Critical L-values of Gross curves

2021/09/05 by Dąbrowski, Andrzej, Jędrzejak, Tomasz, Szymaszkiewicz, Lucjan
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2109.03802

Abstract

Let K = \mathbb Q(√(-q)), where q is a prime congruent to 3 modulo 4. Let A = A(q) denote the Gross curve over the Hilbert class field H of K. In this note we use Magma to calculate the values L(E/H, 1) for all such q's up to some reasonable ranges for all primes q congruent to 7 modulo 8. All these values are non-zero, and using the Birch and Swinnerton-Dyer conjecture, we can calculate hypothetical orders of the Tate-Shafarevich group of A for all such q up to 4831.

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