Explicit formulas for arithmetic support of differential and difference operators

TL;DR

The article develops explicit arithmetic formulas for traces and determinants of formal deformations $D=P+tQ_1+\cdots$ of the simple motivic differential operator $P=\prod_{i=1}^k(x\partial_x-r_i)$, connecting these arithmetic invariants modulo large primes to monodromy data from characteristic-zero solutions. It extends the construction to a $q$-difference setting, derives a parallel set of trace/determinant formulas, and analyzes the space of formal solutions to express determinants in terms of eigenvalues $\lambda_{i,j}$ arising from monodromy-like data. By examining both generic and special deformations (including deformations of the identity), the work provides explicit formulas such as $\det(\rho_{p,\xi,\eta}(D))$ in terms of $\exp(T_\varepsilon\operatorname{Coef}_{x^\varepsilon}(\log(1+P^{-1}Q))x^\varepsilon)$ and $w(\varepsilon)=\prod_i\prod_j(\varepsilon-\lambda_{i,j})$, and it relates these polynomials to analytic monodromy via $\log\mathcal{M}$. The paper also presents adic investigations of the determinant factor $w$ in arithmetic and $q$-analogs, revealing structured denominators tied to primes and cyclotomic polynomials that hint at deeper motivic/arithmetic structures governing the deformations.

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.