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

Authors
  • Gamzon, Adam
Type
Preprint
Publication Date
Feb 27, 2014
Submission Date
Jul 24, 2013
Identifiers
arXiv ID: 1307.6355
Source
arXiv
License
Yellow
External links

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.

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