Stark–Heegner points on modular Jacobians

Abstract

We present a construction which lifts Darmon's Stark–Heegner points from elliptic curves to certain modular Jacobians. Let N be a positive integer and let p be a prime not dividing N. Our essential idea is to replace the modular symbol attached to an elliptic curve E of conductor Np with the universal modular symbol for Γ 0 ( N p ) . We then construct a certain torus T over Q p and lattice L ⊂ T , and prove that the quotient T / L is isogenous to the maximal toric quotient J 0 ( N p ) p - new of the Jacobian of X 0 ( N p ) . This theorem generalizes a conjecture of Mazur, Tate, and Teitelbaum on the p-adic periods of elliptic curves, which was proven by Greenberg and Stevens. As a by-product of our theorem, we obtain an efficient method of calculating the p-adic periods of J 0 ( N p ) p - new .