2008/06/24 by Hossein Movasati, Movasati, Hossein
Mathematics · #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #math.AG #math.CV
paper · pdf · doi:10.48550/arxiv.0806.3926
arxiv created 2008/06/24 · arxiv updated 2009/12/01
In this article we consider the three parameter family of elliptic curves Et: y2-4(x-t1)3+t2(x-t1)+t3=0, t∈\C3 and study the modular holomorphic foliation \Fω in \C3 whose leaves are constant locus of the integration of a 1-form ω over topological cycles of Et. Using the Gauss-Manin connection of the family Et, we show that \Fω is an algebraic foliation. In the case ω=(xdx)/(y), we prove that a transcendent leaf of \Fω contains at most one point with algebraic coordinates and the leaves of \Fω corresponding to the zeros of integrals, never cross such a point. Using the generalized period map associated to the family Et, we find a uniformization of \Fω in T, where T⊂ \C3 is the locus of parameters t for which Et is smooth. We find also a real first integral of \Fω restricted to T and show that \Fω is given by the Ramanujan relations between the Eisenstein series.