On the BSD conjecture for certain families of abelian varieties with rational torsion

Let N and p greater than or equal to 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 J0(N) over 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 greater than or equal to 2 for well-chosen K.

This is joint work with Jun Wang (MCM Beijing).