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# Local torsion on abelian surfaces

#### 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 surface defined over an unramified extension *K* of **Q*** _{ p}* to give a mod

*p*

^{2}condition for detecting when

*A*has a nontrivial

*p*-torsion point defined over

*K*. The second lemma employs crystalline Dieudonné theory to count the number of isomorphism classes of lifts of abelian surfaces over

**F**

*to*

_{p}**Z**/

*p*

^{2}that satisfy the condition 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**

*. Specifically, it shows that if*

_{p}*A*has a nontrivial

*p*-torsion point over a ramified extension

*K*of

**Q**

*and*

_{p}*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 to a series of straightforward estimates.^

#### Subject Area

Applied mathematics|Mathematics

#### Recommended Citation

Gamzon, Adam B, "Local torsion on abelian surfaces" (2012). *Doctoral Dissertations Available from Proquest*. AAI3518232.

https://scholarworks.umass.edu/dissertations/AAI3518232