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

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".

17 pages. Comments welcome!