Dinakar Ramakrishnan
Published 2006-09-16Version 1
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).
Comments: 15 pages, Proceedings of a conference on Algebraic Cycles and Motives (in honor of Jacob Murre) in Leiden, The Netherlands
Algebraic cycles on Hilbert modular fourfolds and poles of L-functions
Quadratic Chabauty for Modular Curves
Geometric realizations of representations for $\text{PSL}(2, \mathbb{F}_p)$ and Galois representations arising from defining ideals of modular curves
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