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$ε$-isomorphisms for rank one $(\varphi,Γ)$-modules over Lubin-Tate Robba rings

Milan Malcic, Rustam Steingart, Otmar Venjakob, Max Witzelsperger

TL;DR

This work extends Nakamura’s $ε$-isomorphism framework to $L$-analytic $(\varphi_L,Γ_L)$-modules over Lubin–Tate Robba rings, formulating and proving an $ε$-isomorphism conjecture for rank-one and trianguline modules via Lubin–Tate deformation. It develops analytic cohomology, a Lubin–Tate version of local Tate duality, and analytic Iwasawa theory to define and study determinant lines and epsilon-constants, linking de Rham specializations to a global interpolation on the Lubin–Tate character variety. Critical advances include the construction of de Rham epsilon-isomorphisms, a determinant-theoretic framework, and a density-based approach to extend the rank-one results to families. The results illuminate how Lubin–Tate analytic structures govern interpolations of local constants and provide a robust framework for future generalizations beyond rank one. The theory has potential implications for $p$-adic local Langlands and families of $L$-analytic Galois representations in Lubin–Tate contexts.

Abstract

Inspired by Nakamura's work (arXiv:1305.0880) on $ε$-isomorphisms for $(\varphi,Γ)$-modules over (relative) Robba rings with respect to the cyclotomic theory, we formulate an analogous conjecture for $L$-analytic Lubin-Tate $(\varphi_L,Γ_L)$-modules over (relative) Robba rings for any finite extension $L$ of $\mathbb{Q}_p.$ In contrast to Kato's and Nakamura's setting, our conjecture involves $L$-analytic cohomology instead of continuous cohomology within the generalized Herr complex. Similarly, we restrict to the identity components of $D_{cris}$ and $D_{dR},$ respectively. For rank one modules of the above type or slightly more generally for trianguline ones, we construct $ε$-isomorphisms for their Lubin-Tate deformations satisfying the desired interpolation property.

$ε$-isomorphisms for rank one $(\varphi,Γ)$-modules over Lubin-Tate Robba rings

TL;DR

This work extends Nakamura’s -isomorphism framework to -analytic -modules over Lubin–Tate Robba rings, formulating and proving an -isomorphism conjecture for rank-one and trianguline modules via Lubin–Tate deformation. It develops analytic cohomology, a Lubin–Tate version of local Tate duality, and analytic Iwasawa theory to define and study determinant lines and epsilon-constants, linking de Rham specializations to a global interpolation on the Lubin–Tate character variety. Critical advances include the construction of de Rham epsilon-isomorphisms, a determinant-theoretic framework, and a density-based approach to extend the rank-one results to families. The results illuminate how Lubin–Tate analytic structures govern interpolations of local constants and provide a robust framework for future generalizations beyond rank one. The theory has potential implications for -adic local Langlands and families of -analytic Galois representations in Lubin–Tate contexts.

Abstract

Inspired by Nakamura's work (arXiv:1305.0880) on -isomorphisms for -modules over (relative) Robba rings with respect to the cyclotomic theory, we formulate an analogous conjecture for -analytic Lubin-Tate -modules over (relative) Robba rings for any finite extension of In contrast to Kato's and Nakamura's setting, our conjecture involves -analytic cohomology instead of continuous cohomology within the generalized Herr complex. Similarly, we restrict to the identity components of and respectively. For rank one modules of the above type or slightly more generally for trianguline ones, we construct -isomorphisms for their Lubin-Tate deformations satisfying the desired interpolation property.
Paper Structure (36 sections, 52 theorems, 378 equations)

This paper contains 36 sections, 52 theorems, 378 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. (phi,Gamma)-Modules over the Robba ring
  4. Definition of the Robba ring $\mathcal{R}$
  5. Frobenius and $\Gamma_L$-action on $\mathcal{R}$
  6. (phi,Gamma)-Modules
  7. Rank one modules and characters
  8. The modules $\mathbf D_{\mathrm{dif}}^{(+)}(M)$
  9. (Analytic) Cohomology groups
  10. Finiteness of analytic Cohomology
  11. Perfectness of analytic Iwasawa cohomology and the Lubin-Tate deformation
  12. Replacing Local Tate duality
  13. Cohomological computations in the character case
  14. Bloch–Kato exponential for analytic (phi,Gamma)-modules
  15. $D_{dR}$ and $D_{cris}$ for analytic (phi,Gamma)-modules
  16. ...and 21 more sections

Key Result

Proposition 3.2

Let $M$ be a $\varphi_L$-module over $\mathcal{R}_F$. Then there exists an $r(M)>0$ such that, for each $0<r\leq r(M)$, there exists a unique finitely generated free $\mathcal{R}_F^r$-submodule $M^r\subseteq M$ satisfying the following properties: In particular, for $0<s\leq r\leq r(M)$, one has

Theorems & Definitions (176)

  • Conjecture : See Conjecture \ref{['conj']}
  • Remark 2.1
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Definition 3.4
  • Remark 3.5
  • Definition 3.6
  • Definition 3.7
  • ...and 166 more