On supersingular elliptic curves and hypergeometric functions

Monks, Keenan

doi:10.2140/involve.2012.5.99

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On supersingular elliptic curves and hypergeometric functions

Keenan Monks

Vol. 5 (2012), No. 1, 99–113

DOI: 10.2140/involve.2012.5.99

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 $_{2} F_{1} (\binom{1 ∕ 2}{} \binom{1 ∕ 2}{1} | z)$ . In this work we study elliptic curves and elliptic integrals with respect to the hypergeometric functions $_{2} F_{1} (\binom{1 ∕ 3}{} \binom{2 ∕ 3}{1} | z)$ and $_{2} F_{1} (\binom{1 ∕ 2}{} \binom{5 ∕ 12}{1} | z)$ , and prove that the supersingular $λ$ -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

Milestones

Received: 12 September 2011

Accepted: 14 September 2011

Published: 28 April 2012

Communicated by Ken Ono

Authors

Keenan Monks
Harvard University 2013 Harvard Yard Mail Center Cambridge 02138 United States

Vol. 5, No. 1, 2012