PreprintCC BY 4.0arXiv
On the Hodge and Tate conjectures for moduli spaces of curves
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.
Related
- Simpson filtrations and closure relations of Hodge moduli spaces
- Cetraro lectures on integrable systems and moduli spaces of curves
- Polynomial point counts for moduli spaces of curves with marked points
- A Murphy's law for supports of forgetful maps between moduli spaces of stable pointed curves
- Motivic Euler characteristics of moduli spaces of curves
BibTeX
@misc{payne2026hodge,
title = {{On the Hodge and Tate conjectures for moduli spaces of curves}},
author = {Sam Payne},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2605.20453},
eprint = {2605.20453},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.20453},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Hodge and Tate conjectures for moduli spaces of curves
AU - Sam Payne
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2605.20453
UR - https://arxiv.org/abs/2605.20453
AB - 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.
ER -