### Description

Abstract: This talk will describe recent joint work in progress with J. Coates, Y. Li and Y. Tian. Let K be the imaginary quadratic field Q(sqrt{-q}), where q is any prime congruent to 7 modulo 16. Let A be the Gross curve defined over the Hilbert class field H of K, with complex multiplication by the ring of integers of K. In their most recent work, Coates and Li found a large family of quadratic twists E of A whose complex L-series L(E/H,s) does not vanish at s=1. We will discuss the p-part of the Birch and Swinnerton-Dyer conjecture for these curves for every prime p which splits in K (in particular, this includes p=2).