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

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.