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Torsion points on $\rm{GL}_2$-type abelian varieties

Jessica Alessandrì, Nirvana Coppola

Abstract

It is well known that the rational torsion of an abelian variety defined over a number field injects into the reduction modulo any sufficiently large prime, so the order of the torsion group divides the greatest common divisor of the sizes of points on the reduction at each prime. Drawing inspiration from Katz's Inventiones paper (1981), we investigate the converse to this for abelian varieties of $\rm GL_2$-type and exhibit a conjectural list of possible torsion orders for modular abelian varieties over $\mathbb Q$ of dimension up to $5$.

Torsion points on $\rm{GL}_2$-type abelian varieties

Abstract

It is well known that the rational torsion of an abelian variety defined over a number field injects into the reduction modulo any sufficiently large prime, so the order of the torsion group divides the greatest common divisor of the sizes of points on the reduction at each prime. Drawing inspiration from Katz's Inventiones paper (1981), we investigate the converse to this for abelian varieties of -type and exhibit a conjectural list of possible torsion orders for modular abelian varieties over of dimension up to .
Paper Structure (6 sections, 7 theorems, 30 equations, 1 table)

This paper contains 6 sections, 7 theorems, 30 equations, 1 table.

Table of Contents

  1. Introduction
  2. Setting
  3. Katz and Lang's problem
  4. Main results
  5. Modular abelian varieties defined over Q
  6. Implementation and conjectural results

Key Result

Theorem 1

Let $A$ be an abelian variety of $\text{GL}_2$-type over a number field $K$. Let $\Sigma$ be the set of primes $\mathfrak{p}$ of $K$ of good reduction for $A$ such that the absolute ramification index of $\mathfrak{p}$ is smaller than $p-1$, where $p$ is the rational prime under $\mathfrak{p}$. Then are divisible by the same primes of good reduction for $A$. More precisely, for any rational prime

Theorems & Definitions (14)

  • Theorem : See Corollary \ref{['cor:main']}
  • Definition 2.1
  • Remark 2.1
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Proposition 4.1
  • proof
  • ...and 4 more