Periods of Hilbert Modular Forms and Rational Points on Elliptic Curves

Let E be a modular elliptic curve over a totally real field. In [7, Chapter 8] the first author formulates a conjecture allowing the construction of canonical algebraic points on E by suitably integrating the associated Hilbert modular form. The main goal of the present paper is to obtain numerical evidence for this conjecture in the first case where it asserts something nontrivial, namely, when E has everywhere good reduction over a real quadratic field. To put our calculations in context, it is useful to recall how, when E is a (modular) elliptic curve over Q, the theory of complex multiplication allows the construction of a distinguished collection of algebraic points on E—the so-called Heegner points which were studied systematically by Birch [2], and provide the setting for the formula of Gross and Zagier [10]. These points are obtained by letting τ be a quadratic irrationality in the Poincaré upper half plane H and considering the images, under the Weierstrass uniformisation associated to an appropriate choice of complex lattice, of expressions of the form