Modular curves and Ramanujan's continued fraction

Bryden R Cais, Brian Conrad

Research output: Contribution to journalArticle

11 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)27-104
Number of pages78
JournalJournal fur die Reine und Angewandte Mathematik
Issue number597
DOIs
StatePublished - Aug 1 2006
Externally publishedYes

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Modular Curves
Ramanujan
Singular Values
Continued fraction
Fermat prime
Polynomials
Geometric proof
Imaginary Quadratic Field
Polynomial
Class Group
Prime factor
Towers
Half line
Odd
Necessary Conditions
Unit
Computing
Model
Family

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: Journal fur die Reine und Angewandte Mathematik, No. 597, 01.08.2006, p. 27-104.

Research output: Contribution to journalArticle

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