On the BSD conjecture for certain families of abelian varieties with rational torsion
Prof. Emmanuel Lecouturier
2022-02-16 15:30-16:30
MCM410&Tencent
Speaker: Prof. Emmanuel Lecouturier (Tsinghua University)
Time: 15:30-16:30 Feb. 16 , 2022 (Wednesday)
Place: MCM410 & Online (Tencent ID: 495 324 142 Password: 1234)
Title: On the BSD conjecture for certain families of abelian varieties with rational torsion
Abstract: Let $N$ and $p$ $\geq 5$ be primes such that $p$ divides $N-1$. In his landmark paper on the Eisenstein ideal, Mazur proved the $p$-part of the BSD conjecture for the $p$-Eisenstein quotient $J^{(p)}$ of $J_0(N)$ over $\mathbf{Q}$. Using recent results and techniques of the work of Venkatesh and Sharifi on the Sharifi conjecture, we prove unconditionally a weak form of the BSD conjecture for $J^{(p)}$ over a quadratic field $K$ (which can be real or imaginary). This includes results in positive analytic rank, as the analytic rank of $J^{(p)}$ over $K$ can be $\geq 2$ for well-chosen $K$. This is joint work with Jun Wang (MCM).
