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Elliptic integrals, since Euler's finding of addition theorem 1751, has beenstudied extensively from various view points. Present paper gives a view pointfrom 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 forthe 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 thering 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 ofnew feature. The goal of the paper is a partial answer to the discriminantconjecture, which claims an existence of certain cusp form of weight 1 withcharacter of topological origin, giving a power root of the discriminant form(Aspects Math., E36,p.\ 265-320.\ 2004). See \S12 Concluding Remarks for moreabout back grounds of the present paper.

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