On the arithmetic of special values of $L$-functions for certain abelian varieties with a rational isogeny

TL;DR

This paper proves a modulo p version of the Birch and Swinnerton-Dyer conjecture for the p-Eisenstein part of certain quadratic twists of a modular Jacobian, extending Mazur's earlier results to even quadratic twists.

Contribution

It establishes a new modulo p BSD conjecture analogue for even quadratic twists of the p-Eisenstein quotient of modular Jacobians, generalizing Mazur's work.

Findings

Proves a modulo p BSD type result for even quadratic twists.

Extends Mazur's results from odd to even quadratic twists.

Provides new insights into the arithmetic of special L-values for abelian varieties.

Abstract

Let $N$ and $p$ be primes $\geq 5$ such that $p \mid \mid N-1$. In this situation, Mazur defined and studied the $p$-Eisenstein quotient $\tilde{J}^{(p)}$ of $J_0(N)$. We prove a kind of modulo $p$ version of the Birch and Swinnerton-Dyer conjecture for the ``$p$-Eisenstein part'' of even quadratic twists of $\tilde{J}^{(p)}$. Our result is the analogue for even quadratic twists of a result of Mazur concerning odd quadratic twists.