A combinatorial interpretation of Serre's conjecture on modular Galois representations

Herremans, Adriaan

doi:10.5802/aif.1980

Cited by 9 publications

(8 citation statements)

References 2 publications

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“…For other uses of the Eicher-Shimura isomorphism together with the Shapiro lemma in the study of modular forms for congruence subgroups see [5,6].…”

Section: A Trace Formula On the Space Of Period Polynomialsmentioning

confidence: 99%

On the trace formula for Hecke operators on congruence subgroups

Popa

2018

Proc. Amer. Math. Soc.

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Dedicated to Don Zagier on the occasion of his 65th birthday.Abstract. We give a new, simple proof of the trace formula for Hecke operators on modular forms for finite index subgroups of the modular group. The proof uses algebraic properties of certain universal Hecke operators acting on period polynomials of modular forms, and it generalizes an approach developed by Don Zagier and the author for the modular group. This approach leads to a very simple formula for the trace on the space of cusp forms plus the trace on the space of modular forms. As applications, we investigate what happens when varying the weight or the level in the trace formula.

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“…For other uses of the Eicher-Shimura isomorphism together with the Shapiro lemma in the study of modular forms for congruence subgroups see [5,6].…”

Section: A Trace Formula On the Space Of Period Polynomialsmentioning

confidence: 99%

On the trace formula for Hecke operators on congruence subgroups

Popa

2018

Proc. Amer. Math. Soc.

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“…As a result, the picture becomes somewhat messy, and moreover, in different expositions different normalizations and distributions of determinants between f and P are adopted. Anyway, we will fix our choices by the following conventions (the same as in [He1], [He2]. )…”

Section: Proposition the Mapmentioning

confidence: 99%

Modular shadows and the Lévy—Mellin ∞—adic transform

Manin¹,

Marcolli²

2008

Modular Forms on Schiermonnikoog

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This paper continues the study of the structures induced on the "invisible boundary" of the modular tower and extends some results of [MaMar1]. We start with a systematic formalism of pseudo-measures generalizing the wellknown theory of modular symbols for SL(2). These pseudo-measures, and the related integral formula which we call the Lévy-Mellin transform, can be considered as an "∞-adic" version of Mazur's p-adic measures that have been introduced in the seventies in the theory of p-adic interpolation of the Mellin transforms of cusp forms, cf. [Ma2]. A formalism of iterated Lévy-Mellin transform in the style of [Ma3] is sketched. Finally, we discuss the invisible boundary from the perspective of non-commutative geometry. Definition.A pseudo-measure on P 1 (R) with values in a commutative group (written additively) W is a function µ : P 1 (Q) × P 1 (Q) → W satisfying the following conditions: for any α, β, γ ∈ P 1 (Q), µ(α, α) = 0, µ(α, β) + µ(β, α) = 0 , (1.1) µ(α, β) + µ(β, γ) + µ(γ, α) = 0 .( 1.2)

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“…and that the submodule H 1 par (Y 1 (N )(C), F k ) is obtained as the subgroup of elements of H 1 (Γ 1 (N ), Sym k−2 (Z 2 )) whose restriction to all unipotent subgroups of Γ 1 (N ) is zero (see, for example, [16, § 12.2]). For more details on this, especially in relation to Serre's conjectures in [39], one may consult [28].…”

Section: Proof the Description Ofmentioning

confidence: 99%

“…)) whose restriction to all unipotent subgroups of Γ 1 (N) is zero (see for example [17, §12.2]). For more details on this, especially in relation to Serre's conjectures in [40], one may consult [28].…”

Section: 1mentioning

confidence: 99%

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Comparison of Integral Structures on Spaces of Modular Forms of Weight Two, and Computation of Spaces of Forms Mod 2 of Weight One. With Appendices by Jean-François Mestre and Gabor Wiese

Edixhoven¹,

Mestre²,

Wiese³

2005

JMJ

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Suppose that φ(f, g) = 0. Then we have 0 = θ(Af + F (g)) = θ(Af ), hence 0 = θ(f ). But then f = 0, as the weight of f is not divisible by p. It follows that F (g) = 0, hence that g = 0. 4.5Now that we know how to characterize the image ofKatz , it becomes time to investigate how to compute this ambient vector space. We want to avoid the problems related to the lifting of elements of S p (Γ 1 (N), ε, F) ′ Katz to characteristic zero with a given character (see [22, §1] and [22, Prop. 1.10], they have to do with what is called Carayol's Lemma). So we describeKatz of characteristic zero forms of weight p with no prescribed character.Proposition 4.6 Let p be a prime number, and N ≥ 1 an integer not divisible by p. Suppose that N = 1 or that p ≥ 5.

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A combinatorial interpretation of Serre's conjecture on modular Galois representations

Cited by 9 publications

References 2 publications

On the trace formula for Hecke operators on congruence subgroups

On the trace formula for Hecke operators on congruence subgroups

Modular shadows and the Lévy—Mellin ∞—adic transform

Comparison of Integral Structures on Spaces of Modular Forms of Weight Two, and Computation of Spaces of Forms Mod 2 of Weight One. With Appendices by Jean-François Mestre and Gabor Wiese

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