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arXiv · 2305.00643

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

Emmanuel Lecouturier·Jun Wang
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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.

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Emmanuel Lecouturier, Jun Wang. 2023-05-01. On the arithmetic of special values of $L$-functions for certain abelian varieties with a rational isogeny. https://arxiv.org/abs/2305.00643

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