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).

中文翻译：

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从原始形式看椭圆积分（类型为$ \ mathrm {A} _2 $，$ \ mathrm {B} _2 $和$ \ mathrm {G} _2 $的周期积分）

自从欧拉发现加定理1751以来，椭圆积分已经从各种角度进行了广泛的研究。本文从类型$ \ mathrm {A} _2 $，$ \ mathrm {B} _2 $和$ \ mathrm {G} _2 $的原始积分中给出了一个观点。我们通过引入类型为$ \ mathrm {A} _2 $，$ \ mathrm {B} _2 $和$ \ mathrm {G} _2 $的广义Eisenstein级数来解决周期图的Jacobi反演问题，它们生成不变函数的环同余子组$ \ Gamma_1（N）（N = 1、2 \ textrm {和} 3）$的周期域上的值。$ \ mathrm {A} _2 $类型的大小写是经典的。输入$ \ mathrm {B} _2 $和$ \ mathrm {G} _2 $类型的案例似乎是新的。本文的目标是对判别猜想的部分回答：显示存在拓扑起源特征的权重$ 1 $的尖端形式，数学方面。，E36，第6页。265-320。2004）。