Non-optimal levels of modl modular representations

Diamond, Fred; Taylor, Richard

doi:10.1007/bf01231768

Cited by 108 publications

(137 citation statements)

References 17 publications

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“…Such a condition is satisfied if the representation ρ arises from a newform in S k (Γ 1 (N p)) with (N, p) = 1. In particular, it is a necessary condition to raise the level of a modular representation from N to N p. In their paper [DT94], Diamond and Taylor prove that this is also sufficient when ρ is assumed to be irreducible.…”

Section: Then the Representation 1⊕χmentioning

confidence: 99%

“…1.1]) shows that a modular ρ indeed arises from a Hecke eigenform of this 'optimal' level. On the other hand, Diamond and Taylor soon thereafter ( [DT94]) completely described the set of prime-to-l integers M > N (ρ) such that ρ arises from a weight-k newform of level M , in terms of the ramification theory of ρ. They called such integers non-optimal levels attached to ρ.…”

Section: Introductionmentioning

confidence: 99%

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On the modularity of reducible mod $l$ Galois representations

Billerey

Menares

2016

Mathematical Research Letters

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Abstract. Given an odd, semisimple, reducible, 2-dimensional mod l Galois representation, we investigate the possible levels of the modular forms giving rise to it. When the representation is the direct sum of the trivial character and a power of the mod l cyclotomic character, we are able to characterize the primes that can arise as levels of the associated newforms. As an application, we determine a new explicit lower bound for the highest degree among the fields of coefficients of newforms of trivial Nebentypus and prime level. The bound is valid in a subset of the primes with natural (lower) density at least 3/4.

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Section: Then the Representation 1⊕χmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the modularity of reducible mod $l$ Galois representations

Billerey

Menares

2016

Mathematical Research Letters

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“…which can be shown to be an isomorphism by the same deformation theory argument in [8,Lemma 6]. This is enough to conclude, because the sheaves e † (ω v ∨ ) and e † ω v are dual of each other.…”

Section: Maass Operatorsmentioning

confidence: 74%

Power Series Expansions of Modular Forms and Their Interpolation Properties

Mori

2011

Int. J. Number Theory

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We define a power series expansion of a holomorphic modular form f in the p-adic neighborhood of a CM point x of type K for a split good prime p. The modularity group can be either a classical conguence group or a group of norm 1 elements in an order of an indefinite quaternion algebra. The expansion coefficients are shown to be closely related to the classical Maass operators and give p-adic information on the ring of definition of f . By letting the CM point x vary in its Galois orbit, the r-th coefficients define a p-adic K × -modular form in the sense of Hida. By coupling this form with the p-adic avatars of algebraic Hecke characters belonging to a suitable family and using a Rankin-Selberg type formula due to Harris and Kudla along with some explicit computations of Watson and of Prasanna, we obtain in the even weight case a p-adic interpolation for the square roots of a family of twisted special values of the automorphic L-function associated with the base change of f to K.

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“…Denote by m φ,p the kernel of θ φ . For any finite set S of prime ideals of O F , denote by T (S) n the subalgebra of T n generated by T q and S q for q n, q ∈ S and U q for q | n, q ∈ S. Define m [18] for F = Q and generalized by [28]. …”

Section: Hilbert Modular Formsmentioning

confidence: 99%

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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Non-optimal levels of modl modular representations

Cited by 108 publications

References 17 publications

On the modularity of reducible mod $l$ Galois representations

On the modularity of reducible mod $l$ Galois representations

Power Series Expansions of Modular Forms and Their Interpolation Properties

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

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