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Control Theorems for Hilbert Modular Varieties

Arshay Sheth

TL;DR

This work extends Hida-theoretic control to Hilbert modular varieties by proving an exact control theorem for the anti-ordinary part of the middle-degree étale cohomology after localization at a suitable Hecke-maximal ideal. The authors combine a Tor descent spectral sequence (built from modules of measures and algebraic representations) with the vanishing results of Caraiani–Tamiozzo to show freeness and weight-localized isomorphisms between Iwasawa and finite-level cohomology with coefficients in algebraic representations. A key outcome is the concentration of étale cohomology in the middle degree after localization and the extension of this vanishing to non-trivial coefficient systems, enabling p-adic variation results for Euler systems such as Asai–Flach. The results generalize Ohta’s modular-curve control to the Hilbert setting and provide a robust framework for constructing and varying Hilbert-family p-adic cohomology classes with potential applications to Bloch–Kato conjectures in the Asai direction.

Abstract

We prove an exact control theorem, in the sense of Hida theory, for the ordinary part of the middle degree étale cohomology of certain Hilbert modular varieties, after localizing at a suitable maximal ideal of the Hecke algebra. Our method of proof builds upon the techniques introduced by Loeffler-Rockwood-Zerbes; another important ingredient in our proof is the recent work of Caraiani-Tamiozzo on the vanishing of the étale cohomology of Hilbert modular varieties with torsion coefficients outside the middle degree. This work will be used in forthcoming work of the author to show that the Asai-Flach Euler system corresponding to a quadratic Hilbert modular form varies in Hida families.

Control Theorems for Hilbert Modular Varieties

TL;DR

This work extends Hida-theoretic control to Hilbert modular varieties by proving an exact control theorem for the anti-ordinary part of the middle-degree étale cohomology after localization at a suitable Hecke-maximal ideal. The authors combine a Tor descent spectral sequence (built from modules of measures and algebraic representations) with the vanishing results of Caraiani–Tamiozzo to show freeness and weight-localized isomorphisms between Iwasawa and finite-level cohomology with coefficients in algebraic representations. A key outcome is the concentration of étale cohomology in the middle degree after localization and the extension of this vanishing to non-trivial coefficient systems, enabling p-adic variation results for Euler systems such as Asai–Flach. The results generalize Ohta’s modular-curve control to the Hilbert setting and provide a robust framework for constructing and varying Hilbert-family p-adic cohomology classes with potential applications to Bloch–Kato conjectures in the Asai direction.

Abstract

We prove an exact control theorem, in the sense of Hida theory, for the ordinary part of the middle degree étale cohomology of certain Hilbert modular varieties, after localizing at a suitable maximal ideal of the Hecke algebra. Our method of proof builds upon the techniques introduced by Loeffler-Rockwood-Zerbes; another important ingredient in our proof is the recent work of Caraiani-Tamiozzo on the vanishing of the étale cohomology of Hilbert modular varieties with torsion coefficients outside the middle degree. This work will be used in forthcoming work of the author to show that the Asai-Flach Euler system corresponding to a quadratic Hilbert modular form varies in Hida families.
Paper Structure (20 sections, 24 theorems, 69 equations)

This paper contains 20 sections, 24 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Main results.
  3. Arithmetic applications.
  4. Comparison with other work.
  5. Acknowledgements
  6. Background on Hilbert modular varieties
  7. Notation.
  8. Shimura varieties for $G$
  9. The Hilbert modular variety $Y_G(U_n)$
  10. The number of components of $Y_G(U_n)$
  11. Hecke action
  12. Betti cohomology
  13. Construction of a Tor spectral sequence
  14. General notation
  15. Algebraic representations
  16. ...and 5 more sections

Key Result

Theorem 1.1

The following hold. (a) We have that $e'_{\mathop{\mathrm{ord}}\nolimits} H^1(N p^{\infty})$ is finite and free as a $\Lambda$-module. (b) For $r \geq 1$ and $k \geq 0$, let $\mathfrak p_{r, k}$ denote the ideal of $\Lambda$ generated by $(1+X)^{p^{r-1}}-(1+p)^{k p^{r-1}}$. Then there is a canonical of $\mathbb Z_p$-modules that is compatible with the action of $G_{\mathbb Q, \Sigma}$ and the Heck

Theorems & Definitions (51)

  • Theorem 1.1: Ohta
  • Theorem : Theorem \ref{['mainthm']}
  • Corollary : Corollary \ref{['lastcor']}
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • ...and 41 more