Explicit Serre weights for two-dimensional Galois representations

Calegari, Frank; Emerton, Matthew; Gee, Toby; Mavrides, Lambros

doi:10.1112/s0010437x17007254

Cited by 7 publications

(23 citation statements)

References 13 publications

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“…Moreover in the formula (24) for such v, we have J = J 1 = J 2 , w = w 1 = w 2 , β 1 ≡ β 2 ≡ 1 mod pO F,p and Nm F/Q (v) ≡ 1 mod p, so that r J,w 0 (T v f ) = 2r J,w 0 (f ). Therefore if r J,w 0 (f ) = 0 for some (J, w), then ρ f (Frob v ) has characteristic polynomial (X − 1) 2 for such v. By the Cebotarev Density Theorem (and class field theory) it follows that ρ f (g) has characteristic polynomial (X − 1) 2 for all g ∈ G K , where K is the strict ray 6 Alternatively, this can be proved by revisiting the construction in Theorem 6.1.1 and observing that if r J,w 0 = 0 for some (J, w), then the liftf is non-cuspidal. One then deduces that the Galois representation ρf is reducible, and hence so is ρ f (possibly after extending scalars in the case p = 2).…”

Section: Normalised Eigenformsmentioning

confidence: 98%

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Minimal weights of Hilbert modular forms in characteristic

Diamond¹,

Kassaei²

2017

Compositio Math.

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Abstract. We consider mod p Hilbert modular forms associated to a totally real field of degree d in which p is unramified. We prove that every such form arises by multiplication by partial Hasse invariants from one whose weight (a d-tuple of integers) lies in a certain cone contained in the set of non-negative weights, answering a question of Andreatta and Goren. The proof is based on properties of the Goren-Oort stratification on mod p Hilbert modular varieties established by

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Section: Normalised Eigenformsmentioning

confidence: 98%

“…Proof. 6 If v is trivial in the strict class group of conductor np, then it follows from the definitions that S v acts trivially on M k,l (U ; E). Moreover in the formula (24) for such v, we have J = J 1 = J 2 , w = w 1 = w 2 , β 1 ≡ β 2 ≡ 1 mod pO F,p and Nm F/Q (v) ≡ 1 mod p, so that r J,w 0 (T v f ) = 2r J,w 0 (f ).…”

Section: Normalised Eigenformsmentioning

confidence: 99%

Minimal weights of Hilbert modular forms in characteristic

Diamond¹,

Kassaei²

2017

Compositio Math.

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“…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 [19]. 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 [5]. We remark that our restriction to the case where K is unramified over Q p is made essentially for simplicity.…”

Section: Introductionmentioning

confidence: 96%

“…We remark that our restriction to the case where K is unramified over Q p is made essentially for simplicity. The methods of this paper, and indeed of [5], are expected to apply to the general case where K/Q p is allowed to be ramified, but the resulting explicit description of the distinguished subspaces is likely to be much more complicated.…”

Section: Introductionmentioning

confidence: 99%

Serre Weights and Wild Ramification in Two-Dimensional Galois Representations

Dembélé

Diamond

Roberts

2016

Forum of Mathematics, Sigma

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A generalization of Serre's Conjecture asserts that if F is a totally real field, then certain characteristic p representations of Galois groups over F arise from Hilbert modular forms. Moreover, it predicts the set of weights of such forms in terms of the local behaviour of the Galois representation at primes over p. This characterization of the weights, which is formulated using p-adic Hodge theory, is known under mild technical hypotheses if p > 2. In this paper we give, under the assumption that p is unramified in F, 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 p-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.

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“…A definition of W (ρ) was first given in [BDJ10] in the case that K/Q l is unramified, and various generalisations and alternative definitions have subsequently been proposed. As a result of the main theorems of [GLS15,CEGM], all of these definitions are equivalent; we refer the reader to the introductions to those papers for a discussion of the various definitions.…”

Section: 2mentioning

confidence: 99%

G-valued local deformation rings and global lifts

Bellovin

Gee

2019

Alg. Number Th.

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We study G-valued Galois deformation rings with prescribed properties, where G is an arbitrary (not necessarily connected) reductive group over an extension of Z l for some prime l. In particular, for the Galois groups of p-adic local fields (with p possibly equal to l) we prove that these rings are generically regular, compute their dimensions, and show that functorial operations on Galois representations give rise to well-defined maps between the sets of irreducible components of the corresponding deformation rings. We use these local results to prove lower bounds on the dimension of global deformation rings with prescribed local properties. Applying our results to unitary groups, we improve results in the literature on the existence of lifts of mod l Galois representations, and on the weight part of Serre's conjecture.

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Explicit Serre weights for two-dimensional Galois representations

Cited by 7 publications

References 13 publications

Minimal weights of Hilbert modular forms in characteristic

Minimal weights of Hilbert modular forms in characteristic

Serre Weights and Wild Ramification in Two-Dimensional Galois Representations

G-valued local deformation rings and global lifts

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