#### Vol. 5, No. 1, 2012

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On supersingular elliptic curves and hypergeometric functions

### Keenan Monks

Vol. 5 (2012), No. 1, 99–113
##### 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 ${}_{2}{F}_{1}\left(\genfrac{}{}{0}{}{1∕2}{}\genfrac{}{}{0}{}{1∕2}{1}|z\right)$. In this work we study elliptic curves and elliptic integrals with respect to the hypergeometric functions ${}_{2}{F}_{1}\left(\genfrac{}{}{0}{}{1∕3}{}\genfrac{}{}{0}{}{2∕3}{1}|z\right)$ and ${}_{2}{F}_{1}\left(\genfrac{}{}{0}{}{1∕2}{}\genfrac{}{}{0}{}{5∕12}{1}|z\right)$, and prove that the supersingular $\lambda$-invariant locus of certain families of elliptic curves are given by these functions.

##### Keywords
elliptic curves, hypergeometric functions
##### Mathematical Subject Classification 2010
Primary: 14H52, 33C05