On the Hodge and Tate conjectures for moduli spaces of curves

TL;DR

This paper reviews recent advances in understanding the cohomology of moduli spaces of stable curves, focusing on the Hodge and Tate conjectures and their implications for algebraic cycles.

Contribution

It demonstrates how boundary stratification techniques verify these conjectures in many cases and discusses future research directions.

Findings

Boundary stratification verifies Hodge and Tate conjectures in many cases

Connections between Galois representations and algebraic cycles are elucidated

Open problems and future directions are outlined

Abstract

We survey recent progress on the cohomology of moduli spaces of stable curves through the lens of the Hodge and Tate conjectures, especially their generalized coniveau forms, which relate Hodge structures and l-adic Galois representations on cohomology to algebraic cycles. We explain how the inductive structure of the boundary stratification verifies these conjectures in a surprisingly wide range of cases, describe the guiding inspiration from arithmetic, and discuss open problems and directions for future research.