Legendre Drinfeld modules and universal supersingular polynomials

TL;DR

This paper introduces a new family of Drinfeld modules analogous to Legendre elliptic curves, providing explicit formulas for periods, supersingular loci, and supersingular polynomials, thus advancing the understanding of their arithmetic properties.

Contribution

It presents a novel family of Drinfeld modules with explicit formulas for periods and supersingular polynomials, establishing connections between these aspects.

Findings

Explicit formulas for periods of the Drinfeld modules.

Formulas for the supersingular locus in the family.

Closed formula for supersingular polynomials in the th-invariant.

Abstract

We introduce a certain family of Drinfeld modules that we propose as analogues of the Legendre normal form elliptic curves. We exhibit explicit formulas for a certain period of such Drinfeld modules as well as formulas for the supersingular locus in that family, establishing a connection between these two kinds of formulas. Lastly, we also provide a closed formula for the supersingular polynomial in the $\jmath$-invariant for generic Drinfeld modules.