On the size of the Shafarevich-Tate group of elliptic curves over function fields

Abstract

Let E be a nonconstant elliptic curve, over a global field K of positive, odd characterisitc. Assuming the finiteness of the Shafarevic-Tate group of E, we show that the order of the Shafarevich-Tate group of E, is given by O^{(N1/2+6log(2)/log(q))}, where N is the conductor of E, q is the cardinality of the finite field of constants of K, and where the constant in the bound depends only on K. The method of proof is to workwith the geometric analog of the Birch-Swinnerton Dyer conjecture for thecorresponding elliptic surface over the finite field, as formulated by Artin-Tate, and to examine the geometry of this elliptic surface.