The Gamma conjecture for $G$-functions

TL;DR

This paper proposes a Gamma conjecture for special G-functions with maximally unipotent monodromy, linking differential equations, mirror symmetry, and algebraic geometry, supported by computational examples.

Contribution

It formulates a new Gamma conjecture for G-functions with specific singularities, connecting it to mirror symmetry and providing computational evidence.

Findings

Formulation of the Gamma conjecture for special G-functions

Numerical evidence supporting the conjecture

Connections established between differential equations and mirror symmetry

Abstract

The Bombieri-Dwork conjecture predicts that the differential equations satisfied by $G$-functions come from geometry. In this paper, we will look at special $G$-functions whose differential equations have a special singularity with maximally unipotent monodromy. We will formulate a Gamma conjecture about such $G$-functions, which has close connections with the mirror symmetry of Calabi-Yau threefolds and the Gamma conjecture in algebraic geometry. We will provide examples to support this conjecture, which involves numerical computations using Mathematica programs.