2019/09/06 by Kyoji Saito, Saito, Kyoji
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1909.02715
openalex publication_date 2019/09/06 · openalex created_date 2019/09/12 · openalex updated_date 2026/07/28
Elliptic integrals, since Euler's finding of addition theorem 1751, has been studied extensively from various view points. Present paper gives a view point from primitive integrals of types A2, B2 and G2 for the three families of elliptic curves of Weierstrass, Jacobi-Legendre and Hesse, respectively. We solve Jacobi inversion problem for the period maps in the sense explained in the introduction (see [Siegel] Chap.1,13) by introducing certain generalized Eisenstein series of types A2, B2 and G2, which generate the ring of invariant functions on the period domain for the congruence subgroups Γ1(N) (N=1,2 and 3). In particular, Eisenstein series of type B2 includes the case of weight two, and Eisenstein series of type G2 includes the cases of weight one and two, which seem to be of new feature. The goal of the paper is a partial answer to the discriminant conjecture, which claims an existence of certain cusp form of weight 1 with character of topological origin, giving a power root of the discriminant form (Aspects Math., E36,p. 265-320. 2004). See \S12 Concluding Remarks for more about back grounds of the present paper.