On modular representations of $$(\bar Q/Q)$$ arising from modular forms

“…Namely we know in this case thatρ has projectively dihedral image, and thus by a result of Hecke is modular (see remark in Section 5.1 of [43]). Then results of [36], [17] prove that it arises from S k(ρ) (Γ 1 (N (ρ))). …”

Section: Theorem 52 (I)mentioning

confidence: 99%

On Serre’s conjecture for 2-dimensional mod p representations of Gal(ℚ∕ℚ)

Khare¹,

Wintenberger

2009

Ann. Math.

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We prove the existence in many cases of minimally ramified p-adic lifts of 2-dimensional continuous, odd, absolutely irreducible, mod p representations ρ of the absolute Galois group of Q. It is predicted by Serre's conjecture that such representations arise from newforms of optimal level and weight.Using these minimal lifts, and arguments using compatible systems, we prove some cases of Serre's conjectures in low levels and weights. For instance we prove that there are no irreducible (p, p) type group schemes over Z. We prove that aρ as above of Artin conductor 1 and Serre weight 12 arises from the Ramanujan Delta-function.In the last part of the paper we present arguments that reduce Serre's conjecture to proving generalisations of modularity lifting theorems of the type pioneered by Wiles.

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“…-If n − = O F , this is the main result of [45,Theorem 1]. When F = Q, see [39,Section 7]. This argument has been extended to Hilbert modular forms when n − = O F by [38,Corollary 4].…”

Section: Congruences Between Modular Formsmentioning

confidence: 94%

On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields

Longo¹

2006

Ann. inst. Fourier

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Let E/F be a modular elliptic curve defined over a totally real number field F and let f be its associated eigenform. This article presents a new method, inspired by a recent work of Bertolini and Darmon, to control the rank of E over suitable quadratic imaginary extensions K/F. In particular, this argument can also be applied to the cases not covered by the work of Kolyvagin and Logachev, that is, when [F : Q] is even and phi not new at any prime

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On modular representations of $$(\bar Q/Q)$$ arising from modular forms

Cited by 392 publications

References 24 publications

The Iwasawa Main Conjectures for GL2

The Iwasawa Main Conjectures for GL2

On Serre’s conjecture for 2-dimensional mod p representations of Gal(ℚ∕ℚ)

On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields

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