Weight of the De Rham-Betti Structures of Abelian Varieties

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.

14 pages, comments welcome! arXiv admin note: text overlap with arXiv:2511.01072