Communications in Number Theory and Physics

Volume 15 (2021)

Gamma functions, monodromy and Frobenius constants

Pages: 91 – 147

DOI: https://dx.doi.org/10.4310/CNTP.2021.v15.n1.a3

Authors

Spencer Bloch

Masha Vlasenko (Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland)

Abstract

In an important paper [8], Golyshev and Zagier introduce what we will refer to as Frobenius constants $\kappa_{\rho,n}$ associated to an ordinary linear differential operator $L$ with a reflection type singularity at $t = c$. For every other regular singularity $t = c^\prime$ and a homotopy class of paths $\gamma$ joining $c^\prime$ and $c$, constants $\kappa_{\rho,n} = \kappa_{\rho,n} (\gamma)$ describe the variation around $c$ of the Frobenius solutions $\phi_{\rho,n} (t)$ to $L$ defined near $t = c^\prime$ and continued analytically along $\gamma$. (Here $\rho \in \mathbb{C}$ are local exponents of $L$ at $t = c^\prime$, see Definition 22 below.) Golyshev and Zagier show in certain cases that the $\kappa_{\rho,n}$ are periods, and they raise the question quite generally how to describe the $\kappa_{\rho,n}$ motivically.

The purpose of this work is to develop the theory (first suggested to us by Golyshev) of motivic Mellin transforms or motivic gamma functions. Our main result (Theorem 30) relates the generating series $\sum^{\infty}_{n=0} \kappa_{\rho,n}(s-\rho)^n$ to the Taylor expansion at $s = \rho$ of a generalized gamma function, which is a Mellin transform of a solution of the dual differential operator $L^\lor$. It follows from this that the numbers $\kappa_{\rho,n}$ are always periods when $L$ is a geometric differential operator (Corollary 31).

Work of the second author was supported by the National Science Centre of Poland (NCN), grant UMO-2016/21/B/ST1/03084.