Supersingular conjectures for the Fricke group

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.