The number of linear factors of supersingular polynomials and sporadic simple groups

TL;DR

This paper explores the relationship between supersingular invariants in specific levels and the orders of certain sporadic simple groups, revealing new coincidences linked through class numbers of imaginary quadratic fields.

Contribution

It establishes new analogies between supersingular invariants in levels 2 and 3 and the orders of the Baby monster and Fischer groups, extending known prime divisibility coincidences.

Findings

Supersingular invariants in levels 2 and 3 relate to the Baby monster and Fischer groups.

Connections are made via class numbers of imaginary quadratic fields.

New coincidences between algebraic invariants and group orders are demonstrated.

Abstract

The set of prime numbers $p$ such that the supersingular $j$-invariants in characteristic $p$ are all contained in the prime field is finite. And it is well known that this set of primes coincides with the set of prime divisors of the order of the Monster simple group. In this paper, we will present analogous coincidences of supersingular invariants in level 2 and 3 and the orders of the Baby monster group and the Fischer's group. The proof uses a connection between the number of supersingular invariants and class numbers of imaginary quadratic fields.