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# Pure and Applied Mathematics Quarterly

## Volume 16 (2020)

### Number 4

### Special Issue: In Honor of Prof. Gert-Martin Greuel’s 75th Birthday

Guest Editors: Igor Burban, Stanislaw Janeczko, Gerhard Pfister, Stephen S.T. Yau, and Huaiqing Zuo

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

Pages: 907 – 966

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n4.a1

#### Author

#### 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).

Received 12 August 2019

Accepted 10 September 2019

Published 13 November 2020