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On the Endomorphism Algebra of Abelian Varieties Associated with Hilbert Modular Forms

Alireza Shavali

Abstract

In this article, we will generalize an explicit formula proved by Quer for the Brauer class of the endomorphism algebra of abelian varieties associated to modular forms of weight 2 to the case of Hilbert modular forms of parallel weight 2, under the condition that the degree of the base field over Q is an odd number.

On the Endomorphism Algebra of Abelian Varieties Associated with Hilbert Modular Forms

Abstract

In this article, we will generalize an explicit formula proved by Quer for the Brauer class of the endomorphism algebra of abelian varieties associated to modular forms of weight 2 to the case of Hilbert modular forms of parallel weight 2, under the condition that the degree of the base field over Q is an odd number.

Paper Structure

This paper contains 4 sections, 20 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Endomorphism Ring and Galois representation
  3. Twisted Algebras and Galois Cohomology
  4. Computing the Brauer Class

Key Result

Lemma 1

For each embedding $\sigma$ one has $\mathrm{End}_{{\mathbb Q}_{\ell}[G_M]}(V_{\sigma}) = {\mathbb Q}_{\ell}$. In particular, $V_{\sigma}$ is absolutely irreducible as a $G_M$-representation.

Theorems & Definitions (32)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • proof
  • Lemma 3
  • proof
  • Corollary 2
  • proof
  • ...and 22 more