Infinitesimal deformations of opers in positive characteristic and the de Rham cohomology of symmetric products

TL;DR

This paper extends the Eichler-Shimura isomorphism to positive characteristic, analyzing de Rham cohomology of opers and their symmetric products, revealing a decomposition analogous to classical results in a new algebraic setting.

Contribution

It establishes an Eichler-Shimura-type decomposition for dormant opers in positive characteristic using de Rham cohomology and symmetric product structures.

Findings

Deformation space of a $G$-oper decomposes into cohomology groups of symmetric products.

An Eichler-Shimura-type decomposition is obtained for dormant opers on stable curves.

Basic properties of de Rham cohomology groups and deformation spaces are formulated.

Abstract

The Eichler-Shimura isomorphism describes a certain cohomology group with coefficients in a space of polynomials by using holomorphic modular/cusp forms. It determines a canonical decomposition of the corresponding de Rham cohomology group associated to a specific oper on a Riemann surface. One purpose of the present paper is to establish its analogue for opers in positive characteristic. We first discuss some basic properties on the (parabolic) de Rham cohomology groups and deformation spaces of $G$-opers (where $G$ is a semisimple algebraic group of adjoint type) in a general formulation. In particular, it is shown that the deformation space of a $G$-oper induced from an $\mathrm{SL}_2$-oper decomposes into a direct sum of the (parabolic) de Rham cohomology groups of its symmetric products. As a consequence, we obtain an Eichler-Shimura-type decomposition for dormant opers on general pointed stable curves by considering a transversal intersection of related spaces in the de Rham moduli space.