Georgia Institute of Technology College of Sciences

In joint work with Joan Birman and Bill Menasco, we describe a new finite set of geodesics connecting two given vertices of the curve complex. As an application, we give an effective algorithm for distance in the curve complex.

We study the Legendre elliptic curve E: y^2=x(x+1)(x+t) over the field F_p(t) and its extensions K_d=F_p(mu_d*t^(1/d)). When d has the form p^f+1, in previous work we exhibited explicit points on E which generate a group V of large rank and finite index in the full Mordell-Weil group E(K_d), and we showed that the square of the index is the order of the Tate-Shafarevich group; moreover, the index is a power of p. In this talk we will explain how to use p-adic cohomology to compute the Tate-Shafarevich group and the quotient E(K_d)/V as modules over an appropriate group ring.