# Third kind elliptic integrals and 1-motives

## TL;DR

This paper explores the relationship between the Generalized Grothendieck's Conjecture of Periods and elliptic integrals of the first, second, and third kind in the context of 1-motives with semi-abelian varieties involving elliptic curves and tori.

## Contribution

It extends previous work by analyzing 1-motives with semi-abelian varieties as non-trivial extensions of elliptic curves by tori, introducing elliptic integrals of the third kind.

## Key findings

- Equivalence of the conjecture to transcendental statements involving elliptic integrals of all three kinds.
- Introduction of elliptic integrals of the third kind for period matrix computation.
- Extension of the conjecture's applicability to more complex 1-motives.

## Abstract

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.