Apéry series and Mellin transforms of solutions of differential equations

27 February 2020
Spencer Bloch

One can study periods of algebraic varieties by a process of "fibering out" in which the variety is fibred over a punctured curve $f:X->U$. I will explain this process and how it leads to the classical Picard Fuchs (or Gauss-Manin) differential equations. Periods are computed by integrating solutions of Picard Fuchs over suitable closed paths on $U$. One can also couple (i.e.tensor) the Picard Fuchs connection to given connections on $U$. For example, $t^s$ with $t$ a unit on $U$ and $s$ a parameter is a solution of the connection on $\mathscr{O}_U$ given by $\nabla(1) = sdt/t$. Our "periods" become integrals over suitable closed chains on $U$ of $f(t)t^sdt/t$. Golyshev called the resulting functions of $s$ "motivic Gamma functions". 
Golyshev and Zagier studied certain special Picard Fuchs equations for their proof of the Gamma conjecture in mirror symmetry in the case of Picard rank 1. They write down a generating series, the Apéry series, the knowledge of the first few terms of which implied the gamma conjecture. We show their Apéry series is the Taylor series of a product of the motivic Gamma function times an elementary function of $s$. In particular, the coefficients of the Apéry series are periods up to inverting $2\pi i$. We relate these periods to periods of the limiting mixed Hodge structure at a point of maximal unipotent monodromy. This is joint work with M. Vlasenko. 

  • Number Theory Seminar