De Rham-Betti Groups of Type IV Abelian Fourfolds

Abstract

We determine the de Rham--Betti (dRB) groups of several classes of abelian varieties over $\overline{\mathbb{Q}}$. We prove that $G_{\mathrm{dRB}}(A)=\mathrm{MT}(A)$ for every simple abelian fourfold of type IV. At present, much of what is known about the dRB structures of abelian varieties derives from W\"ustholz's Analytic Subgroup Theorem, whose applications primarily control linear relations among periods of $\mathrm{H}^{1}(A)$ and divisor-class information. In comparison with the theory of Hodge structures, our understanding of dRB structures remains limited. We therefore adopt an approach different from the method of Moonen-Zarhin for determining the Mumford-Tate groups of these abelian varieties. Depending on the endomorphism type of the abelian fourfold, we use Galois-theoretic analysis, van Geemen's half-twist construction, and positivity constraints arising from polarizations, as appropriate, to exclude proper reductive subgroups of the corresponding Mumford-Tate groups as candidates for the dRB groups. We also use results on periods due to Gross and Chudnovsky. This article is an expansion of the second part of the author's PhD thesis https://pure.uva.nl/ws/files/311471255/Thesis.pdf; see also https://arxiv.org/abs/2511.01072 by the author.