×

Gamma functions, monodromy and Frobenius constants. (English) Zbl 1455.14019

V. V. Golyshev and D. Zagier [Izv. Math. 80, No. 1, 24–49 (2016; Zbl 1369.14054); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 80, No. 1, 27–54 (2016)] have introduced 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 \(t=c'\) and a homotopy class of paths \(\gamma\) joining \(c'\) and \(c\), constants \(\kappa_{\rho,n}(\gamma)\) describe the variation around \(c\) of the Frobenius solutions to \(L\) defined near \(t=c'\) and continued analytically along \(\gamma\). The purpose of this work is to develop a theory (suggested by Golyshev) of motivic Mellin transforms or motivic gamma functions. The main result relates the generating series \(\sum_{n=0}^{\infty}\kappa_{\rho,n}(s-\rho)^n\) to the Taylor expansion at \(s=\rho\) of a generalized gamma function, which is a Mellin transform of a solution of the differential operator dual to \(L\). It follows from this that the numbers \(\kappa_{\rho,n}\) are always periods when \(L\) is a geometric differential operator.

MSC:

14D07 Variation of Hodge structures (algebro-geometric aspects)
11G35 Varieties over global fields
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14F42 Motivic cohomology; motivic homotopy theory

Citations:

Zbl 1369.14054
PDFBibTeX XMLCite
Full Text: DOI arXiv