An elliptic curve \(E\) over a field \(K\) of characteristic \(p>0\) is called supersingular if the group \(E(\overline{K})\) has no \(p\)-torsion. This condition depends only on the \(j\)-invariant of \(E\) and it is well known that there are only finitely many supersingular \(j\)-invariants in \(\mathbb{F}_p\).
The authors of the paper under review describe several different ways of constructing canonical polynomials in \(\mathbb{Q} [j]\) whose reduction modulo \(p\) gives the supersingular polynomial \[ ss_p(j):= \prod_{\substack{ E/\overline{\mathbb{F}}_p\\ E\text{ supersingular }}} (j-j(E))\in \mathbb{F}_p[j]. \] These polynomials are of three kinds:
A. Polynomials coming from modular forms of weight \(p-1\).
Four special modular forms of weight \(p-1\) are defined and, if \(f\) is one of these four forms, the coefficients of the associated polynomial \(\widetilde{f}\) are \(p\)-integral and \[ ss_p(j)= \pm j^\delta (j-1728)^\varepsilon \widetilde{f}(j) \bmod p\qquad (\delta\in \{0,1,2\},\;\varepsilon\in \{0,1\}). \] B. The Atkin orthogonal polynomials.
This description was found by Atkin more than ten years ago but proofs have never been published. Atkin has defined a sequence of polynomials \(A_n(j)\in \mathbb{Q}[j]\), one in each degree \(n\), as the orthogonal polynomials with respect to a special scalar product. The coefficients of \(A_n\) are rational numbers in general but they are \(p\)-integral for primes \(p> 2n\). In particular if \(n_p\) is the degree of the supersingular polynomial \(ss_p\), then \(A_{n_p}\) has \(p\)-integral coefficients and we have the congruence \[ ss_p(j) \equiv A_{n_p}(j) \pmod p \] as well as recursion relation, closed formula and differential equation of \(A_n\).
The proofs here are simpler than those of Atkin.
C. Other orthogonal polynomials coming from hyperelliptic series.
This is a partially expository paper.
For the entire collection see [Zbl 0881.00035].