Quantum j-invariant in positive characteristic I: Definition and Convergence

TL;DR

This paper introduces the quantum j-invariant in positive characteristic, focusing on its definition and convergence properties within local function fields, laying groundwork for future research in this area.

Contribution

It defines the quantum j-invariant in positive characteristic and addresses foundational questions of its convergence, correcting a previous theorem without affecting overall results.

Findings

Defined the quantum j-invariant in positive characteristic

Analyzed convergence properties of the invariant

Corrected a key theorem in the initial publication

Abstract

We introduce the quantum $j$-invariant in positive characteristic as a multi-valued, modular-invariant function of a local function field. In this paper, we concentrate on basic definitions and questions of convergence. Note: This version contains a correction to the published version of Theorem 3. The error that was found is not in any way serious, but its correction does change slightly the statement of Theorem 3 appearing in the published version. Otherwise, it has no impact on this article nor any of its sequels.