Research Papers

Research Summary
  • Description of recent and current projects. The overview is suitable for a general mathematical audience. For more information, please see the papers below.

Publications
  • "Curves with prescribed period and index over local fields", J. Algebra, 2007 (314) no. 1, 157--167.
    Using rigid analytic techniques, we show that there are curves of every genus, period, and index over a local field, subject to divisibility conditions proved by Lichtenbaum, and also assuming that the base field does not have characteristic 2.
  • "Period, index, and potential Sha" (with Pete L. Clark), accepted for publication to Algebra and Number Theory.
    Given a fixed elliptic curve E over a number field K, we show that the Tate-Shafarevich group is arbitrarily large when one replaces K with a P-extension, for a given integer P>1. More specifically, we show that given any integer r, there is a P-extension L/K such that the P-torsion in the Tate-Shafarevich group of E/L has order at least r.
  • "Period and index of genus one curves over number fields", submitted for publication to Math. Annalen on 3/09.
    We solve the period-index problem for genus 1 curves over number fields; that is, we show that given an elliptic curve E over a number field K, and any positive integers P, D with D dividing P, that there is a genus 1 curve over K with Jacobian E, period P, and index PD. By results of Lichtenbaum, this is the strongest possible result for genus 1 curves over number fields.

In progress
  • "Tate-Shafarevich and Brauer groups of split elliptic threefolds", (with C. Schoen).
    Forthcoming.
  • "Rationality of theta characteristics"
    A theta characteristic is a square root of the canonical bundle. We give some criteria for determining when a curve over a local field has a rational theta characteristic; in particular, curves with good reduction always do.

Miscellenea
  • An interesting elliptic pencil
    We exhibit a nonisotrivial pencil of genus 1 curves over the rational affine line which is generically smooth, has points over every completion, but for which no fiber has points over every completion (hence has no rational points). This construction was inspired by a question of Pete L. Clark.
  • Elliptic curves: an overview
    Notes for a seminar talk giving an introductory survey to elliptic curves. Accessible to a graduate-level audience.
  • Cheat sheet for elliptic curves
    • Basic formulas about elliptic curves, taken from Silverman's book.