On the BSD conjecture for series of elliptic curves with complex multiplication (Q5905590)

This article is a research announcement. It reports a number of results in the area of the Birch and Swinnerton-Dyer conjecture for elliptic curves. In the case of an elliptic curve over \(\mathbb{Q}\) with complex multiplication the Hasse-Weil \(L\)-function of which does not vanish at 1, the Birch and Swinnerton-Dyer conjecture is known to hold up to factors of 2 and 3. These results prove the complete conjecture for families of such elliptic curves. This entails showing that the \(L\)-function for the curves in these families are in fact non-zero at the critical value of \(s=1\), and then proving that the factors of 2 and 3 are correct.