### Abstract

We use arithmetic models of modular curves to establish some properties of Ramanujan's continued fraction. In particular, we give a new geometric proof that its singular values are algebraic units that generate specific abelian extensions of imaginary quadratic fields, and we use a mixture of geometric and analytic methods to construct and study an infinite family of two-variable polynomials over ℤ that are related to Ramanujan's function in the same way that the classical modular polynomials are related to the classical j-function. We also prove that a singular value on the imaginary axis, necessarily real, lies in a radical tower in ℝ only if all odd prime factors of its degree over ℚ are Fermat primes; by computing some ray class groups, we give many examples where this necessary condition is not satisfied.

Original language | English (US) |
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Pages (from-to) | 27-104 |

Number of pages | 78 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Issue number | 597 |

DOIs | |

State | Published - Aug 1 2006 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Journal fur die Reine und Angewandte Mathematik*, (597), 27-104. https://doi.org/10.1515/CRELLE.2006.063