Algebraic hypergeometric transformations of modular origin

It is shown that Ramanujan's cubic transformation of the Gauss hypergeometric function ₂F₁ arises from a relation between modular curves, namely the covering of X0(3) by X0(9). In general, when 2 ≤ N ≤ 7, the N-fold cover of X0(N) by X0(N²) 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 X₀(6),X₀(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.