Serre weights and wild ramification in two-dimensional Galois representations
A generalization of Serre’s Conjecture asserts that if is a totally real field, then certain characteristic representations of Galois groups over arise from Hilbert modular forms. Moreover it predicts the set of weights of such forms in terms of the local behavior of the Galois representation at primes over . This characterization of the weights, which is formulated using -adic Hodge theory, is known under mild technical hypotheses if . In this paper we give, under the assumption that is unramified in , a conjectural alternative description for the set of weights. Our approach is to use the Artin–Hasse exponential and local class field theory to construct bases for local Galois cohomology spaces in terms of which we identify subspaces that should correspond to ones defined using -adic Hodge theory. The resulting conjecture amounts to an explicit description of wild ramification in reductions of certain crystalline Galois representations. It enables the direct computation of the set of Serre weights of a Galois representation, which we illustrate with numerical examples. A proof of this conjecture has been announced by Calegari, Emerton, Gee and Mavrides.
2010 Mathematics Subject Classification:11F80 (primary), and 11S15, 11S25 (secondary)
is a continuous, odd, irreducible representation, then arises from a Hecke eigenform in the space of cusp forms of some weight and level . Serre in fact formulated a refined version of the conjecture specifying the minimal such and subject to the constraints and ; a key point is that the weight depends only on the restriction of to a decomposition group at , and the level on ramification away from . The equivalence between the weaker version of the conjecture and its refinement was already known through the work of many authors for , and finally settled for as well by Khare and Wintenberger.
Buzzard, Jarvis and one of the authors  considered a generalization of Serre’s conjecture to the setting of Hilbert modular forms for a totally real number field and formulated an analogous refinement for representations assuming is unramified in ; versions without this assumption are given in  and . The equivalence between the conjecture and its refinement was proved, assuming and a Taylor–Wiles hypothesis on , in a series of papers by Gee and several sets of co-authors culminating in  and , with an alternative to the latter provided by Newton . Generalizations to higher-dimensional Galois representations have also been studied by Ash, Herzig and others beginning with ; see  for recent development.
One of the main difficulties in even formulating refined versions of generalizations of Serre’s conjecture is in prescribing the weights; the approach taken in  and subsequent papers, at least if is wildly ramified at primes over , is to do this in terms of Hodge–Tate weights of crystalline lifts of . The main purpose of this paper is to make the recipe for the set of weights more explicit. In view of the connection between Serre weights and crystalline lifts, this amounts to a conjecture in explicit -adic Hodge theory about wild ramification in reductions of crystalline Galois representations.
We now explain this in more detail. Let be a totally real number field, its ring of integers, a non-zero ideal of , the set of embeddings and suppose with all and of the same parity. A construction completed by Taylor in  then associates a -adic Galois representation to each Hecke eigenform in the space of Hilbert modular cusp forms of weight and level . One then expects that every continuous, irreducible, totally odd
is modular in the sense that it is arises as the reduction of such a Galois representation. One further expects that the prime-to- part of the minimal level from which arises is its Artin conductor, but the prediction of the possible weights is more subtle. If is unramified in , then a recipe is given in  in terms of the restrictions of to decomposition groups at primes over . An interesting feature of this recipe not so apparent over is the dependence of the conjectured weights on the associated local extension class when the restriction at is reducible. If
for some characters , then the resulting short exact sequence
defines a class in
where and . The class is well-defined up to a scalar in , in the sense that another choice of basis with respect to which has the form (1) yields a non-zero scalar multiple of . Alternatively, we may view as the class in defined by the cocycle obtained by writing
The space has dimension at least , with equality unless is trivial or cyclotomic. Whether is modular of a particular weight depends on whether this extension class lies in a certain distinguished subspace of whose definition relies on constructions from -adic Hodge theory. If and is wildly ramified at , then the associated extension class is a non-trivial element of a space of dimension at least two, making it difficult to determine the set of weights without a more explicit description of the distinguished subspaces.
We address the problem in this paper by using local class field theory and the Artin–Hasse exponential to give an explicit basis for the space of extensions (Corollary 5.2), in terms of which we provide a conjectural alternate description of the distinguished subspaces (Conjecture 7.2). We point out that a related problem is considered by Abrashkin in ; in particular, the results of  imply cases of our conjecture where the distinguished subspaces can be described using the ramification filtration on .
An earlier version of this paper was posted on the arXiv in March 2016. At the time, we reported that a proof of Conjecture 7.2 under certain genericity hypotheses would be forthcoming in the Ph.D thesis of Mavrides . In fact, Conjecture 7.2 has now been proved completely by Calegari, Emerton, Gee, and Mavrides in a preprint posted to the arXiv in August 2016 . We remark that our restriction to the case where is unramified over is made essentially for simplicity. The methods of this paper, and indeed of , are expected to apply to the general case where is allowed to be ramified, but the resulting explicit description of the distinguished subspaces is likely to be much more complicated.
