Level lowering for modular mod $\ell$ representations over totally real fields

Jarvis, Frazer

doi:10.1007/s002080050255

Cited by 25 publications

(23 citation statements)

References 9 publications

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“…Level-lowering theorems of Fujiwara [32], Rajaei [57] and the third author [44,45] can be viewed as partial results in the direction of Conjectures 4.7 and 4.9, of which they are also consequences. Proposition 4.13.…”

Section: Mod ℓ Langlands Correspondencesmentioning

confidence: 99%

On Serre's conjecture for mod ℓ Galois representations over totally real fields

2010

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Abstract. In 1987 Serre conjectured that any mod ℓ two-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation of this conjecture to 2-dimensional representations of the absolute Galois group of a totally real field where ℓ is unramified. The hard work is in formulating an analogue of the "weight" part of Serre's conjecture. Serre furthermore asked whether his conjecture could be rephrased in terms of a "mod ℓ Langlands philosophy". Using ideas of Emerton and Vignéras, we formulate a mod ℓ local-global principle for the group D * , where D is a quaternion algebra over a totally real field, split above ℓ and at 0 or 1 infinite places, and show how it implies the conjecture.

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Section: Mod ℓ Langlands Correspondencesmentioning

confidence: 99%

On Serre's conjecture for mod ℓ Galois representations over totally real fields

2010

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“…This result is provided over Q by [DT,Theorem 2], which establishes that the action of G Q on a certain module factors through Gal(Q ab /Q). The analogue when F = Q still hold: see [Ja,Section 6]. …”

Section: Congruences Between Modular Forms and The Euler Systemmentioning

confidence: 99%

Anticyclotomic Iwasawa’s Main Conjecture for Hilbert modular forms

Longo¹

2012

Comment. Math. Helv.

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“…Eisenstein ideals. We generalize the notion of Eisenstein ideals to F as in [Jar1]. Let d ⊂ O F be an integral ideal, and set …”

Section: Mod P Hilbert Modular Formsmentioning

confidence: 99%

“…In the case F = Q, this is Proposition 2 of [DT], and the proof is essentially the same. See also the comments in section 3 of [Jar1]. Let ρ m be reducible.…”

Section: Mod P Hilbert Modular Formsmentioning

confidence: 99%

Weights of Galois representations associated to Hilbert modular forms

Schein¹

2008

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. Let F be a totally real field, p ≥ 3 a rational prime unramified in F , and p a place of F over p. Let ρ : Gal(F /F ) → GL2(Fp) be a two-dimensional mod p Galois representation which is assumed to be modular of some weight and whose restriction to a decomposition subgroup at p is irreducible. We specify a set of weights, determined by the restriction of ρ to inertia at p, which contains all the modular weights for ρ. This proves part of a conjecture of Diamond, Buzzard, and Jarvis, which provides an analogue of Serre's epsilon conjecture for Hilbert modular forms mod p.

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Level lowering for modular mod $\ell$ representations over totally real fields

Cited by 25 publications

References 9 publications

On Serre's conjecture for mod ℓ Galois representations over totally real fields

On Serre's conjecture for mod ℓ Galois representations over totally real fields

Anticyclotomic Iwasawa’s Main Conjecture for Hilbert modular forms

Weights of Galois representations associated to Hilbert modular forms

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