Hasse invariants for the Clausen elliptic curves

Gauss’s [inline-graphic not available: see fulltext] hypergeometric function gives periods of elliptic curves in Legendre normal form. Certain truncations of this hypergeometric function give the Hasse invariants for these curves. Here we study another form, which we call the Clausen form, and we prove that certain truncations of [inline-graphic not available: see fulltext] and [inline-graphic not available: see fulltext] in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {F}_{p}[x]$\end{document} are related to the characteristic p Hasse invariants.