TL;DR
This paper demonstrates that Tate classes on Siegel modular 3-folds are generated by images of Hilbert modular surfaces and Shimura curves, using a detailed analysis of period pairings between cohomological forms and these surfaces.
Contribution
It establishes a new understanding of the structure of Tate classes on Siegel 3-folds, linking them explicitly to Hilbert modular surfaces and Shimura curves.
Findings
Tate classes are spanned by images of Hilbert modular surfaces at degree 2.
Tate classes are also spanned by images of Shimura curves at degree 4.
The proof involves analyzing period pairings between cohomological forms and these surfaces.
Abstract
This article proves that Tate classes on Siegel modular 3-folds are spanned by the images of Hilbert modular surfaces at degree 2 and by the images of Shimura curves at degree 4. The proof involves a careful study of the period pairing between degree-4 rapidly decreasing cohomological forms and Hilbert modular surfaces.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Algebra and Geometry · Algebraic Geometry and Number Theory