Unlikely intersections with Bessel functions and Laguerre polynomials II

Abstract: Several problems in arithmetic geometry identify a class of "special points" and seek to describe the set of special points satisfying a given system of algebraic equations. Two classical examples are the Manin-Mumford conjecture concerning torsion points in abelian varieties, and the Andre-Oort conjecture concerning CM points in Shimura varieties. We propose a new variation on this theme, where the role of special points is played by the zeros of special functions such as the Bessel function, or of classical orthogonal polynomials. The Bessel function problem is related to a conjecture of Fuglede in harmonic analysis from 1974, and the orthogonal polynomial problem to a conjecture of Stieltjes from 1890.

In the first talk we will recall the Manin-Mumford conjecture and introduce the Bessel function and orthogonal polynomial analogs. We will sketch the proof of classical Manin-Mumford using o-minimality and how it can be adapted to obtain various results in this new context. This naturally leads to the more "exotic" o-minimal structure of multisummable functions, as opposed to the structure R_{an,exp} used in classical applications. In the second talk we will discuss new functional transcendence results that are needed to complete the argument: a variant of the Ax-Schanuel theorem for the special functions appearing in this context. This naturally leads to the study of differential Galois groups for irregular-singular systems, as opposed to the regular-singular systems used in classical applications.

All the results are based on joint work with Avner Kiro and some also with Gady Kozma.

number theory

Audience: researchers in the topic

BC-MIT number theory seminar