### 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 X0(3) by X0(9). In general, when 2 ≤ N ≤ 7, the N-fold cover of X0(N) by X0(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.

Original language | English (US) |
---|---|

Pages (from-to) | 3859-3885 |

Number of pages | 27 |

Journal | Transactions of the American Mathematical Society |

Volume | 359 |

Issue number | 8 |

DOIs | |

State | Published - Aug 2007 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Algebraic hypergeometric transformations of modular origin.** / Maier, Robert S.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 359, no. 8, pp. 3859-3885. https://doi.org/10.1090/S0002-9947-07-04128-1

}

TY - JOUR

T1 - Algebraic hypergeometric transformations of modular origin

AU - Maier, Robert S

PY - 2007/8

Y1 - 2007/8

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=67651029530&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67651029530&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-07-04128-1

DO - 10.1090/S0002-9947-07-04128-1

M3 - Article

AN - SCOPUS:67651029530

VL - 359

SP - 3859

EP - 3885

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 8

ER -