Non-vanishing theorems for central $L$-values of some elliptic curves with complex multiplication

John Coates, Yongxiong Li

Published 2018-11-19, updated 2018-12-05Version 2

The paper uses Iwasawa theory at the prime $p=2$ to prove non-vanishing theorems for the value at $s=1$ of the complex $L$-series of the family of elliptic curves, first studied systematically by B. Gross, with complex multiplication by the full ring of integers of the field $K = \mathbb{Q} (\sqrt{-q})$, where $q$ is any prime $\equiv 7 \mod 8$. Some of these non-vanishing theorems were proven earlier by D. Rohrlich using complex analytic methods, but others are completely new. It is essential for the proofs to study the Iwasawa theory of the higher dimensional abelian variety with complex multiplication which is obtained by taking the restriction of scalars to $K$ of the Gross elliptic curve.