Explicit formulas for arithmetic support of differential and difference operators

TL;DR

This paper derives explicit formulas for the arithmetic support of deformed differential and q-difference operators, linking their properties to monodromy in characteristic zero, with implications for understanding their algebraic and geometric structures.

Contribution

It provides explicit formulas for the arithmetic support of deformed differential and q-difference operators, connecting these to monodromy in characteristic zero.

Findings

Formulas for arithmetic support of differential operator deformations

Extension of results to q-difference operators

Connection between arithmetic support and monodromy

Abstract

We compute arithmetic support of the formal deformations $D=P+tQ_1+t^2Q_2+...$ of the differential operator $P=(x\partial_x-r_1)...(x\partial_x-r_k)$, where $r_1,...,r_k\in\mathbb{Q}$ for sufficiently large primes $p$ in terms of the monodromy of $D$ in characteristic zero. An analog of these results is also provided in the case of $q$-difference operators.