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Modular Curves, Modular Surfaces, and Modular Fourfolds

Dinakar Ramakrishnan
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Abstract

This article surveys some recent work of the author on Hilbert modular fourfolds X. After some preliminaries on the cohomology and special, codimension 2 cycles Z on X of Hirzebruch-Zagier type, a proof of the Tate conjecture for X over abelian fields is exposed. It is followed by a description of a method to construct classes in the Bloch's Chow group CH^3(X,1) by using Hecke translates of the cycles Z as above with suitable intersections in translates of modular curves. The article ends with the introduction of a modular Gersten complex for a general Shimura variety X and the corresponding groups CH^p_{mod}(X) and CH^p_{mod}(X,1).

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Dinakar Ramakrishnan. 2006-09-16. Modular Curves, Modular Surfaces, and Modular Fourfolds. https://arxiv.org/abs/math/0609459

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