Algebraic hypergeometric transformations of modular origin

Abstract

It is shown that Ramanujan's cubic transformation of the Gauss hypergeometric function 2 F 1 arises from a relation between modular curves, namely the covering of X 0 (3) by X 0 (9). In general, when 2≤ N≤ 7, the N-fold cover of X 0 (N) by X 0 (N 2 ) gives rise to an algebraic hypergeometric transformation. The N = 2,3,4 transformations are arithmetic-geometric mean iterations, but the N = 5,6,7 transformations are new. In the final two cases the change of variables is not parametrized by rational functions, since X 0 (6),X 0 (7) are of genus 1. Since their quotients X 0 + (6),X 0 + (7) under the Fricke involution (an Atkin-Lehner involution) are of genus 0, the parametrization is by two-valued algebraic functions. The resulting hypergeometric transformations are closely related to the two-valued modular equations of Fricke and H. Cohn.