Supersingular conjectures for the Fricke group

TL;DR

This paper proves several conjectures related to supersingular polynomials for the Fricke group, including formulas and prime splitting criteria, connecting number theory with sporadic simple groups.

Contribution

It provides proofs for conjectures on supersingular polynomials for specific Fricke groups, linking their properties to group theory and resultants.

Findings

Formula for the square of supersingular polynomial mod p

Prime splitting criteria linked to sporadic simple groups

Validation of Nakaya's conjectures

Abstract

A proof is given of several conjectures from a recent paper of Nakaya concerning the supersingular polynomial $ss_p^{(N*)}(X)$ for the Fricke group $\Gamma_0^*(N)$, for $N \in \{2, 3, 5, 7\}$. One of these conjectures gives a formula for the square of $ss_p^{(N*)}(X)$ (mod $p$) in terms of a certain resultant, and the other relates the primes $p$ for which $ss_p^{(N*)}(X)$ splits into linear factors (mod $p$) to the orders of certain sporadic simple groups.