Local torsion on abelian surfaces with real multiplication by $\mathbf{Q}(\sqrt{5})$

TL;DR

This paper extends results about torsion points from elliptic curves to abelian surfaces with real multiplication by Q(√5), showing finiteness of primes with certain torsion properties and linking to Galois representation deformation theory.

Contribution

It proves an averaging theorem for torsion points on abelian surfaces with real multiplication by Q(√5), generalizing prior elliptic curve results and connecting to modular Galois representations.

Findings

Finiteness of primes with nontrivial torsion points over bounded degree extensions

Establishment of an average torsion result for abelian surfaces with real multiplication

Connection between torsion properties and deformation theory of Galois representations

Abstract

Fix an integer $d>0$. In 2008, David and Weston showed that, on average, an elliptic curve over $\mathbf{Q}$ picks up a nontrivial $p$-torsion point defined over a finite extension $K$ of the $p$-adics of degree at most $d$ for only finitely many primes $p$. This paper proves an analogous averaging result for principally polarized abelian surfaces over $\mathbf{Q}$ with real multiplication by $\mathbf{Q}(\sqrt{5})$ and a level-$\sqrt{5}$ structure. Furthermore, we indicate how the result on abelian surfaces with real multiplication by $\mathbf{Q}(\sqrt{5})$ relates to the deformation theory of modular Galois representations.