Legendre Drinfeld modules and universal supersingular polynomials (Q2876604)

Drinfeld modules are function fields analogues of elliptic curves, particularly in rank \(2\). There are many similarities between these two objects but also a few deep differences. One motivation of this paper is to explore the parallelism between the two theories when studying the relation between the periods and supersingular polynomials. To motivate the author's definition of Legendre Drinfeld modules, let \(\Lambda\subset {\mathbb C}_{\infty}\) be a rank \(2\) \({\mathbb F}_q[T]\)-lattice. Here \({\mathbb C}_{\infty}\) denotes the completion of an algebraic closure of \({\mathbb F}_q(T)_{ \infty}\), the completion of \({\mathbb F}_q(T)\) at the infinite prime \(1/T\). Attached to \(\Lambda\) we have the exponential function \(e_{\Lambda}(z)=z\prod_{0\neq \lambda\in \Lambda}(1-\frac{z}{\lambda})\).NEWLINENEWLINEWe have that \(\prod_{\lambda\in \Lambda/ T\Lambda}(x-e_{\Lambda}(\frac{\lambda}{T}))\) is a polynomial of the form \(x+g(\Lambda)x^q+\Delta(\Lambda) x^{q^2}\). Let \(\phi^{\Lambda}\) be the Drinfeld module corresponding to \(\Lambda\), which is given by \(\phi^{\Lambda}_T:=T+g(\Lambda)\tau + \Delta(\Lambda)\tau^2\). Then \(\ker (\phi^{\Lambda}_T)\) can be viewed as the function field analogue of the \(2\)-torsion of an elliptic curve \(E\). The special family of Drinfeld modules \(\phi_T=T-(T+\Delta)\tau +\Delta \tau^2\) is proposed as an analogue of the Legendre normal form elliptic curves.NEWLINENEWLINEThe author exhibits explicit formulas for a certain period of these Drinfeld modules. Also, formulas for the supersingular locus in that family are found, establishing a connection between these two kinds of formulas. Finally, it is proved a closed formula for the supersingular polynomial in the \(j\)-invariant for generic Drinfeld modules.