Chow motives of universal families over some Shimura surfaces

TL;DR

This paper establishes an absolute Chow-Kuenneth decomposition for the motive of universal abelian family over certain Shimura surfaces and proves the Hodge conjecture for their general fibers, extending prior results.

Contribution

It provides the first such decomposition for these motives and confirms the Hodge conjecture in this context, advancing understanding of algebraic cycles on Shimura varieties.

Findings

Chow-Kuenneth decomposition established for universal families over Shimura surfaces

Hodge conjecture proven for general fibers of these families

Extension of Ribet's results to new cases

Abstract

We prove an absolute Chow-Kuenneth decomposition for the motive of universal families A of abelian varieties over some compact Shimura surface. We furthermore prove the Hodge conjecture for general fibres of A, extending results of Ribet.