Supersingular loci from traces of Hecke operators

Abstract

A classical observation of Deligne shows that, for any prime p ≥ 5 p \geq 5 , the divisor polynomial of the Eisenstein series E p − 1 ( z ) E_{p-1}(z) mod p p is closely related to the supersingular polynomial at p p , S p ( x ) ≔ ∏ E / F ¯ p supersingular ( x − j ( E ) ) ∈ F p [ x ] . \begin{align*} S_p(x) ≔\prod _{E/\overline {\mathbb {F}}_p \text { supersingular}}(x-j(E))\,\, \in \mathbb {F}_p[x]. \end{align*} Deuring, Hasse, and Kaneko and Zagier [Supersingular j-invariants, hypergeometric series, and Atkin’s orthogonal polynomials, Providence, RI, 1998, pp. 97–126] found other families of modular forms which also give the supersingular polynomial at p p . In a new approach, we prove an analogue of Deligne’s result for the Hecke trace forms T k ( z ) T_k(z) defined by the Hecke action on the space of cusp forms S k S_k . We use the Eichler-Selberg trace formula to identify congruences between trace forms of different weights mod p p , and then relate their divisor polynomials to S p ( x ) S_p(x) using Deligne’s observation.