Analytic rank (awaiting review)

The analytic rank of a modular curve is the order of vanishing of its L-function at its central point, which is equal to the sums of the analytic ranks of the L-functions of the simple modular abelian varieties corresponding to Galois orbits of modular forms that are the isogeny factors of its Jacobian.

The Birch and Swinnerton-Dyer conjecture for modular abelian varieties implies that the analytic rank is equal to the Mordell-Weil rank of the Jacobian.