#### Date of Award

5-2012

#### Document Type

Open Access Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Degree Program

Mathematics

#### First Advisor

Tom Weston

#### Second Advisor

Paul Gunnells

#### Third Advisor

Siman Wong

#### Subject Categories

Mathematics | Statistics and Probability

#### Abstract

Fix an integer *d *> 0. In 2008, Chantal David and Tom 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 dissertation is an extension of that work, investigating the frequency with which a principally polarized abelian surface A over Q with real multiplication by Q adjoin a squared-root of 5 has a nontrivial *p*-torsion point defined over *K*. Averaging by height, the main result shows that *A* picks up a nontrivial *p*-torsion point over *K* for only finitely many *p*.

The proof of our main theorem primarily rests on three lemmas. The first lemma uses the reduction-exact sequence of an abelian survace defined over an unramified extension *K *of Q*p* to give a mod *p*^{2 }condition for detecting when *A* has a nontrival *p*-torsion point defined over *K*. The second lemma employs crystalline Dieudonne theory to count the number of isomorphism classes of lifts of abelian surfaces over F*p* to Z/*p ^{p} *that satisfy the conditions from our first lemma. Finally, the third lemma addresses the issue of the assumption in the first lemma that

*K*is an unramified extension of Q

_{p.}Specifically, it shows that if

*A*has a nontrival

*p*-torsion point over a ramified extension

*K*of Q

*p*

*and*

*p*- 1 >

*d*then this

*p*-torsion point is actually defined over the maximal unramified subextension of

*K.*We then combine these algebraic results to reduce the main analytic calculation toa series of straightforward estimates.

#### Recommended Citation

Gamzon, Adam, "Local Torsion on Abelian Surfaces" (2012). *Open Access Dissertations*. 549.

https://scholarworks.umass.edu/open_access_dissertations/549