On supersingular elliptic curves and hypergeometric functions

Abstract

The Legendre family of elliptic curves has the remarkable property that both its periods and its supersingular locus have descriptions in terms of the hypergeometric function ${}_2{F}_1\left(\genfrac{}{}{0ex}{}{1∕2}{}\genfrac{}{}{0ex}{}{1∕2}{1}|z\right)$. In this work we study elliptic curves and elliptic integrals with respect to the hypergeometric functions ${}_2{F}_1\left(\genfrac{}{}{0ex}{}{1∕3}{}\genfrac{}{}{0ex}{}{2∕3}{1}|z\right)$ and ${}_2{F}_1\left(\genfrac{}{}{0ex}{}{1∕2}{}\genfrac{}{}{0ex}{}{5∕12}{1}|z\right)$, and prove that the supersingular $\lambda$-invariant locus of certain families of elliptic curves are given by these functions.