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

Abstract

Fix an integer d ≥ 1. In 2008, David and Weston showed that, on average, an elliptic curve over 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 A over Q with real multiplication by [Formula: see text] and a level-[Formula: see text] structure. Furthermore, we indicate how the result on abelian surfaces with real multiplication relates to the deformation theory of modular Galois representations.