arXiv math/06110412006-11math.NTmath.CA

On Rationally Parametrized Modular Equations

Robert S. Maier

备注：: 57 pages, 19 tables
发表信息：: J. Ramanujan Math. Soc. 24 (2009), 1-73

英文摘要：

Many rationally parametrized elliptic modular equations are derived. Each comes from a family of elliptic curves attached to a genus-zero congruence subgroup $Γ_0(N)$, 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.

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