A p-adic construction of ATR points on Q-curves

In this note we consider certain elliptic curves defined over real quadratic fields isogenous to their Galois conjugate. We give a construction of algebraic points on these curves defined over almost totally real number fields. The main ingredient is the system of Heegner points arising from Shimura curve uniformizations. In addition, we provide an explicit p-adic analytic formula which allows for the effective, algorithmic calculation of such points

THE pp-ADIC UNIFORMISATION OF MODULAR CURVES BY pp-ARITHMETIC GROUPS (Profinite monodromy, Galois representations, and Complex functions)

This is a transcription of the author's lecture at the Kyoto conference "Profinite monodromy, Galois representations, and complex functions" marking Yasutaka Ihara's 80th birthday. Much of it, notably the material in the last section, is the fruit of an ongoing collaboration with Jan Vonk. In his important work on "congruence monodromy problems", Professor Ihara proposed that the group Gamma :=SL_{2}(mathbb{Z}[1/p]) acting on the product of a Drinfeld and a Poincaré upper half-plane provides a congenial framework for describing the ordinary locus of the j-line in characteristic p. In Ihara's picture, which rests on Deuring's theory of the canonical lift, the ordinary points of the j-line are essentially in bijection with conjugacy classes in Gamma that are hyperbolic at p and elliptic at infty. The present note explains how the classes that are elliptic at p and hyperbolic at infty form the natural domain for a kind of p-adic uniformisation of the modular curve X_{0}(p), leading to a conjectural analogues of Heegner points, elliptic units, and singular moduli defined over ring class fields of real quadratic fields

On Heegner Points for primes of additive reduction ramifying in the base field

Let E be a rational elliptic curve, and K be an imaginary quadratic field. In this article we give a method to construct Heegner points when E has a prime bigger than 3 of additive reduction ramifying in the field K. The ideas apply to more general contexts, like constructing Darmon points attached to real quadratic fields which is presented in the appendix

Plectic Jacobians

Looking for a geometric framework to study plectic Heegner points, we define a collection of abelian varieties - called plectic Jacobians - using the middle degree cohomology of quaternionic Shimura varieties (QSVs). The construction is inspired by the definition of Griffiths' intermediate Jacobians and rests on Nekovar-Scholl's notion of plectic Hodge structures. Moreover, we construct exotic Abel--Jacobi maps sending certain zero-cycles on QSVs to plectic Jacobians