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 (2014)

  • Issue: 1, page 45-71
  • ISSN: 1958-7236

Abstract

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Let F be an imaginary quadratic field and 𝒪 its ring of integers. Let 𝔫 ⊂ 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 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 𝔽 ¯ p ⊗ d e t e for some e ≥ 0 ; except possibly in some few cases.

How to cite

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Mohamed, Adam. "Weight reduction for cohomological mod $p$ modular forms over imaginary quadratic fields." Publications mathématiques de Besançon (2014): 45-71. <http://eudml.org/doc/275780>.

@article{Mohamed2014,
abstract = {Let $F$ be an imaginary quadratic field and $ \mathcal\{O\}$ its ring of integers. Let $ \mathfrak\{ n\} \subset \mathcal\{ O\} $ be a non-zero ideal and let $ p&gt; 5$ be a rational inert prime in $F$ and coprime with $ \mathfrak\{ n\}$. Let $ V$ be an irreducible finite dimensional representation of $ \overline\{\mathbb\{F\}\}_\{p\}[\{\rm GL\}_2(\mathbb\{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 $ \overline\{\mathbb\{F\}\}_\{p\}\otimes det^e$ for some $ e \ge 0$; except possibly in some few cases.},
affiliation = {Universität Duisburg-Essen, Institut für Experimentelle Mathematik, Ellernstr 29, 45326 Essen, Germany},
author = {Mohamed, Adam},
journal = {Publications mathématiques de Besançon},
keywords = {Modular forms modulo $p$; imaginary quadratic fields; Hecke operators; Serre weight; modular forms modulo },
language = {eng},
number = {1},
pages = {45-71},
publisher = {Presses universitaires de Franche-Comté},
title = {Weight reduction for cohomological mod $p$ modular forms over imaginary quadratic fields},
url = {http://eudml.org/doc/275780},
year = {2014},
}

TY - JOUR
AU - Mohamed, Adam
TI - Weight reduction for cohomological mod $p$ modular forms over imaginary quadratic fields
JO - Publications mathématiques de Besançon
PY - 2014
PB - Presses universitaires de Franche-Comté
IS - 1
SP - 45
EP - 71
AB - Let $F$ be an imaginary quadratic field and $ \mathcal{O}$ its ring of integers. Let $ \mathfrak{ n} \subset \mathcal{ O} $ be a non-zero ideal and let $ p&gt; 5$ be a rational inert prime in $F$ and coprime with $ \mathfrak{ n}$. Let $ V$ be an irreducible finite dimensional representation of $ \overline{\mathbb{F}}_{p}[{\rm GL}_2(\mathbb{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 $ \overline{\mathbb{F}}_{p}\otimes det^e$ for some $ e \ge 0$; except possibly in some few cases.
LA - eng
KW - Modular forms modulo $p$; imaginary quadratic fields; Hecke operators; Serre weight; modular forms modulo
UR - http://eudml.org/doc/275780
ER -

References

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  16. G. Wiese, On the faithfulness of parabolic cohomology as a Hecke module over a finite field, J. Reine Angew. Math. ( 2007), 79-103. Zbl1126.11028MR2337642

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