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.
ASJC Scopus subject areas
- Applied Mathematics