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Dr. Lennart Gehrmann: Plectic Jacobians

December 3 at 13:3015:00 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

Details

Date:
December 3
Time:
13:30 – 15:00 CET

Venue

Heidelberg, Mathematikon, SR A