Date of Award
Open Access Dissertation
Doctor of Philosophy (PhD)
Mathematics | Statistics and Probability
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 Qp to give a mod p2 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 Fp to Z/pp 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 Qp. Specifically, it shows that if A has a nontrival p-torsion point over a ramified extension K of Qp 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.
Gamzon, Adam, "Local Torsion on Abelian Surfaces" (2012). Open Access Dissertations. 549.