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arXiv · 2511.01072

De Rham-Betti Groups of Type IV Abelian Varieties

Zekun Ji
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Abstract

We study the de Rham-Betti structure of a simple abelian variety of type IV. We will take a Tannakian point of view inspired by André. The main results are that the de Rham-Betti groups of simple CM abelian fourfolds and simple abelian fourfolds over $\overline{\mathbb{Q}}$ whose endomorphism algebra is a degree 4 CM-field coincide with their Mumford-Tate groups. The method of proof involves a thorough investigation of the reductive subgroups of the Mumford-Tate groups of these abelian varieties, inspired by Kreutz-Shen-Vial. The condition that the underlying abelian variety is simple and the condition that the de Rham-Betti group is an algebraic group defined over $\mathbb{Q}$ are also used in a crucial way. The proof is different from the method of computing Mumford-Tate groups of these abelian varieties by Moonen-Zarhin. We will also study a family of de Rham-Betti structures, in the formalism proposed by Saito-Terasoma. For such families with geometric origin, we will characterize properties of fixed tensors of the de Rham-Betti group associated with such a family.

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Zekun Ji. 2025-11-02. De Rham-Betti Groups of Type IV Abelian Varieties. https://arxiv.org/abs/2511.01072

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