The Tate-Shafarevich group of the Legendre curve

Algebra Seminar
Monday, May 5, 2014 - 3:00pm for 1 hour (actually 50 minutes)
Skiles 006
Professor Doug Ulmer – Georgia Tech – douglas.ulmer@math.gatech.edu
Salvador Barone
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.