Supersingular conjectures for the Fricke group

Abstract

A proof is given of several conjectures from a recent paper of Nakaya concerning the supersingular polynomial $s{s}_p^{(N\ast )}(X)$ for the Fricke group ${\Gamma }_0^{\ast }(N)$, for $N\in \{2,3,5,7\}$. One of these conjectures gives a formula for the square of $s{s}_p^{(N\ast )}(X)$ (mod $p$) in terms of a certain resultant, and the other relates the primes $p$ for which $s{s}_p^{(N\ast )}(X)$ splits into linear factors (mod $p$) to the orders of certain sporadic simple groups.

27 pages, 3 tables