The now-proved Conjecture 7.2 immediately yields an alternate description of the set of Serre weights for . Combining this with the predicted modularity of gives Conjecture 7.3, for which we have gathered extensive computational evidence. Indeed the appeal of our description is that one can compute the set of Serre weights directly from . In this paper, we illustrate this computation systematically in several examples with quadratic and . A sequel paper  will support Conjecture 7.3 via a much broader range of examples and elaborate on computational methods. In particular, the examples provided in  illustrate subtle features of the recipe for the weights arising only when is highly non-generic, with particular attention to the case . Such examples were instrumental in leading us to Conjecture 7.2 in its full generality.
This paper is structured as follows: In Section 2 we recall the general statement of the weight part of Serre’s conjecture for unramified at . In Sections 3, 4 and 5, we study the space of extensions in detail, arriving at an explicit basis in terms of the Artin–Hasse exponential. In Sections 6 and 7, we use this basis to give our conjectural description of the distinguished subspaces appearing in the definition of the set of Serre weights. We illustrate this description in more detail in the quadratic case in Section 8, and with numerical examples for in Sections 9 and 10. We remark that aside from these examples and the discussion of Serre’s conjecture at the end of Sections 2 and 7, the setting for the paper is entirely local.
2. Serre weights
2.1. Notation and general background
Let be an unramified extension of with ring of integers and residue field , and let . We fix algebraic closures and of and , and let denote the set of embeddings . We let denote the algebraic closure obtained as the residue field of the ring of integers of , and we identify with the set of embeddings via the canonical bijection.
For a field , we write for the absolute Galois group of . We let denote the inertia subgroup of , i.e. the kernel of the natural surjection . We write for the absolute (arithmetic) Frobenius elements on and on , and for the arithmetic Frobenius element of . We let denote the Artin map, normalized in the standard way, so the image of any uniformizer of in is .
Recall that the fundamental character is defined by
where is any root of and . Then the composite of with the Artin map is the homomorphism sending to and any element of to its reduction mod . Replacing by a root of for alters by an unramified character, so in fact is independent of the choice of uniformizer of . For each , we define the associated fundamental character to be .
A Serre weight (for ) is an irreducible -representation of . Recall that these are precisely the representations of the form
where , and for each . Moreover we can assume that for each and that for some , in which case the resulting representations are also inequivalent.
Let be a continuous representation. The next two subsections recall from  the definition of the set of Serre weights associated to .
2.2. Serre weights associated to a reducible representation
Suppose first that is reducible and write . The isomorphism class of is then determined by the ordered pair and a cohomology class , where we set . We first define a set
For each pair we will define a subspace , but we first need to recall the notion of labelled Hodge–Tate weights.
Recall that if is an -dimensional vector space over and is a crystalline (hence de Rham) representation, then is a free module of rank over endowed with an (exhaustive, separated) decreasing filtration by (not necessarily free) -submodules. Writing , we have a corresponding decomposition where each is an -dimensional filtered vector space over . For each , the multiset of -labelled Hodge–Tate weights of are the integers with multiplicity . In particular if is a crystalline character, then it has a unique -labelled Hodge–Tate weight for each . One finds that the vector determines up to an unramified character, and that .
Returning to the definition of , suppose that . Let be the crystalline lift of with -labelled Hodge–Tate weight (resp. ) for (resp. ) such that , and similarly let be the crystalline lift of with -labelled Hodge–Tate weight (resp. ) for (resp. ) such that . We then let denote the set of extension classes associated to reductions of crystalline extensions of by . We then set except in the following two cases (continuing to denote by ):
If is cyclotomic, and , then ;
if is trivial and , then where is the set of unramified homomorphisms .
Finally we define by the rule
Thus if and only if where is defined as the union of the over
(so depends on the ordered pair , and it is understood to be the empty set if ).
We remark that has dimension at least , with equality holding unless is trivial or cyclotomic (see [4, Lemma 3.12]). Moreover in a typical situation (for example if with for all ), the projection from to the set of subsets of is bijective, and the projection to the set of Serre weights is injective (see [4, Section 3.2]). In that case has cardinality and hence so does that of if . On the other hand, one would expect that for “most” , the class does not lie in any of the proper subspaces for , so that contains a single Serre weight.
It is not however true in general that the projection from to the set of Serre weights is injective, i.e., may have cardinality greater than , in which case it is not immediate from the definition of that it is a subspace of . However it is proved in  if and , then there is an element such that , so that is in fact a subspace. Indeed the proof of Theorem 9.1 of  shows that if and , then has a crystalline lift with -labelled Hodge–Tate weights , if and only if and . It follows that for all (using that in the exceptional case where is cyclotomic and ), and that if and only if has a crystalline lift with -labelled Hodge–Tate weights .
The main aim of this paper is to use local class field theory to give a more explicit description of , and to use this description to define subspaces which we conjecture coincide with the (even for ).
