Stark-Heegner points and the cohomology of quaternionic Shimura varieties

Abstract

Let F be a totally real field of narrow class number one, and let E/F be a modular, semistable elliptic curve of conductor N≠(1). Let K/F be a non-CM quadratic extension with (DiscK,N)=1 such that the sign in the functional equation of L(E/K,s) is −1. Suppose further that there is a prime p|N that is inert in K. We describe a p-adic construction of points on E which we conjecture to be rational over ring class fields of K/F and satisfy a Shimura reciprocity law. These points are expected to behave like classical Heegner points and can be viewed as new instances of the Stark-Heegner point construction of [5]. The key idea in our construction is a reinterpretation of Darmon's theory of modular symbols and mixed period integrals in terms of group cohomology