NON-VANISHING THEOREMS FOR QUADRATIC TWISTS OF ELLIPTIC CURVES

摘要

In this paper, we use rather classical results on modular symbols to prove that, for certain families of elliptic curves defined over Q, there always exists a large class of explicit quadratic twists whose complex L-series does not vanish at s = 1. We also prove the 2-part of the conjecture of Birch and Swinnerton-Dyer for many of these quadratic twists.