On the Hasse invariants of the Tate normal forms $E_5$ and $E_7$

Abstract

A formula is proved for the number of linear factors over ${\mathbb{F}}_l$ of the Hasse invariant of the Tate normal form $E_5(b)$ for a point of order $5$, as a polynomial in the parameter $b$, in terms of the class number of the imaginary quadratic field $K=\mathbb{Q}(\sqrt{-l})$, proving a conjecture of the author from 2005. A similar theorem is proved for quadratic factors with constant term $-1$, and a theorem is stated for the number of quartic factors of a specific form in terms of the class number of $\mathbb{Q}(\sqrt{-5l})$. These results are shown to imply a recent conjecture of Nakaya on the number of linear factors over ${\mathbb{F}}_l$ of the supersingular polynomial $s{s}_l^{(5\ast )}(X)$ corresponding to the Fricke group ${\Gamma }_0^{\ast }(5)$. The degrees and forms of the irreducible factors of the Hasse invariant of the Tate normal form $E_7$ for a point of order $7$ are determined, which is used to show that the polynomial $s{s}_l^{(N\ast )}(X)$ for the group ${\Gamma }_0^{\ast }(N)$ has roots in ${\mathbb{F}}_{l^2}$, for any prime $l\ne N$, when $N\in \{2,3,5,7\}$.

37 pages. In version 5, minor corrections have been made