In our PH.D. thesis we have showed that the Generalized Grothendieck's Conjecture of Periods applied to 1-motives, whose underlying semi-abelian variety is a product of elliptic curves and of tori, is equivalent to a transcendental conjecture involving elliptic integrals of the first and second kind, and logarithms of complex numbers. In this paper we investigate the Generalized Grothendieck's Conjecture of Periods in the case of 1-motives whose underlying semi-abelian variety is a non trivial extension of a product of elliptic curves by a torus. This will imply the introduction of elliptic integrals of the third kind for the computation of the period matrix of M and therefore the Generalized Grothendieck's Conjecture of Periods applied to M will be equivalent to a transcendental conjecture involving elliptic integrals of the first, second and third kind.

中文翻译：

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第三类椭圆积分和1-动机

在我们的 PH.D. 我们已经证明了广义格洛腾迪克周期猜想适用于 1 动机，其潜在的半阿贝尔变体是椭圆曲线和圆环的乘积，等价于涉及第一类和第二类椭圆积分的超越猜想，并且复数的对数。在本文中，我们研究了在 1-动机的情况下的广义格洛腾迪克周期猜想，其潜在的半阿贝尔变体是圆环的椭圆曲线乘积的非平凡扩展。这将意味着引入第三类椭圆积分来计算 M 的周期矩阵，因此应用于 M 的广义格洛腾迪克周期猜想将等价于涉及第一类椭圆积分的超越猜想，