## Experimental Mathematics

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

### 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.

#### 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