Algebraic Hypergeometric Transformations of Modular Origin
Abstract
Description
It is shown that Ramanujan's cubic transformation of the Gauss hypergeometric function ${}_2F_1$ arises from a relation between modular curves, namely the covering of $X_0(3)$ by $X_0(9)$. In general, when $2\le N\le 7$ 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 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.
Final version, 27 pages, accepted by Transactions of the AMS. Some typos and equation formatting problems fixed
Final version, 27 pages, accepted by Transactions of the AMS. Some typos and equation formatting problems fixed