Frobenius intertwiners for q-difference equations

TL;DR

This paper explores Frobenius intertwiners for q-difference equations related to quantum hypergeometric equations on cotangent bundles over projective spaces, connecting them to p-adic gamma functions and Dwork's p-adic differential equations.

Contribution

It provides explicit formulas for Frobenius intertwiners in q-hypergeometric equations and links these to p-adic gamma functions and Dwork's p-adic hypergeometric structures.

Findings

Explicit formula for Frobenius intertwiner constant term using p-adic q-gamma function.

Connection between q-difference equations and p-adic hypergeometric equations.

Closed formulas for p-adic constants in terms of p-adic zeta functions.

Abstract

We consider a class of $q$-hypergeometric equations describing the quantum difference equation for the cotangent bundles over projective spaces $X=T^{*}\mathbb{P}^{n-1}$ . We show that over $\mathbb{Q}_p$ these equations are equipped with the Frobenius action $(q,z)\to (q^p,z^p)$. We obtain an explicit formula for the constant term of the Frobenius intertwiner in terms of the $p$-adic $q$-gamma function of Koblitz. In the limit $q\to 1$ we arrive at the Frobenius structures for the $p$-adic hypergeometric and Bessel differential equations studied by Dwork. In particular, we find closed formulas for $p$-adic constants appearing in works of Dwork and Sperber in terms of $p$-adic zeta functions.