De Rham-Betti Groups of Some Abelian Fourfolds

2026/07/25 by Zekun Ji
Mathematics · #math.AG

paper · pdf

Abstract

We determine the de Rham--Betti (dRB) groups for several classes of abelian varieties over ℚ. We prove that G_dRB(A)=MT(A) for simple CM abelian fourfolds and abelian fourfolds with quartic CM endomorphism field. Due to the current limited knowledge of dRB structures, we adopt an approach different from Moonen--Zarhin's method of determining Mumford-Tate groups. Instead, we use two results by Gross and Chudnovsky, Galois-theoretic analysis and positivity constraints arising from polarizations to exclude certain reductive subgroups of Mumford-Tate groups as candidates for dRB groups. This article is based on 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.

Citations

• De Rham-Betti Groups of Type IV Abelian Varieties
• De Rham-Betti classes with coefficients
• Some remarks concerning the Grothendieck Period Conjecture
• Hodge classes on abelian varieties of low dimension

Related

• De Rham-Betti Groups of Type IV Abelian Varieties
• De Rham-Betti classes with coefficients
• Galois representations, Mumford-Tate groups and good reduction of abelian varieties
• On some abelian varieties of type IV
• A Survey of the Hodge Conjecture for Abelian Varieties