Numerical study of refined conjectures of the BSD type

TL;DR

This paper experimentally investigates Mazur and Tate's refined BSD-type conjectures using SageMath, finding discrepancies in the original statements but confirming that slight modifications can make them hold.

Contribution

The paper provides the first experimental analysis of Mazur and Tate's conjectures, suggesting necessary modifications for their validity.

Findings

Discrepancies found in original conjecture statements

Modified conjectures appear to hold in experiments

Uses SageMath for computational verification

Abstract

In 1987, Mazur and Tate stated conjectures which, in some cases, resemble the classical Birch-Swinnerton-Dyer conjecture and its $p$-adic analog. We study experimentally three conjectures stated by Mazur and Tate using SageMath. Our findings indicate discrepancies in some of the original statements of some of the conjectures presented by Mazur and Tate. However, a slight modification on the statement of these conjectures does appear to hold.