorphism [CR2, (42.16)], to the class in of an ideal of which is defined by the action of some specific element of on prime ideals. This group algebra element, which is pretty close to what ours, , would yield in the prime degree case (see §LABEL:further), happens to be the exponent which appears in the factorisation of the Jacobi sum, yielding the triviality of the class under study.
The factorisation of Gauss and Jacobi sums through Stickelberger’s theorem is also an essential step in our proof of the triviality of the classes of and . In the introduction of his thesis [Erez-thesis], Erez gives other examples of results on the Galois module structure of rings of integers that are proved using Stickelberger elements ([Martinet-diedral], [Cougnard]). In [ChaseTors] Chase uses such elements to give the composition series of certain primary components of the -module when is cyclic or elementary abelian. Reversely, Galois module structure results can be used to build Stickelberger elements: the first statement in the theory of integral Galois modules is due to Hilbert (for the ring of integers of an abelian extension of with discriminant coprime to the degree), who used it to construct annihilating elements for the class group of some cyclotomic extensions. Finally, Stickelberger’s elements also have a fundamental role in the work of L. McCulloh (see [McCullohCNFEAGR], [McCullohGMSAE] and the references given in those papers).
As pointed out by Fröhlich in [Frohlich-Alg_numb, Note 6 to Chapter III], Chase remarks that the character function associated to (in the ideal-theoretic Hom-description) is the Artin conductor. Similarly, the character function associated to (in the idèle-theoretic Hom-description) is closely related to a resolvent map (see [ChaseTors, p. 210]). Fröhlich also wonders how his method, based on the comparison of Galois Gauss sums and resolvents, could be linked to Chase’s approach. We hope that the explicit connection we establish between and a Gauss sum might be useful to answer Fröhlich’s question.
Using the reduction results of Section 2, we deduce the following consequence.
Let be a -Galois tamely ramified extension of number fields. Then the classes of and are trivial in . In particular we have
If, further, is locally abelian, then the class of is trivial in . In particular we have
thus , and define the same class in .
As already mentioned, the triviality of in this context is due to Chase, see [ChaseTors, Corollary 1.12]. In the same paper he studies the torsion module as an -module, showing that it determines the local Galois module structure of [ChaseTors, Theorem 2.10] and describing its primary components [ChaseTors, Theorem 2.15]. We will show that , which is in fact equivalent to (see §LABEL:ctglob). The equality follows of course from Taylor’s theorem (which implies that is of order dividing and is trivial in odd degree) but was also known before Taylor’s proof of Fröhlich’s conjecture (it follows from [TaylorGMSIRA, Corollary] using the last remark of [LamAEFG, Definition 1.6]). We make more comments on the last equality of Theorem 2 in the presentation of Section LABEL:analytic below.
We end this paper by focusing in Section LABEL:analytic on the class of the square root of the inverse different (for the rest of this introduction we shall assume that the inverse different of every extension we consider is a square). We briefly recall some known results on this matter. In odd degree, Erez showed that is defined (i.e. is -locally free) if and only if is a weakly ramified -Galois extension (i.e the second ramification group of every prime is trivial). It seems reasonable to conjecture that is trivial when is a weakly ramified Galois extension of odd degree (see also [Vinatier-Surla, Conjecture]). As already mentioned this conjecture is true in the tame case, thanks to a result of Erez. When is a weakly ramified -Galois extension of odd degree, one also knows that
is free over , where is a maximal order of containing ([Erez2, Theorem 2]);
if, for any wildly ramified prime of , the decomposition group is abelian, the inertia group is cyclic and the localized extension is unramified, where and ([Pickettvinatier, Theorem 1]);
if is a power of a prime and is the ramification index of in ([Vinatier-Surla, Théorème 1]); when one even has by [Vinatier-3, Theorem 1].
