Effective computation of ATR Darmon points

ATR points were introduced by Henri Darmon as a conjectural construction of
algebraic points on certain elliptic curves for which the Heegner
point method is not available. They can be explicitly computed in
particular examples, and Darmon and Logan gathered some numerical evidence
supporting the conjecture. However, due to some restrictions on the
algorithm, they only numerically tested the construction on elliptic curves
which all happen to be also equipped with classical Heegner points. In a
joint work with Marc Masdeu we improve the algorithm used by Darmon and
Logan in order to provide more numerical evidence towards Darmon's
conjecture. In particular, we numerically verify it on an elliptic curve
for which no other construction of algebraic points is known. In this talk
I will describe ATR points and discuss some of the computational issues
encountered in their effective calculation.