Small Algebraic Central Values of Twists of Elliptic $L$-Functions

TL;DR

This paper investigates heuristic predictions and computational evidence for small algebraic central values of twisted elliptic curve $L$-functions, exploring their implications for an analogue of the Brauer-Siegel theorem over cyclic extensions.

Contribution

It introduces new heuristic predictions for small algebraic central values of twisted elliptic $L$-functions and provides computational evidence supporting these predictions.

Findings

Heuristic models accurately predict small algebraic central values.

Computational data supports the predicted distribution of these values.

Implications for the Brauer-Siegel type results in cyclic extensions.

Abstract

We consider heuristic predictions for small non-zero algebraic central values of twists of the $L$-function of an elliptic curve $E/\mathbb{Q}$ by Dirichlet characters. We provide computational evidence for these predictions and consequences of them for instances of an analogue of the Brauer-Siegel theorem associated to $E/\mathbb{Q}$ extended to chosen families of cyclic extensions of fixed degree.