Locally Analytic Vectors in p-adic Analysis

• Locally analytic vectors are defined as elements in a p-adic Banach representation whose orbit maps are locally analytic, replacing K-finite vectors in nonarchimedean settings.
• They facilitate rigorous connections between p-adic Banach representations, (ϕ,Γ)-modules, and cohomology theories, underpinning methods such as analytic descent and resolution techniques.
• Their applications extend to the p-adic Langlands program, period ring geometry, and modern p-adic Hodge theory, offering new insights into automorphic cohomology and representation classification.

Locally analytic vectors form a foundational concept in $p$-adic analysis, nonarchimedean representation theory, and $p$-adic Hodge theory, representing a robust replacement for $K$-finite vectors in settings where the Galois or symmetry group is a $p$-adic Lie group of arbitrary dimension. Their rigorous study connects $p$-adic Banach representations, $(\varphi,\Gamma)$-modules, and cohomology theories, playing a critical role in recent advances in multi-variable $p$-adic Hodge theory, the $p$-adic Langlands program, and the geometric theory of period rings.

1. Definition and Construction

Let $G$ be a $p$-adic Lie group of dimension $d$ (over $\mathbb{Q}_p$ or a finite extension), and $W$ a Banach representation of $G$ over a $p$-adic field. The subspace of locally analytic vectors $W^{\mathrm{la}}$ consists of those $w\in W$ such that the orbit map

$$\mathrm{orb}_w\colon G \to W,\quad g \mapsto g\cdot w$$

is locally $p$-adic analytic. This is formalized by choosing a sufficiently small pro-$p$ open subgroup $H\subset G$, an analytic chart $c:H\to\mathbb{Z}_p^{d}$, and requiring that for some $\{w_k\}_{k\in\mathbb{N}^d} \subset W$ with $\|p^{n|k|}w_k\|\to 0$,

$$h\cdot w = \sum_{k\in\mathbb{N}^d} c(h)^k w_k,\quad \forall h\in H,$$

where the series converges in $W$ (Berger et al., 2014). The union over all such analytic neighborhoods defines the LB-space structure of $W^{\mathrm{la}}$.

This definition extends to modules and sheaves over $p$-adic analytic spaces and period rings, and admits binomial/Mahler expansions in integral and characteristic $p$ settings (Porat, 2024).

2. Analytic Structure: Scalers, Operators, and Functorial Properties

The field of locally analytic scalars, e.g., $\hat{K}_\infty^{\mathrm{la}}$ in the context of infinite $p$-adic Lie extensions $K_\infty/K$, is defined as the union of rigid-analytic fixed fields under open subgroups

$$\hat{K}_\infty^{\mathrm{la}} = \bigcup_{n\geq 1} \{x \in \hat{K}_\infty : g \mapsto g\cdot x\ \text{is analytic on }\Gamma_n\},$$

where $\Gamma_n$ are small uniform subgroups (Berger et al., 2014). This field is a Fréchet–Stein algebra of dimension $d-1$ over $K$ in the $d$-dimensional Galois setting.

On $W^{\mathrm{la}}$, the (multi-variable) Sen operators, defined via Lie algebra derivations,

$$D_v(f) = \left.\frac{d}{dt}(\exp(tv)\cdot f)\right|_{t=0},\quad v\in\mathrm{Lie}\,G,$$

realize the infinitesimal $p$-adic Galois/Lie group action in all directions.

These operators generalize the classical Sen and Tate operators: $$\Theta_{\text{Sen}} = \lim_{\gamma\to 1}\frac{\gamma-1}{\chi_{\text{cycl}}(\gamma)-1}$$ for $\mathrm{dim}\,G=1$ (cyclotomic case) (Berger et al., 2014).

Functorially, the locally analytic vector functor is left-exact and admits a vanishing theorem for its higher derived functors under suitable decompletion or Tate–Sen conditions (Porat, 2024, Porat, 2022).

3. Cohomological Theorems and Resolutions

Locally analytic vectors play a critical role in the cohomology of $p$-adic Lie groups. A central result (Fust, 2023) is:

Comparison Theorem: For an admissible Banach representation $V$ of a $p$-adic reductive group $G$, the inclusion

$V^{\mathrm{la}}\hookrightarrow V$

induces isomorphisms on all continuous group cohomology groups: $H^i(G, V^{\mathrm{la}}) \cong H^i(G, V),$ with the canonical Hausdorff topology and unique finest locally convex structure when $H^i$ is finite-dimensional.

