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 $N\ne (1)$. Let $K/F$ be a non-CM quadratic extension with $(\mathrm{Disc}K,N)=1$ such that the sign in the functional equation of $L(E/K,s)$ is $-1$. Suppose further that there is a prime $\mathfrak{p}|N$ that is inert in $K$. We describe a $\mathfrak{p}$-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