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Local torsion on abelian surfaces
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 p2 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 Fp to Z/p2 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 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 to a series of straightforward estimates.^
Gamzon, Adam B, "Local torsion on abelian surfaces" (2012). Doctoral Dissertations Available from Proquest. AAI3518232.