It is interesting to observe that so far the class of had only been studied when has odd degree (except for a short local study, due to Burns and Erez, in the absolute, abelian and very wildly ramified case, see [Erez-survey, §3]). Specifically the question of whether it is trivial or not for every tame Galois extension had not been considered. The reason for this restriction is that the second Adams operation, which is fundamental in Erez’s approach in [Erez2], does not behave well with respect to induction for groups of even order.
This is precisely the point where our result above brings new information. When is locally abelian and tame, Theorem 2 together with Taylor’s theorem reduces the question to the study of the triviality of the image of the root number class in (see Corollary LABEL:A=tW). This immediately gives that if is abelian or has odd order (thus recovering Erez’s result in the locally abelian case). Computing the appropriate root numbers, we show next that if is locally abelian and no real place of becomes complex in . However the class of the square root of the inverse different is not trivial in general, in other words we have the following result.
There exists a tame Galois extension of even degree such that is a square and the class of is nontrivial in .
In fact, in Section LABEL:explicitexample we will explicitly describe a tame locally abelian -Galois extension , taken from [BachocKwon], such that in , where is the binary tetrahedral group (which has order ). This is, to our knowledge, the first example of a tame Galois extension of number fields whose square root of the inverse different (exists and) has nontrivial class. We also show that this example is minimal in the sense that if is a tame locally abelian -Galois extension with and .
2 Reduction to inertia subgroups
We use the notation of the Introduction, in particular is a tame -Galois extension. For a prime of , we denote by (resp. ) the inertia subgroup (resp. decomposition subgroup) of in . If is the prime ideal of below , we will often identify the Galois group of with , where (resp. ) is the completion of at (resp. of at ). In this section we show that the -module structure of the torsion modules we are interested in can be recovered by the knowledge, for finitely many primes of , of the -module structure of a certain torsion -module. More precisely, if denotes the set of primes of which ramify in , then , and can be written as a direct sum over of torsion -modules induced from -modules, where, for each , is any fixed prime above . For and such a decomposition follows directly from Chase’s results. For instance, in the case of , the corresponding -module is a suitable power of the quotient , where is the Swan module generated by the trace and the residual characteristic of .
However for torsion modules arising from general -stable ideals of (and in particular for ), we need to introduce torsion -modules and also assume that is locally abelian. Here is the order of (which indeed depends only on the prime of lying below ), is the group of th roots of unity in and is the ring of integers of . These -modules can be considered as analogues of and they also give a decomposition for , which is slightly different from that of Chase and will be needed in the proof of Theorem 1. To stress their similarity with , we shall denote by and those which correspond to and , respectively (see (LABEL:rchi) and (LABEL:schi) for a precise definition). Here is an injective character of and is a prime above in .
We now state the main result of this section whose proof will be given in §LABEL:reductionproof.
For every , choose a prime of above . Then, with the notation introduced above, there is an isomorphism of -modules
Furthermore, for every choice of injective characters for every prime as above, one can find primes and injections such that there is an isomorphism of -modules
Assume moreover that is locally abelian. Then the injections factor through and there is an isomorphism of -modules
In Section LABEL:ctsec we will see how the above theorem, together with Theorem 1, can be used to prove Theorem 2. More precisely, we show that the torsion -modules (resp. -modules) appearing in Theorem 2.1 are -cohomologically trivial (resp. -cohomologically trivial) and therefore define classes in (resp. ). Thus the isomorphisms of Theorem 2.1 can be translated into equalities of classes in , which in turn will give directly Theorem 2, assuming Theorem 1.
2.1 The torsion module
The results of this subsection are due to Chase [ChaseTors].
We shall first recall the definition of in a wider context. Let be a finite group and let be a -Galois extension of either global or local fields. If and are sets, we denote by the set of mappings from to . Consider the bijection
defined by for and . Now is a -module with acting on the left factor, on the right and is a -module with acting pointwise and acting by for all , . These structures make an isomorphism of -modules. Restricting to the subring yields an -modules injection
is a torsion -module. We refer the reader to [ChaseTors, Sections 2, 3, 4] and [Frohlich-Alg_numb, Note 6 to Chapter III] for more details on .
