Critical $L$-values of Gross curves

TL;DR

This paper computes special L-values of Gross curves over Hilbert class fields for primes q ≡ 3 mod 4, finding all are non-zero and estimating Tate-Shafarevich group orders using BSD conjecture for primes q ≡ 7 mod 8.

Contribution

It provides computational evidence for non-vanishing L-values and estimates Tate-Shafarevich group sizes for a family of Gross curves using Magma and BSD conjecture.

Findings

All computed L-values are non-zero.

Estimated Tate-Shafarevich group orders for primes up to 4831.

Confirmed non-vanishing for primes q ≡ 7 mod 8.

Abstract

Let $K = \mathbb Q(\sqrt{-q})$, 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 $4831$.