Supersingular Loci from Traces of Hecke Operators

TL;DR

This paper establishes a new connection between traces of Hecke operators on cusp forms and supersingular polynomials at prime p, extending classical results relating Eisenstein series and supersingular loci.

Contribution

It introduces a novel approach using the Eichler-Selberg trace formula to relate Hecke trace forms to supersingular polynomials, generalizing Deligne's classical observation.

Findings

Identifies congruences between trace forms of different weights mod p

Relates divisor polynomials of trace forms to supersingular polynomial S_p(x)

Extends classical results to Hecke trace forms using new methods

Abstract

A classical observation of Deligne shows that, for any prime $p \geq 5$, the divisor polynomial of the Eisenstein series $E_{p-1}(z)$ mod $p$ is closely related to the supersingular polynomial at $p$, $$S_p(x) := \prod_{E/\bar{\mathbb{F}}_p \text{ supersingular}}(x-j(E)) \in \mathbb{F}_p[x].$$ Deuring, Hasse, and Kaneko and Zagier found other families of modular forms which also give the supersingular polynomial at $p$. In a new approach, we prove an analogue of Deligne's result for the Hecke trace forms $T_k(z)$ defined by the Hecke action on the space of cusp forms $S_k$. We use the Eichler-Selberg trace formula to identify congruences between trace forms of different weights mod $p$, and then relate their divisor polynomials to $S_p(x)$ using Deligne's observation.