$L$-derivatives of the fixed elliptic curve over rank-one imaginary quadratic fields

TL;DR

This paper establishes a one-sided central limit theorem for the logarithms of L-derivatives of a fixed elliptic curve over rank-one imaginary quadratic fields, revealing distributional properties similar to known results and showing many derivatives are not small.

Contribution

It extends the understanding of L-derivative distributions for elliptic curves over imaginary quadratic fields, providing new probabilistic results analogous to classical theorems.

Findings

Proves a one-sided central limit theorem for L-derivatives

Shows many L-derivatives are not small

Establishes distributional properties similar to Radziwi extl{}l ext{}l and Soundararajan's results

Abstract

There is a one-sided central limit theorem for the logarithms of $L$-derivatives of a fixed rational non-CM elliptic curve $E$ over imaginary quadratic fields of rank one, analogous to a result of Radziwi\l\l\ and Soundararajan. There are also many $L$-derivatives that are not small.