Non-vanishing of central \(L\)-values of the Gross family of elliptic curve

We prove non-vanishing theorems for the central values of \(L\)-series of certain quadratic twists of the Gross elliptic curve with complex multiplication by the imaginary quadratic field \(\mathbb{Q}(\sqrt{-q})\), where \(q\) is any prime congruent to \(7\) modulo \(8\). This completes the non-vani...

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Bibliographic Details
Published inarXiv.org
Main Authors Kezuka, Yukako, Yong-Xiong, Li
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 15.05.2023
Subjects
Curves
Fields (mathematics)
Number theory
Scalars
Theorems
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Summary:We prove non-vanishing theorems for the central values of \(L\)-series of certain quadratic twists of the Gross elliptic curve with complex multiplication by the imaginary quadratic field \(\mathbb{Q}(\sqrt{-q})\), where \(q\) is any prime congruent to \(7\) modulo \(8\). This completes the non-vanishing theorems proven by Coates and the second author in which the primes \(q\) were taken to be congruent to \(7\) modulo \(16\). From this, we obtain the finiteness of the Mordell-Weil group and the Tate-Shafarevich group for these curves. For a special prime \(\mathfrak{P}\) lying above the prime \(2\), we also prove a rank zero converse theorem and the \(\mathfrak{P}\)-part of the Birch-Swinnerton-Dyer conjecture for the higher-dimensional abelian varieties obtained by restriction of scalars.
ISSN:2331-8422