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 $A$.

中文翻译：

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具有复数乘法的某些椭圆曲线的中心L值的不消失定理

假设$ q $是任何素数$ \ equiv 7 \ mod 16 $，$ K = \ mathbb {Q}（\ sqrt {-q}）$，而$ H $是$ K $的希尔伯特类字段。设$ A / H $为在$ H $上定义的总椭圆曲线，并乘以$ K $的整数环。我们证明了存在一个大的显式无限的二次扭曲$ A $家族，其复杂的$ L $系列在$ s = 1 $时不消失。当$ q> 7 $时，这个不消失的定理是全新的。它的证明至关重要地取决于我们早先的论文中为岩泽理论建立的结果，即阿贝尔变种$ B / K $的质数$ p = 2 $，即标量从$ H $到$ K $的限制。椭圆曲线$ A $。