The rationality of Stark-Heegner points over genus fields of real quadratic fields


We study the algebraicity of Stark-Heegner points on a modular elliptic curve $E$. These objects are $p$-adic points on $E$ given by the values of certain $p$-adic integrals, but they are conjecturally defined over ring class fields of a real quadratic field $K$. The present article gives some evidence for this algebraicity conjecture by showing that linear combinations of Stark-Heegner points weighted by certain genus characters of $K$ are defined over the predicted quadratic extensions of $K$. The non-vanishing of these combinations is also related to the appropriate twisted Hasse-Weil $L$-series of $E$ over $K$, in the spirit of the Gross-Zagier formula for classical Heegner points.


Massimo Bertolini

Dipartimento di Matematica
Università degli Studi di Milano
Via Saldini, 50
20133 Milano

Henri Darmon

Department of Mathematics
McGill University
805 Shebrooke Street West
Montreal, QC  H3A 2K6