Mathematics – Number Theory
Scientific paper
2006-11-02
J. Ramanujan Math. Soc. 24 (2009), 1-73
Mathematics
Number Theory
57 pages, 19 tables
Scientific paper
Many rationally parametrized elliptic modular equations are derived. Each comes from a family of elliptic curves attached to a genus-zero congruence subgroup $\Gamma_0(N)$, as an algebraic transformation of elliptic curve periods, parametrized by a Hauptmodul (function field generator). The periods satisfy a Picard-Fuchs equation, of hypergeometric, Heun, or more general type; so the new modular equations are algebraic transformations of special functions. When N=4,3,2 they are modular transformations of Ramanujan's elliptic integrals of signatures 2,3,4. This gives a modern interpretation to his theories of integrals to alternative bases: they are attached to certain families of elliptic curves. His anomalous theory of signature 6 turns out to fit into a general Gauss-Manin rather than a Picard-Fuchs framework.
No associations
LandOfFree
On Rationally Parametrized Modular Equations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On Rationally Parametrized Modular Equations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On Rationally Parametrized Modular Equations will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-292112