A numerical approach toward the $p$-adic Beilinson conjecture for elliptic curves over $\mathbb{Q}$

TL;DR

This paper introduces a new numerical algorithm based on rigid cohomology and $F$-isocrystals to verify the $p$-adic Beilinson conjecture for elliptic curves over $Q$, advancing computational methods in number theory.

Contribution

It reformulates the $p$-adic Beilinson conjecture for elliptic curves over $Q$ and develops a novel algorithm for its numerical verification.

Findings

Algorithm successfully verifies cases of the conjecture

Utilizes rigid cohomology and $F$-isocrystals for computations

Provides a computational framework for the $p$-adic Beilinson conjecture

Abstract

Restricting ourselves to elliptic curves over $\mathbb{Q}$, we reformulate the $p$-adic Beilinson conjecture due to Perrin-Riou, which is customized to our computational approach. We then develop a new algorithm for numerical verifications of the $p$-adic Beilinson conjecture, which is based on the theory of rigid cohomology and $F$-isocrystals.