A view on elliptic integrals from primitive forms (Period integrals of type $\mathrm{A_2, B_2}$ and $\mathrm{G_2}$

Abstract

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 $\mathrm{A_2}, \mathrm{B_2}$ and $\mathrm{G_2}$ 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 $\mathrm{A_2}, \mathrm{B_2}$ and $\mathrm{\mathrm{G_2}}$, which generate the ring of invariant functions on the period domain for the congruence subgroups $\Gamma_1(N)$ ($N=1,2$ and $3$). In particular, Eisenstein series of type $\mathrm{B_2}$ includes the case of weight two, and Eisenstein series of type $\mathrm{G_2}$ 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.