2.3. Serre weights associated to an irreducible representation
While the focus of this paper is on the case where is reducible, for completeness we recall the definition of in the case where is irreducible. We let denote the quadratic unramified extension of , the residue field of , the set of embeddings of in , and the natural projection . For , we let denote the corresponding fundamental character of . Note that if is irreducible, then it is necessarily tamely ramified and in fact induced from a character of . We define by the rule:
It is true in this case as well that typically has cardinality (see [4, Section 3.1]). Moreover the result of  characterizing in terms of reductions of crystalline representations (for ) holds in the irreducible case as well.
2.4. The case
To indicate the level of complexity hidden in the general recipe for weights, we describe the set more explicitly in the classical case . Replacing by a twist, we can assume has the form or for some with (where is the mod cyclotomic character). In the first case we find that (with the two weights coinciding if ). In the second case we may further assume (after twisting) that for some character . Since the space is one-dimensional unless is trivial or cyclotomic, one does not need much information about the spaces in order to determine ; indeed all one needs is that:
unless is cyclotomic and ;
is the peu ramifiée subspace if is cyclotomic, i.e., the subspace corresponding to under the Kummer isomorphism .
It then follows (see ) that
We remark that the first case above is the most typical and the next one arises in the setting of “companion forms.” The remaining cases take into account special situations that arise when or its inverse is trivial or cyclotomic.
2.5. Serre’s conjecture over totally real fields
We now recall how Serre weights arise in the context of Galois representations associated to automorphic forms. Let be a totally real field in which is unramified. Let denote its ring of integers and the set of primes of dividing . For each , we let , and the set of embeddings . The irreducible -representations of are then of the form: where each is a Serre weight for .
The representation is modular of weight if and only if for all .
We refer the reader to  for the definition of modularity of weight and its relation to the usual notion of weights of Hilbert modular forms. We just remark that is modular of some weight if and only if is modular in the usual sense that for some Hilbert modular eigenform , and that the set of weights for which is modular determines the possible cohomological weights and local behavior at primes over of those eigenforms (see [4, Prop. 2.10]).
Under the assumption that is modular (of some weight), Conjecture 2.1 can be viewed as the generalization of the weight part of Serre’s Conjecture and has been proved under mild technical hypotheses (for ) in a series of papers by Gee and coauthors culminating in , together with the results of either Gee and Kisin  or Newton . Moreover their result holds without the assumption that is unramified in using the description of in terms of reductions of crystalline representations.
Finally we remark that Conjecture 2.1 is known in the case . In this case the modularity of is a theorem of Khare and Wintenberger [17, 18], and the weight part follows from prior work of Gross, Edixhoven and others (see [4, Thm. 3.20]); it amounts to the statement that if , then arises from a Hecke eigenform of weight and level prime to if and only if .
3. The ramification filtration on cohomology
In this section we use the upper numbering of ramification groups to define filtrations on the Galois cohomology groups parametrizing the extensions of characters under consideration.
3.1. Definition of the filtration
Continue to let denote a finite unramified extension of of degree with residue field , and let be any character. Recall from [23, IV.3] that has a decreasing filtration by closed subgroups where , for , and is the wild ramification subgroup . We define an increasing filtration on by setting
for . Note that for , and that
Let be a cocycle representing a class in . Since is trivial, the restriction of defines a homomorphism ; so if , then if and only if for all . In particular, if and only if ; since , it follows that for .
3.2. Computation of the jumps in the filtration
For any , we set . Since , the compactness of and continuity of the cocycle imply that in fact
We will now compute the jumps in the filtration, i.e., the dimension of
for every and .
We must first introduce some notation. Choose an embedding , let where is the absolute Frobenius on . Recall that denotes the character defined by
where is any root of in , and set for . We may then write where for integers satisfying for . Moreover this expression is unique if we further require (in the case that is the cyclotomic character) that some for some . We extend the definition of to all integers by setting if . We define to be the tame signature of ; thus the tame signature of is an element of the set
Define an action of on by the formula
Note that if has tame signature , then has tame signature , as does where is the (outer) automorphism of defined by conjugation by a lift of to . We define be the period of to be the cardinality of its orbit under , and the absolute niveau of to be the period of its tame signature. (Note that the orbit of the tame signature of under is independent of the choice of .)
For , we define
so that and .
Let for . Then unless or . Moreover if and , then for some integer not divisible by . More precisely, if has tame signature of period and the integers are defined by (5), then:
if is trivial and otherwise;
if , then
if , then
if is cyclotomic, and otherwise.
Proof. We let denote the value claimed for in the statement. Note that if , then is the number of such that , where is the set of such that and for some with . Moreover is in bijection with the set of such that , and if , then . Therefore
Similarly if , then is the number of such that and ; moreover if , then , so
It follows that
Therefore , so it suffices to prove that for all , and we need only consider such that .
For , the inflation-restriction exact sequence
shows that has dimension if is trivial, and otherwise, so that . We may therefore assume that and that is an integer. Moreover either and is not divisible by , or .
Let where and is an unramified extension of