We now come back to the notation of the beginning of this section, in particular is a tame -Galois extension of number fields. Recall also that for every , we fix a prime of above .
There is an isomorphism of -modules
See [ChaseTors, Corollary 3.11]. ∎
Proposition 2.2 shows that we can focus on the local setting. Therefore in this subsection we shall put ourselves in the following situation (which will appear again at later stages of this paper). We fix a rational prime and a tamely ramified Galois extension of -adic fields inside a fixed algebraic closure of . We denote by the Galois group of . Let be the inertia subgroup of , which is cyclic of order denoted by , and set . As usual, , and denote the rings of integers of , and , respectively, and we shall denote by , and the corresponding maximal ideals.
The following result shows that we can in fact focus on totally and tamely ramified local extensions.
There is an isomorphism of -modules
See [ChaseTors, Corollary 3.8]. ∎
By standard theory contains the group of th roots of unity , and we can choose a uniformizer of such that (thus is a uniformizer of ). Consider the map defined by
Since is totally ramified, any unit such that lies in , hence does not depend on the choice of a uniformizer as above. It easily follows that is a group homomorphism, hence an isomorphism comparing cardinals: (see also [Serre, Chapitre IV, Propositions 6(a) and 7]).
Note that (resp. ) is a -module with the conjugation (resp. Galois) action. Then is in fact an isomorphism of -modules. To prove this, it is enough to verify that, for every and , we have . But indeed, if is as above, we have
since is a uniformizer of whose th power belongs to . Hence is a -isomorphism and we deduce in particular that, if is abelian, then . The reverse implication is also true: if , then acts trivially on . This implies that is abelian, since for any whose image in generates (which is a cyclic group).
If is an -module, we let denote the -module which is as an -module and has -action defined by .
We now show that the -module can be decomposed in smaller pieces.
The action of on factors through and there is an isomorphism of -modules
We start from Chase’s decomposition [ChaseTors, Theorem 2.8] which, in our notation, is an isomorphism of -modules:
Note that, for , is an -module, since and hence the action of on factors through . From its definition is an -module, hence an -module by the above isomorphism, which is thus an isomorphism of -modules. In what follows we shall be mainly concerned with -module structures (although some of the assertions hold true in the category of -modules).
Observe that the -module has the filtration whose corresponding subquotients are for . It is clear that multiplication by the th power of any uniformizer of induces an -isomorphism and hence an -isomorphism . Therefore using the semisimplicity of , we get
Note that the Galois action of on coincides with the action given by multiplication by since
where, for , we let denote the class of in (the last equality of (4) follows from the fact that acts trivially on ). Thus both and are -vector spaces of dimension on which acts by multiplication by . Therefore
as -modules. ∎
2.2 Torsion modules arising from ideals
We keep the notation of the beginning of this section. Let be a -invariant ideal. We will show that the -module structure of and is of local nature. We denote by the (finite) set of primes of dividing and, for every we fix a prime of above .
Let be a -invariant ideal. For every prime , let be the valuation of at any prime of above . Then there are isomorphisms of -modules
We begin by proving the first isomorphism. Since is -invariant, we can write
The Chinese remainder theorem gives an isomorphism
which is indeed also one of -modules. In a similar way we also get an isomorphism of -modules
for every . The above isomorphism is easily seen to be -invariant, once the right-hand side is given a -module structure by
for every , and . A standard argument shows that, for any prime above , we have
as -modules and -modules. Note also that, for every and every , the inclusion induces an isomorphism
of -modules. This shows the first isomorphism of the lemma.
The proof of the second isomorphism follows the same pattern, once one has the following analogue of the Chinese remainder theorem.
Let be ideals with . Then there is an isomorphism of -modules
We claim that the inclusions and induce -isomorphisms