Experimental Mathematics

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

Patricia L. Quattrini

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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.

Article information

Source
Experiment. Math., Volume 15, Issue 3 (2006), 355-366.

Dates
First available in Project Euclid: 5 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.em/1175789764

Mathematical Reviews number (MathSciNet)
MR2264472

Zentralblatt MATH identifier
1204.11080

Subjects
Primary: 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50]
Secondary: 11Y70: Values of arithmetic functions; tables

Keywords
Elliptic curves Tate-Shafarevich groups modular forms

Citation

Quattrini, Patricia L. On the Distribution of Analytic ${\sqrt{|\sha|}}$ Values on Quadratic Twists of Elliptic Curves. Experiment. Math. 15 (2006), no. 3, 355--366. https://projecteuclid.org/euclid.em/1175789764


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