TY - JOUR

T1 - Modular curves and Ramanujan's continued fraction

AU - Cais, Bryden

AU - Conrad, Brian

N1 - Funding Information:
The first author was supported by an NSF VIGRE grant and the second author was supported by NSF grant DMS-0093542 and a grant from the Alfred P. Sloan Foundation during this work. The authors thank K. Conrad for providing us with a proof of Lemma D.1 and thank W. Stein for his publically available computer cluster Meccah that was used for the numerical computations. We are also greatly indebted to J. Parson for many enlightening and insightful discussions concerning Appendix C.

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.

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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 -