Adam Gamzon | Searcharxiv

arXiv2014

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

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.

math.NT↗

arXiv2013

Unobstructed Hilbert modular deformation problems

Consider the semisimple mod p reduction of the Galois representation associated to a Hilbert newform f by Carayol and Taylor. This paper discusses how, under certain conditions on f, the universal ring for deformations of this residual representation with fixed determinant is unobstructed for almost all primes. We follow the approach of Weston, who carried out a similar program for classical modular forms in 2004. As such, the problem essentially comes down to verifying that various local invariants vanish at all places dividing p or the level of the newform. We conclude with an explicit example illustrating how one can in principle find a lower bound on p such that the universal ring for deformations of the residual representation attached to f with fixed determinant is unobstructed for all primes over p.

math.NT↗