The Hasse invariant of the Tate normal form $E_7$ and the supersingular polynomial for the Fricke group $\Gamma_0^*(7)$

TL;DR

This paper derives formulas for the factorization of the Hasse invariant of a specific elliptic curve form over finite fields, linking these to class numbers of quadratic fields and addressing conjectures about supersingular polynomials.

Contribution

It provides explicit formulas for factor counts of the Hasse invariant in terms of class numbers and proposes conjectural formulas for other factors, connecting to recent conjectures in the field.

Findings

Formula for linear and cubic factors of the Hasse invariant over finite fields.

Conjectural formulas for quadratic and sextic factors based on class numbers.

Implication of these formulas for a recent conjecture on supersingular polynomials.

Abstract

A formula is proved for the number of linear factors and irreducible cubic factors over $\mathbb{F}_l$ of the Hasse invariant $\hat H_{7,l}(a)$ of the Tate normal form $E_7(a)$ for a point of order $7$, as a polynomial in the parameter $a$, in terms of the class number of the imaginary quadratic field $K=\mathbb{Q}(\sqrt{-l})$. Conjectural formulas are stated for the numbers of quadratic and sextic factors of $\hat H_{7,l}(a)$ of certain specific forms in terms of the class number of $\mathbb{Q}(\sqrt{-7l})$, which are shown to imply a recent conjecture of Nakaya on the number of linear factors over $\mathbb{F}_l$ of the supersingular polynomial $ss_l^{(7*)}(X)$ corresponding to the Fricke group $\Gamma_0^*(7)$.