# Non‐vanishing theorems for central L‐values of some elliptic curves with complex multiplication

@article{Coates2018NonvanishingTF,
title={Non‐vanishing theorems for central L‐values of some elliptic curves with complex multiplication},
author={John Coates and Yongxiong Li},
journal={arXiv: Number Theory},
year={2018}
}
• Published 2018
• Mathematics
• arXiv: Number Theory
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… Expand
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Critical $L$-values for some quadratic twists of gross curves
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Let $K=\Bbb Q(\sqrt{-q})$, where $q$ is a prime congruent to $3$ modulo $4$. Let $A=A(q)$ denote the Gross curve. Let $E=A^{(-\beta)}$ denote its quadratic twist, with $\beta=\sqrt{-q}$. The curveExpand
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