A view on elliptic integrals from primitive forms (period integrals of type $\mathrm{A}_2$, $\mathrm{B}_2$ and $\mathrm{G}_2$)

Kyoji Saito (Research Institute for Mathematical Sciences, Kyoto University, Sakyoku Kitashirakawa, Kyoto, Japan; and Laboratory of AGHA, Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, Russian Federation)

Abstract

Elliptic integrals, since Euler’s finding of addition theorem 1751, has been studied extensively from various view points. The present paper gives a view point from primitive integrals of types $\mathrm{A}_2$, $\mathrm{B}_2$ and $\mathrm{G}_2$. We solve Jacobi inversion problem for the period maps by introducing generalized Eisenstein series of types $\mathrm{A}_2$, $\mathrm{B}_2$ and $\mathrm{G}_2$, which generate the ring of invariants functions on the period domain for the congruence subgroups $\Gamma_1 (N) (N = 1, 2 \textrm{ and } 3)$. Type $\mathrm{A}_2$ case is classical. Type $\mathrm{B}_2$ and type $\mathrm{G}_2$ cases seems to be new. The goal of the paper is a partial answer to the discriminant conjecture: to show an existence of the cusp form of weight $1$ with character of topological origin, which is a power root of the discriminant form (Aspects Math., E36, p. 265–320. 2004).

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Received 12 August 2019

Accepted 10 September 2019

Published 13 November 2020