- This event has passed.
Dr. Lennart Gehrmann: Plectic Jacobians
December 3 at 13:30 – 15:00 CET
Heegner points play an important role in our understanding of the arithmetic of modular elliptic curves. These points, that arise from CM points on Shimura curves, control the Mordell-Weil group of elliptic curves of rank 1.
The work of Bertolini, Darmon and their schools has shown that p-adic methods can be successfully employed to generalize the definition of Heegner points to quadratic extensions that are not necessarily CM.
Numerical evidence strongly supports the belief that these so-called Stark-Heegner points completely control the Mordell-Weil group of elliptic curves of rank 1.
Inspired by Nekovar and Scholl’s plectic conjectures, Michele Fornea and I recently proposed a plectic generalization of Stark–Heegner points: a cohomological construction of elements in the completed tensor product of local points of elliptic curves that should control Mordell-Weil groups of higher rank.
In this talk, focusing on the quadratic CM case, I will present an alternative speculative framework that can be used to cast the definition of plectic Stark-Heegner points in geometric terms.
More precisely, given a variety X that admits uniformization by a product of p-adic upper half planes I will construct:
– a subgroup of the group of zero-cycles of X, called plectic zero cycles of X
– a topolgoical group, called the plectic Jacobian of X
– a plectic Abel-Jacobi map, i.e. a map from plectic zero cycles to the plectic Jacobian