From Equivariant Blow-Ups to Modular Symbols

Part of the beauty of mathematics is the interplay between its different branches. In this lecture, Maxim Kontsevich will talk about a recent discovery of an unexpected relation between questions in birational algebraic geometry and the theory of automorphic forms. Computer experiments with an easy-looking system of equations played an essential role in the discovery.

Kontsevich’s recent work on this topic has been with Yuri Tschinkel and Vasily Pestun. At the origin of this work was a question of constructing birational invariants of algebraic varieties endowed with an action of a finite group (e.g., a cyclic group generated by an automorphism of finite order). Based on a simple ansatz that ignores almost all non-trivial geometric information except spectra of group action at fixed points, they arrived at an overdetermined system of linear equations. To their surprise, the system has sporadic non-trivial solutions giving, for instance, an invariant of 3-dimensional manifolds with a birational automorphism of order 43. The existence of such a solution is explained by the existence of a certain arithmetic object (a motive, or an automorphic form).