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

TL;DR

This paper explores elliptic integrals through primitive forms associated with types A2, B2, and G2, solving the Jacobi inversion problem using generalized Eisenstein series and contributing to the understanding of the discriminant conjecture.

Contribution

It introduces generalized Eisenstein series for types A2, B2, and G2, and applies them to solve the Jacobi inversion problem for specific elliptic curve families, advancing the theory of invariant functions.

Findings

Solved Jacobi inversion problem for period maps.

Constructed Eisenstein series generating invariant function rings.

Provided partial insights into the discriminant conjecture.

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.