DAGA Seminar: Quaternionic Darmon points and arithmetic applications

Tuesday 6 December 2011 - Matteo Longo


Tuesday, December 6, 2011, h. 10:15-11:15, room 2AB40
Matteo Longo (Padova)
"Quaternionic Darmon points and arithmetic applications"

In 2001 H. Darmon introduced the notion of ''Stark-Heegner points'' on rational elliptic curves. These are local points on elliptic curves (defined over the algebraic closure of Q_p). A deep conjecture by Darmon states that they are actually global points (defined over abelian extensions of real quadratic fields). Also, they are non-torsion if and only if the special value of the derivative of the relevant complex L-function of the elliptic curve is non-zero. These conjectures generalize what is known in the case of imaginary quadratic extensions (where the role of Start-Heegner points is played by classical Heegner points). In this talk I will present a construction of Darmon-style points on elliptic curves in a more general setting than that originally considered by Darmon. I will also explain in which cases Darmon's conjectures can actually be proved and, if time permits, I will offer an application to the Birch and Swinnerton-Dyer conjecture for elliptic curves. These results have been obtained in collaboration with V. Rotger and S. Vigni.

Rif. int. A. Bertapelle

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