Speaker: Amnon Besser (Ben Gurion University)
Title: p-adic height pairings on abelian varieties and Quadratic Chabauty.
Abstract: Quadratic Chabauty is a recent method for finding integral or rational points on certain curves. In particular, the version of the theory developed by Balakrishnan and Dogra was recently used successfully to find the rational points on a certain modular curve related with a conjecture of Serre. In this talk I will report on Joint work in progress with Steffen Mueller and Padmavarthi Srinivasan, which simplifies the method significantly using the theory of p-adic height pairing on abelian varieties.
The classical theory of real valued heights was recast, starting with the work of Zhang in the 90's, as a theory of "adelic" metrics on line bundles. Using techniques from the theories of iterated integrals developed by Coleman and Vologodsky and the theory of line bundles on abelian varieties we can give an analogous theory of p-adic adelic metrics, leading to p-adic heights. Quadratic Chabauty is shown to come from a situation where a non-trivial line bundle on the Jacobian of a curve X pulls back to a trivial line bundle on X. The resulting computations for finding rational points are far simpler than previously known.
Perhaps the most surprising aspect of the theory is the way that the recent understanding of Vologodsky integration by Katz and Litt gives a totally new way of understanding the contributions to the height at primes of bad reduction.
In this talk I will largely treat p-adic interated integrals as a black box so that relevant background consists mostly of algebraic geometry and a bit of combinatorics.
URL associated with Seminar
https://research.math.osu.edu/arithgeoseminar/ArithGeomSeminar.html
