Algebraic Hypergeometric Transformations of Modular Origin

Details Algebraic Hypergeometric Transformations of Modular Origin Algebraic Hypergeometric Transformations of Modular Origin

2005-01-24

arxiv.org/abs/math/0501425v3

Trans. Amer. Math. Soc. 359 (2007), 3859-3885

Mathematics

Number Theory

Final version, 27 pages, accepted by Transactions of the AMS. Some typos and equation formatting problems fixed

Scientific paper

10.1090/S0002-9947-07-04128-1

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.

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