Number theory seminar in memory of F. Momose

Explicit computations with ATR points.


Marc Masdeu


In his book on rational points on modular elliptic curves, Henri Darmon gives a construction of a supply of algebraic points predicted by the Birch and Swinnerton-Dyer conjecture, in cases where the Heegner point construction does not work. One of these cases arises with ”Almost Totally Real” (ATR) extensions, and Darmon and Logan gathered some numerical evidence supporting the conjecture. However, all the curves for which they construct algebraic points are isogenous to to their Galois conjugates, and in that situation one might hope for a variation of the Heegner point construction to still work. In a joint project with Xavier Guitart (UPC) we improve the algorithm used by Darmon and Logan in order to provide more numerical evidence in support of Darmon’s conjecture. In particular, we find approximations to algebraic points on curves for which no other construction is available, not even conjecturally. In the talk I will describe the problems that one faces when computing these points, and how we have overcome them.


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