Nonvanishing theorems for central $L$values of some elliptic curves with complex multiplication II
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 nonvanishing 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 $A$.
 Publication:

arXiv eprints
 Pub Date:
 April 2019
 arXiv:
 arXiv:1904.05756
 Bibcode:
 2019arXiv190405756C
 Keywords:

 Mathematics  Number Theory;
 11G05;
 11G40