# Critical $L$-values for some quadratic twists of Gross curves

## TL;DR

This paper computes critical L-values for quadratic twists of Gross curves over Hilbert class fields, providing numerical evidence related to the Birch and Swinnerton-Dyer conjecture for various primes q.

## Contribution

It extends previous computational work by calculating L-values for a broader range of primes q and their quadratic twists, offering new data for BSD conjecture analysis.

## Key findings

- All computed L-values are non-zero.
- Calculated hypothetical orders of Sha for multiple q values.
- Extended previous computational results to new primes q.

## Abstract

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 curve $E$ is defined over the Hilbert class field $H$ of $K$. We use Magma to calculate the values $L(E/H,1)$ for all such $q$'s up to some reasonable ranges (different for $q\equiv 7 \, \text{mod} \, 8$ and $q\equiv 3 \, \text{mod} \, 8$). All these values are non-zero, and using the Birch and Swinnerton-Dyer conjecture, we can calculate hypothetical orders of $\sza(E/H)$ in these cases. Our calculations extend those given by J. Choi and J. Coates [{\it Iwasawa theory of quadratic twists of $X_0(49)$}, Acta Mathematica Sinica(English Series) {\bf 34} (2017), 19-28] for the case $q=7$.