### 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) |
---|---|

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 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

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

**Modular curves and Ramanujan's continued fraction.** / Cais, Bryden R; Conrad, Brian.

Research output: Contribution to journal › Article

*Journal fur die Reine und Angewandte Mathematik*, no. 597, pp. 27-104. https://doi.org/10.1515/CRELLE.2006.063

}

TY - JOUR

T1 - Modular curves and Ramanujan's continued fraction

AU - Cais, Bryden R

AU - Conrad, Brian

PY - 2006/8/1

Y1 - 2006/8/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33750176124&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750176124&partnerID=8YFLogxK

U2 - 10.1515/CRELLE.2006.063

DO - 10.1515/CRELLE.2006.063

M3 - Article

AN - SCOPUS:33750176124

SP - 27

EP - 104

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

SN - 0075-4102

IS - 597

ER -