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.
@article{1175789764,
author = {Quattrini, Patricia L.},
title = {On the Distribution of Analytic ${\sqrt{|\sha|}}$ Values on Quadratic Twists of Elliptic Curves},
journal = {Experiment. Math.},
volume = {15},
number = {1},
year = {2006},
pages = { 355-366},
language = {en},
url = {http://dml.mathdoc.fr/item/1175789764}
}
Quattrini, Patricia L. On the Distribution of Analytic ${\sqrt{|\sha|}}$ Values on Quadratic Twists of Elliptic Curves. Experiment. Math., Tome 15 (2006) no. 1, pp. 355-366. http://gdmltest.u-ga.fr/item/1175789764/