The Hasse invariant of the Tate normal form $E_5$ and the class number of $\mathbb{Q}(\sqrt{-5l})$

TL;DR

This paper establishes a relationship between the factorization of the Hasse invariant of a specific elliptic curve form and the class number of quadratic fields, providing explicit formulas for factors over finite fields.

Contribution

It reveals a direct link between the factorization patterns of the Hasse invariant and class numbers of quadratic fields, extending understanding of supersingular polynomials and elliptic curve invariants.

Findings

Number of irreducible quartic factors is a linear function of class number h(-5l).

Similar relation holds for quadratic factors under different congruences.

Provides a formula for linear factors of supersingular polynomial over finite fields.

Abstract

It is shown that the number of irreducible quartic factors of the form $g(x) = x^4+ax^3+(11a+2)x^2-ax+1$ which divide the Hasse invariant of the Tate normal form $E_5$ in characteristic $l$ is a simple linear function of the class number $h(-5l)$ of the field $\mathbb{Q}(\sqrt{-5l})$, when $l \equiv 2,3$ modulo $5$. A similar result holds for irreducible quadratic factors of $g(x)$, when $l \equiv 1, 4$ modulo $5$. This implies a formula for the number of linear factors over $\mathbb{F}_p$ of the supersingular polynomial $ss_p^{(5*)}(x)$ corresponding to the Fricke group $\Gamma_0^*(5)$.