2009 Number Theory Seminar - 10

In [Mazur, Visualizing elements of order three in the Shafarevich-Tate group, Asian J. Math. 3 (1999), no. 1, 221-232], Mazur proves that any element in Sha[3](E/K) can be made visible in an abelian surface. Mazur did not comment on whether this surface can be taken to be a Jacobian of a genus 2 curve, however. His construction does not prove that it can, but with an easy observation, one can actually see that it is possible. In this talk, I will explain what Mazur proved and what the easy observation is. This is joint work with Sander Dahmen (SFU/UBC). Note that we will move to room K9509 for this second talk.