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Algebraic hypergeometric transformations of modular origin


Author: Robert S. Maier
Journal: Trans. Amer. Math. Soc. 359 (2007), 3859-3885
MSC (2000): Primary 11F03, 11F20, 33C05
DOI: https://doi.org/10.1090/S0002-9947-07-04128-1
Published electronically: March 7, 2007
MathSciNet review: 2302516
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Abstract: 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\leqslant N\leqslant 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 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.


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Additional Information

Robert S. Maier
Affiliation: Departments of Mathematics and Physics, University of Arizona, Tucson, Arizona 85721
Email: rsm@math.arizona.edu

DOI: https://doi.org/10.1090/S0002-9947-07-04128-1
Received by editor(s): January 24, 2005
Received by editor(s) in revised form: July 18, 2005
Published electronically: March 7, 2007
Additional Notes: The author was supported in part by NSF Grant No. PHY-0099484.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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