[Paper Review] Hodge-Iwasawa Theory I

Xin T. Tong|arXiv (Cornell University)|Jun 5, 2020

Algebraic Geometry and Number Theory30 references4 citations

TL;DR

This paper introduces Hodge-Iwasawa theory as a simultaneous generalization of relative $p$-adic Hodge theory and noncommutative Iwasawa theory, unifying frameworks for studying $p$-adic representations, étale local systems, and Iwasawa cohomology in families. It constructs deformed period rings and sheaves over ind-Fréchet algebras and schematic relative Fargues-Fontaine curves, establishing a Waldhausen $K$-theoretic framework where the cohomological functor induces a null-homotopic map on $K$-theory, yielding a new perspective on $oldsymbol{\epsilon}$-isomorphism conjectures.

ABSTRACT

In this paper, we are going to establish a simultaneous generalization of the relative Iwasawa theory proposed by Kedlaya-Pottharst and the relative $p$-adic Hodge theory after Kedlaya-Liu. We call this Hodge-Iwasawa theory in the sense that one could apply the theory to study noncommutative Iwasawa cohomology and noncommutative Iwasawa theories in families and meanwhile one could apply the theory to study the deformation theory of étale local systems or families of representations of fundamental groups or the equivariant constructible $p$-adic sheaves, with more sophisticated point of view coming from Kato, Fukaya-Kato. We follow closely the approach of Kedlaya-Liu to study the corresponding modules and sheaves over the corresponding deformed version of the period rings and period sheaves.

Motivation & Objective

• To unify relative $p$-adic Hodge theory and noncommutative Iwasawa theory in a common framework.
• To generalize the study of $p$-adic representations and étale local systems to families over deformed period rings.
• To develop a cohomological and $K$-theoretic framework for noncommutative, equivariant $p$-adic sheaves using Waldhausen categories.
• To extend the Tamagawa-Iwasawa theory to higher homotopical and geometric settings via Hodge-Iwasawa modules.
• To provide a geometric and motivic interpretation of $\epsilon$-isomorphism conjectures in the $p$-adic setting.

Proposed method

• Constructs Hodge-Iwasawa sheaves and vector bundles over ind-Fréchet algebras and deformed schematic relative Fargues-Fontaine curves.
• Applies the approach of Kedlaya-Liu to deformed period rings and sheaves, generalizing $\widetilde{\Pi}_{L,A}$-modules in the Iwasawa context.
• Uses Fréchet-Stein and rigid analytic deformations of relative Fargues-Fontaine curves to model families of $p$-adic representations.
• Defines a Waldhausen exact functor $R\Gamma_\emptyset(X_\sharp, \cdot)$ from perfect complexes on $X_\sharp$ to perfect $T$-modules.
• Applies finiteness and cohomological dimension results from [KL] to show that the induced $K$-theory map is homotopic to zero.
• Reduces the problem to finite fields via Jacobson radical quotients and semi-simple reductions, following [FK, Proposition 2.1.3].

Experimental results

Research questions

• RQ1 How can relative $p$-adic Hodge theory and noncommutative Iwasawa theory be simultaneously generalized in a coherent framework?
• RQ2 What is the role of deformed period rings and sheaves in modeling families of $p$-adic representations and étale local systems?
• RQ3 How do Waldhausen $K$-theory and the $\epsilon$-isomorphism conjecture interact in the $p$-adic setting for constructible sheaves?
• RQ4 Can the cohomological functor $R\Gamma_\emptyset(X_\sharp, \cdot)$ induce a null-homotopic map on $K$-theory in the $p$-adic context?
• RQ5 What is the higher homotopical and geometric interpretation of Tamagawa-Iwasawa theory for Hodge-Iwasawa modules?

Key findings

• The Waldhausen $K$-theory map induced by the cohomological functor $R\Gamma_\emptyset(X_\sharp, \cdot)$ is homotopic to the zero map over $\mathrm{Spa}(\mathbb{Q}_p, \mathfrak{o}_{\mathbb{Q}_p})$.
• The null-homotopy result holds for the category $\mathbb{D}_{\mathrm{perf}}(X_\sharp, T)$ of perfect complexes on a deformed rigid space $X_\sharp$ with coefficients in a ring $T$.
• The proof relies on finiteness and cohomological dimension theorems from [KL, 8.1, 10.1] in the $p$-adic setting over $\mathbb{Q}_p^{\mathrm{ur}}$.
• The reduction to finite fields is achieved by quotienting out the Jacobson radical of finite quotients $T/I$, enabling the use of semi-simple and finite field cases.
• The framework generalizes the $\epsilon$-isomorphism conjectures to the $p$-adic setting, extending results from [FK], [F], [BB], [LZ], and [Nak3].
• The theory provides a new geometric and motivic perspective on noncommutative Iwasawa cohomology and constructible $p$-adic sheaves via Hodge-Iwasawa modules.