Algebraic hypergeometric transformations of modular origin
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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.References
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Additional Information
- Robert S. Maier
- Affiliation: Departments of Mathematics and Physics, University of Arizona, Tucson, Arizona 85721
- MR Author ID: 118320
- ORCID: 0000-0002-1259-1341
- Email: rsm@math.arizona.edu
- 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.
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2302516