Local torsion on abelian surfaces

Abstract

LOCAL TORSION ON ABELIAN SURFACES MAY 2012 ADAM B. GAMZON, B.A., UNIVERSITY OF CONNECTICUT M.S., UNIVERSITY OF MASSACHUSETTS AMHERST Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Tom Weston 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( √ 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 Qp 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 Dieudonne theory to count the number of isomorphism classes of lifts of abelian surfaces over Fp to 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 Qp. Specifically, it shows that if A has a nontrivial p-torsion point over a ramified extension K of Qp