Quantum $j$-Invariant in Positive Characteristic II: Formulas and Values at the Quadratics

TL;DR

This paper provides explicit formulas for the quantum j-invariant at quadratic elements in positive characteristic, showing the finiteness of its values and their boundedness, advancing understanding in this area.

Contribution

It derives explicit limit formulas for the quantum j-invariant at quadratic elements and proves the finiteness and boundedness of its values in positive characteristic.

Findings

Explicit formulas for $j^{ m qt}(f)$ at quadratic $f$

Number of values of $j^{ m qt}(f)$ is finite

None of the values of $j^{ m qt}(f)$ is infinite

Abstract

The multi-valued quantum $j$-invariant in positive characteristic is studied at quadratic elements. For every quadratic $f$, an explicit expression for each of the values of $j^{\rm qt}(f)$ is given as a limit of rational functions of $f$. It is proved that the number of values of $j^{\rm qt}(f)$ is finite and that none of these values is $\infty$.