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2006 On the Distribution of Analytic ${\sqrt{|\sha|}}$ Values on Quadratic Twists of Elliptic Curves
Patricia L. Quattrini
Experiment. Math. 15(3): 355-366 (2006).

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.

Citation

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Patricia L. Quattrini. "On the Distribution of Analytic ${\sqrt{|\sha|}}$ Values on Quadratic Twists of Elliptic Curves." Experiment. Math. 15 (3) 355 - 366, 2006.

Information

Published: 2006
First available in Project Euclid: 5 April 2007

zbMATH: 1204.11080
MathSciNet: MR2264472

Subjects:
Primary: 11F33
Secondary: 11Y70

Keywords: Elliptic curves , modular forms , Tate-Shafarevich groups

Rights: Copyright © 2006 A K Peters, Ltd.

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