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Restrictions on endomorphism rings of jacobians and their minimal fields of definition

Pip Goodman

Abstract

Zarhin has extensively studied restrictions placed on the endomorphism algebras of Jacobians $J$ for which the Galois group associated to their 2-torsion is insoluble and 'large' (relative to the dimension of $J$). In this paper we examine what happens when this Galois group merely contains an element of 'large' prime order. In doing so we obtain a partial converse to a result by Guralnick and Kedlaya on the endomorphism field.

Restrictions on endomorphism rings of jacobians and their minimal fields of definition

Abstract

Zarhin has extensively studied restrictions placed on the endomorphism algebras of Jacobians for which the Galois group associated to their 2-torsion is insoluble and 'large' (relative to the dimension of ). In this paper we examine what happens when this Galois group merely contains an element of 'large' prime order. In doing so we obtain a partial converse to a result by Guralnick and Kedlaya on the endomorphism field.

Paper Structure

This paper contains 1 section, 17 theorems, 12 equations, 1 table.

Table of Contents

  1. Bounds on the endomorphism field

Key Result

Lemma 2.12

Let $l$ and $p = 2g+1$ be distinct primes such that $g$ is odd and the multiplicative order of $l$ modulo $p$ is $g$. Let $V$ be a symplectic space of dimension $2g$ over $\mathbb{F}_l$, and $\rho : \mathbb{Z}/p\mathbb{Z} \rightarrow V$ a faithful representation which preserves the symplectic pairin

Theorems & Definitions (34)

  • Lemma 2.12
  • proof
  • Lemma 2.13
  • Lemma 2.14
  • proof
  • Theorem 2.15
  • proof
  • Theorem 2.16
  • proof
  • Lemma 3.1
  • ...and 24 more