Experimental Evidence on a Refined Conjecture of the BSD type

TL;DR

This paper provides computational evidence supporting a refined conjecture related to elliptic curves of rank one, exploring the behavior of the conjecture for primes of ordinary reduction.

Contribution

It offers the first experimental verification of Mazur and Tate's refined BSD conjecture for specific elliptic curves and primes.

Findings

Support for the refined conjecture in tested cases

Evidence consistent with theoretical predictions

Insights into the behavior of elliptic curves of rank one

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve of level $N$ and rank equal to $1$. Let $p$ be a prime of ordinary reduction. We experimentally study conjecture $4$ of B. Mazur and J. Tate in his article "Refined Conjectures of the Birch and Swinnerton-Dyer Type". We report the computational evidence.