Residue of special functions of Anderson $A$-modules at the characteristic graph

TL;DR

This paper explores the relationship between the period lattice and special functions of Anderson $A$-modules, introducing a canonical residue interpretation and a new framework for sheaves of meromorphic functions in rigid analytic geometry.

Contribution

It introduces a canonical interpretation of the inverse map as a residue morphism and develops the concept of costability for sheaves of meromorphic functions on the rigid analytic plane.

Findings

A modification of the inverse map is shown to be canonical via residue interpretation.

The framework of costability is introduced for sheaves of $E(bC_ infty)$-valued meromorphic functions.

The approach unifies various phenomena observed in different contexts.

Abstract

Let $E$ be an Anderson $A$-module over $\mathbb{C}_{\infty}$. The period lattice of $E$ is related to its module of special functions by means of a non-canonical isomorphism introduced by the authors in [GM21]. In this paper, we explain how a modification of the inverse map is canonical by interpreting it as a residue morphism along the characteristic graph. This phenomenon has already been observed in various situations. The main innovation of this text is that of costability (costable admissible opens, costable site, etc.) which provides a convenient framework to develop the notion of sheaves of $E(\mathbb{C}_{\infty})$-valued meromorphic functions on the rigid analytic plane.