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

TL;DR

This paper proves the existence of infinitely many quadratic twists of a specific elliptic curve with complex multiplication over certain fields, ensuring their L-series do not vanish at s=1, advancing non-vanishing results for these curves.

Contribution

It introduces a new non-vanishing theorem for the L-series of quadratic twists of CM elliptic curves over fields with prime q ≡ 7 mod 16, extending previous results to larger q.

Findings

Established an explicit infinite family of non-vanishing quadratic twists.

Connected non-vanishing results to Iwasawa theory at prime p=2.

Extended non-vanishing theorems to cases where q > 7.

Abstract

Let $q$ be any prime $\equiv 7 \mod 16$, $K = \mathbb{Q}(\sqrt{-q})$, and let $H$ be the Hilbert class field of $K$. Let $A/H$ be the Gross elliptic curve defined over $H$ with complex multiplication by the ring of integers of $K$. We prove the existence of a large explicit infinite family of quadratic twists of $A$ whose complex $L$-series does not vanish at $s=1$. This non-vanishing theorem is completely new when $q > 7$. Its proof depends crucially on the results established in our earlier paper for the Iwasawa theory at the prime $p=2$ of the abelian variety $B/K$, which is the restriction of scalars from $H$ to $K$ of the elliptic curve $A$.