On the Mazur--Tate conjecture for prime conductor and Mazur's Eisenstein ideal

TL;DR

This paper extends de Shalit's work on the Mazur--Tate conjecture for modular Jacobians to include Eisenstein primes, providing new theoretical insights and applications involving supersingular invariants.

Contribution

It completes the proof of the Mazur--Tate conjecture at Eisenstein primes and introduces the use of generalized cuspidal 1-motives for this purpose.

Findings

Extended the Mazur--Tate conjecture to Eisenstein primes.

Derived an elementary combinatorial identity involving supersingular j-invariants.

Utilized generalized cuspidal 1-motives as a key tool.

Abstract

In 1995, Ehud de Shalit proved an analogue of a conjecture of Mazur--Tate for the modular Jacobian $J_0(p)$. His main result was valid away from the Eisenstein primes. We complete the work of de Shalit by including the Eisenstein primes, and give some applications such as an elementary combinatorial identity involving discrete logarithms of difference of supersingular $j$-invariants. An important tool is our recent work on the so called "generalized cuspidal $1$-motive".