Presented By: Department of Mathematics
RTG Seminar on Number Theory Seminar
The arithmetic of plectic Jacobians
Note: the main talk goes from 3-3:45pm and then will continue again after a 15 minute break.
Heegner points play a pivotal role in our understanding of the arithmetic of modular elliptic curves. They control the Mordell-Weil group of elliptic curves of rank 1, and they arise as CM points on Jacobians of Shimura curves.
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. Notably, Guitart, Masdeu and Sengun have defined and numerically computed Stark-Heegner (SH) points in great generality. Their computations strongly support the belief that SH points completely control the Mordell-Weil group of elliptic curves of rank 1.
Inspired by Nekovar and Scholl's plectic conjectures, Lennart Gehrmann and I recently proposed a plectic generalization of SH points: a cohomological construction of local points on elliptic curves that should control Mordell-Weil groups of higher rank. In this talk, focusing on the quadratic CM case, we will present an alternative speculative framework that can be used to cast the definition of plectic Heegner points in geometric terms. Moreover, we will provide some evidence for our conjectures by showing that higher derivatives of p-adic L-functions compute plectic points.
Join Zoom Meeting
https://umich.zoom.us/j/95185733075
Meeting ID: 951 8573 3075
Passcode: umrtg Speaker(s): Michele Fornea (Columbia University)
Heegner points play a pivotal role in our understanding of the arithmetic of modular elliptic curves. They control the Mordell-Weil group of elliptic curves of rank 1, and they arise as CM points on Jacobians of Shimura curves.
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. Notably, Guitart, Masdeu and Sengun have defined and numerically computed Stark-Heegner (SH) points in great generality. Their computations strongly support the belief that SH points completely control the Mordell-Weil group of elliptic curves of rank 1.
Inspired by Nekovar and Scholl's plectic conjectures, Lennart Gehrmann and I recently proposed a plectic generalization of SH points: a cohomological construction of local points on elliptic curves that should control Mordell-Weil groups of higher rank. In this talk, focusing on the quadratic CM case, we will present an alternative speculative framework that can be used to cast the definition of plectic Heegner points in geometric terms. Moreover, we will provide some evidence for our conjectures by showing that higher derivatives of p-adic L-functions compute plectic points.
Join Zoom Meeting
https://umich.zoom.us/j/95185733075
Meeting ID: 951 8573 3075
Passcode: umrtg Speaker(s): Michele Fornea (Columbia University)
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