On Rationally Parametrized Modular Equations

By Robert S. Maier. 2006

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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.

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Citations in the corpus

• The elliptic dilogarithm for the sunset graph (2013, methodology): “16) Using the result of Maier in [31], we can derive the expression for the Γ1(6) invariant Hauptmodul, t as modular form function of q := exp(2iπτ) with τ the period ratio τ = $c/$r”