Chow motives of elliptic modular surfaces and threefolds

TL;DR

This paper proves conjectures related to the structure of Chow groups for elliptic modular threefolds, extending known results from surfaces and providing more precise insights into their algebraic cycles.

Contribution

It establishes the existence and structure of a filtration on Chow groups for elliptic modular threefolds, advancing the understanding of algebraic cycles in higher-dimensional varieties.

Findings

Proved conjectures on Chow group filtrations for elliptic modular threefolds

Extended results from surfaces to threefolds in the context of algebraic cycles

Provided more precise descriptions of Chow groups for these varieties

Abstract

The main result of this paper is the proof for elliptic modular threefolds of conjectures on the existence and structure of a filtration on the Chow groups of smooth projective varieties. In the form we prove them these conjectures were formulated by the second-named author, but as Jannsen has shown, these conjectures are equivalent to a conjecture of Beilinson. These conjectures have previously been proved for surfaces in general, but for elliptic modular surfaces we obtain more precise results which are then used in proving the conjectures for elliptic modular threefolds.