On the Distribution of Analytic ${\sqrt{|\sha|}}$ Values on Quadratic Twists of Elliptic Curves

Abstract

The aim of this paper is to analyze the distribution of analytic (and signed) square roots of $\smallsha$ values on imaginary quadratic twists of elliptic curves.

Given an elliptic curve $E$ of rank zero and prime conductor $N$, there is a weight-$\frac32$ modular form $g$ associated with it such that the $d$-coefficient of $g$ is related to the value at $s=1$ of the $L$-series of the $(-d)$-quadratic twist of the elliptic curve $E$. Assuming the Birch and Swinnerton-Dyer conjecture, we can then calculate for a large number of integers $d$ the order of $\smallsha$ of the $(-d)$-quadratic twist of $E$ and analyze their distribution.