Weight of the De Rham-Betti Structures of Abelian Varieties

TL;DR

This paper proves that abelian varieties over algebraic numbers have de Rham-Betti groups containing m as homotheties, ruling out certain classes in odd cohomology and generalizing previous results.

Contribution

It establishes the presence of m as homotheties in the de Rham-Betti group of any abelian variety over bar, extending prior findings.

Findings

De Rham-Betti group of abelian varieties over bar contains m as homotheties.

No non-zero dRB classes exist in odd-degree cohomology of such abelian varieties.

Generalizes earlier results in the literature.

Abstract

In this note, we prove that for any abelian variety defined over $\overline{\mathbb{Q}}$, its de Rham-Betti (dRB) group necessarily contains $\mathbb{G}_{m}$ as the group of homotheties. Consequently, this rules out the existence of non-zero dRB classes in odd-degree cohomology groups of abelian varieties over $\overline{\mathbb{Q}}$. This generalises results of the first part of arXiv:2511.01072.