Weight reduction for cohomological mod $p$ modular forms over imaginary quadratic fields

Adam Mohamed

doi:10.5802/pmb.4

Weight reduction for cohomological mod

p

modular forms over imaginary quadratic fields

Adam Mohamed ¹

¹ Universität Duisburg-Essen, Institut für Experimentelle Mathematik, Ellernstr 29, 45326 Essen, Germany

Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2014), pp. 45-71.

Abstract
Résumé

Let $F$ be an imaginary quadratic field and $𝒪$ its ring of integers. Let $𝔫 \subset 𝒪$ be a non-zero ideal and let $p > 5$ be a rational inert prime in $F$ and coprime with $𝔫$ . Let $V$ be an irreducible finite dimensional representation of ${\bar{𝔽}}_{p} [{GL}_{2} (𝔽_{p^{2}})]$ . We establish that a system of Hecke eigenvalues appearing in the cohomology with coefficients in $V$ already lives in the cohomology with coefficients in ${\bar{𝔽}}_{p} \otimes d e t^{e}$ for some $e \geq 0$ ; except possibly in some few cases.

Soient $F$ un corps quadratique imaginaire et $𝒪$ son anneau d’entiers. Soient $𝔫 \subset 𝒪$ un idéal non nul et $p > 5$ un nombre premier inerte dans $F$ copremier avec $𝔫$ . Soit $V$ une représentation irréductible de dimension finie de ${\bar{𝔽}}_{p} [{GL}_{2} (𝔽_{p^{2}})]$ . Nous établissons qu’un système de valeurs propres de Hecke appartenant au groupe de cohomologie â ?¡ coefficients dans $V$ appartient aussi au groupe de cohomologie â ?¡ coefficients dans ${\bar{𝔽}}_{p} \otimes d e t^{e}$ pour $e \geq 0$ à l’exception, éventuellement, de quelques cas.

Received: 2013-10-13
Published online: 2015-04-13

MR Zbl

DOI: 10.5802/pmb.4

Classification: 11F75, 11F67, 11F25, 11F41
Keywords: Modular forms modulo $p$, imaginary quadratic fields, Hecke operators, Serre weight

Author's affiliations:

Adam Mohamed ¹

¹ Universität Duisburg-Essen, Institut für Experimentelle Mathematik, Ellernstr 29, 45326 Essen, Germany

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Adam Mohamed. Weight reduction for cohomological mod $p$ modular forms over imaginary quadratic fields. Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2014), pp. 45-71. doi : 10.5802/pmb.4. https://pmb.centre-mersenne.org/articles/10.5802/pmb.4/

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