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

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