Algebraic Hypergeometric Transformations of Modular Origin
| dc.creator | Maier, Robert S. | |
| dc.date | 2005-01-24 | |
| dc.date | 2006-03-24 | |
| dc.date.accessioned | 2026-07-07T08:09:29Z | |
| dc.date.available | 2026-07-07T08:09:29Z | |
| dc.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. | |
| dc.description | Final version, 27 pages, accepted by Transactions of the AMS. Some typos and equation formatting problems fixed | |
| dc.identifier | https://arxiv.org/abs/math/0501425 | |
| dc.identifier | http://arxiv.org/abs/math/0501425 | |
| dc.identifier | Trans. Amer. Math. Soc. 359 (2007), 3859-3885 | |
| dc.identifier | doi:10.1090/S0002-9947-07-04128-1 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/131556 | |
| dc.subject | Number Theory | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | 11F03; 11F20, 33C05 | |
| dc.title | Algebraic Hypergeometric Transformations of Modular Origin | |
| dc.type | text |