Formes modulaires de Hilbert modulo $p$ et valeurs d'extensions entre caractères galoisiens

Breuil, Christophe; Diamond, Fred

doi:10.24033/asens.2230

Cited by 38 publications

(91 citation statements)

References 29 publications

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“…This immediate from Lemmas 6.1, 6.2, 7.1 and 7.2, and a standard level lowering recipe derived in (Freitas and Siksek, 2015a, Section 2.3) from the work of Jarvis, Fujiwara and Rajaei. Alternatively, one could use modern modularity lifting theorems which integrate level lowering with modularity lifting, as for example in Breuil and Diamond (2014).…”

Section: Proof Of Theoremmentioning

confidence: 99%

Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ+ y2m= zp

Anni¹,

Siksek²

2016

Algebra Number Theory

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Abstract. Let K be a real abelian field of odd class number in which 5 is unramified. Let S 5 be the set of places of K above 5. Suppose for every non-empty proper subset S ⊂ S 5 there is a totally positive unit u ∈ O K such that q∈S Norm Fq/F 5 (u mod q) = 1. We prove that every semistable elliptic curve over K is modular, using a combination of several powerful modularity theorems and class field theory. We deduce that if K is a real abelian field of conductor n < 100, with 5 ∤ n and n = 29, 87, 89, then every semistable elliptic curve E over K is modular.Let ℓ, m, p be prime, with ℓ, m ≥ 5 and p ≥ 3. To a putative non-trivial primitive solution of the generalized Fermat x 2ℓ +y 2m = z p we associate a Frey elliptic curve defined over Q(ζp) + , and study its mod ℓ representation with the help of level lowering and our modularity result. We deduce the non-existence of non-trivial primitive solutions if p ≤ 11, or if p = 13 and ℓ, m = 7.

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Section: Proof Of Theoremmentioning

confidence: 99%

Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ+ y2m= zp

Anni¹,

Siksek²

2016

Algebra Number Theory

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“…(We refer to Remark 4.5.8 for variations on assumption (ii). The stronger assumptions appearing in Remark 4.5.9 are analogous to the multiplicity one conditions appearing in [BD14].) Interestingly, the argument proving Theorem A also lets us deduce the hardest part of Theorem D, namely the existence of the shadow weights F (a, c, b − p + 1), F (b + p − 1, a, c) in the two exceptional cases.…”

mentioning

confidence: 87%

“…By Stickelberger's theorem (see e.g. [BD14], Théorème 2.5.1) one has ord p (κ 1 ) = 0, ord p (κ 2 ) = 1 and…”

Section: Let Us Definementioning

confidence: 99%

On mod local-global compatibility for in the ordinary case

Herzig¹,

Le²,

Morra³

2017

Compositio Math.

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Abstract. Suppose that F/F + is a CM extension of number fields in which the prime p splits completely and every other prime is unramified. Fix a place w|p of F . Suppose that r : Gal(F /F ) → GL 3 (Fp) is a continuous irreducible Galois representation such that r| Gal(F w /Fw) is upper-triangular, maximally non-split, and generic. If r is automorphic, and some suitable technical conditions hold, we show that r| Gal(F w /Fw) can be recovered from the GL 3 (Fw)-action on a space of mod p automorphic forms on a compact unitary group. On the way we prove results about weights in Serre's conjecture for r, show the existence of an ordinary lifting of r, and prove the freeness of certain Taylor-Wiles patched modules in this context. We also show the existence of many Galois representations r to which our main theorem applies.

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“…Minimal level case. We recall the minimal level case as in [6] and more generally in [13]. Fix a place v|p.…”

Section: 3mentioning

confidence: 99%

“…We can use the action of Hecke operators away from v on π D (r) to define a local factor π D v (r) at v, which is an admissible smooth representation of GL 2 (F v ), and is supposed to be the right candidate in the mod p local Langlands correspondence, and many important properties about it have been established, see e.g. [14], [6], [13]. Our main result is the following theorem.…”

Section: Introductionmentioning

confidence: 99%

Multiplicity one for the $\mathrm{mod} \: p$ cohomology of Shimura curves: the tame case

Wang

2018

Mathematical Research Letters

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Let F be a totally real field, p an unramified place of F dividing p and r : Gal(F /F ) → GL 2 (Fp) a continuous irreducible modular representation. The work of Buzzard, Diamond and Jarvis [9] associates to r an admissible smooth representation of GL 2 (Fp) on the mod p cohomology of Shimura curves attached to indefinite division algebras which split at p. When r| Gal(Fp/Fp) is tamely ramified and generic (and under some additional technical assumptions), we determine the subspace of invariants of this representation under the principal congruence subgroup of level p. In particular, the subspace depends only on r| Gal(Fp/Fp) and satisfies a multiplicity one property.Theorem 1.1. Assume that F v is unramified over Q p and ρ is tamely ramified and generic. Under certain assumptions (see Cor. 3.9), we have πRemark 1.2. After the first version of the paper was written, we are informed that Le, Morra and Schraen obtain a similar result independently [19]. Both the proofs use results of [13] as a global input, however, the local representation theory part is quite different.We also prove a similar result when D is a definite quaternion algebra unramified at all places over p. In [13], the subspace of pro-p-Iwahori fixed vectors of π D v (r) is determined, for ρ generic, but could be reducible non-split. The proof of our theorem uses the construction of [13] as a main tool. We intend to extend the result to reducible non-split ρ in future work.The organization of the paper is as follows. In Section 2, we give all the local results that we need, especially the local criterion Corollary 2.29. Note that the criterion does not apply to ρ reducible non-split (see Remark 2.27). In Section 3, we deduce our main theorem from our local results and results of [13].

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Formes modulaires de Hilbert modulo $p$ et valeurs d'extensions entre caractères galoisiens

Cited by 38 publications

References 29 publications

Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ+ y2m= zp

Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ+ y2m= zp

On mod local-global compatibility for in the ordinary case

Multiplicity one for the $\mathrm{mod} \: p$ cohomology of Shimura curves: the tame case

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