Resolution techniques, such as the analytic variant of the Schneider–Stuhler complex or Chevalley–Eilenberg-type resolutions, provide explicit projective resolutions in categories of locally analytic representations (Agrawal et al., 2024).

4. Multivariate $(\varphi,\Gamma)$-Module Theory and Analytic Descent

The analytic theory of $(\varphi,\Gamma)$-modules, crucial in $p$-adic Hodge theory, generalizes to arbitrary deeply ramified $p$-adic Lie extensions by passing from $K$-finite vectors to locally analytic vectors (Berger et al., 2014, Berger, 2013, Poyeton, 2022). Specifically,

• For $V$ a $G_K$-representation, the space of locally analytic vectors in $(\widetilde{\mathbf{B}}\otimes V)^{H_K}$ provides the correct module of coefficients for generalizing the classical overconvergent $(\varphi,\Gamma)$-module theory.
• Descent along locally analytic vectors gives rise to a full equivalence between suitable $p$-adic representations and étale $(\varphi,\Gamma)$-modules over the corresponding analytic period rings, under which all higher derived functors of the analytic vector functor vanish (Porat, 2024, Porat, 2022).
• In mixed characteristic and integral settings, binomial/Mahler expansions yield a robust theory matching and interpolating characteristic $0$- and $p$-phenomena (Porat, 2024, Poyeton, 2024).

5. Applications in Hodge Theory and $p$-Adic Automorphic Representations

Locally analytic vectors underpin several key phenomena:

• Hodge–Tate weights and Sen theory: The eigenvalues of the Sen operator on the module of locally analytic vectors recover classical Hodge–Tate weights (Berger et al., 2014, Porat, 2022). In higher rank, the full matrix of Sen operators gives the "infinitesimal character".
• $(\varphi,\tau)$-module overconvergence: Analyticity in both cyclotomic and Kummer–type directions proves overconvergence for associated modules (Gao et al., 2018).
• Classification of representations: In the $p$-adic local Langlands correspondence, locally analytic vectors in unitary Banach representations of $\mathrm{GL}_2(\mathbb{Q}_p)$ are explicitly classified in terms of extensions between principal and special series (Liu et al., 2011).
• Automorphic cohomology and geometric models: The structure of locally analytic vectors in completed cohomology and period sheaf cohomology allows for precise geometric realization of $p$-adic local Langlands functors, Jacquet–Langlands correspondences, and classicality theorems for modular and Shimura curves (Li et al., 20 Jan 2026, Qiu et al., 15 May 2025, Pan, 2020, Pan, 2022, Dospinescu et al., 2024).

6. Lubin–Tate and Non-Cyclotomic Extensions

The structure of locally analytic vectors in period rings and $(\varphi,\Gamma)$-modules is sensitive to the nature of the $p$-adic Lie extension. For Lubin–Tate extensions, locally analytic scalars assemble from power series in the logarithms $x_\tau$ of Lubin–Tate characters: $$\hat{K}_\infty^{\mathrm{la}} = \text{closure of } K_\infty[x_\tau \mid \tau\neq\mathrm{Id}]$$ with analytic coordinates attached to embeddings $\tau: F\to\mathbb{C}_p$ (Berger et al., 2014, Berger, 2013).

For general non-cyclotomic, non-Lubin–Tate cases, results and counter-examples show either collapse to the base field (no non-trivial analytic parameters) or the existence of analytic lifts only in the presence of a cyclotomic or Lubin-Tate direction (Poyeton, 2022, Poyeton, 2024).

7. Current Directions and Open Problems

Research continues on:

• The full structure of locally analytic vectors for general, possibly noncommutative, $p$-adic Lie extensions, and their impact on $(\varphi,\Gamma)$-module theory (see (Poyeton, 2024) for discussions of conjectures and counterexamples).
• The role of locally analytic vectors in the moduli of trianguline (i.e., "triangular") representations and the construction of period rings for trianguline periods (Poyeton, 2022).
• Extensions to integral and mixed characteristic settings, with vanishing theorems for higher derived analytic functors, and their applications to fields-of-norms and the geometry of the Fargues–Fontaine curve (Porat, 2022, Porat, 2024).
• Analyticity criteria and decomposition theories in completed cohomology, especially in the context of the $p$-adic Langlands program for higher rank groups (Qiu et al., 15 May 2025, Dospinescu et al., 2024, Li et al., 20 Jan 2026).

Locally analytic vectors and their associated modules thus serve as an indispensable bridge between $p$-adic representation theory, nonarchimedean harmonic analysis, and arithmetic geometry, providing the analytic structures necessary to interpolate and extend core features of classical and $p$-adic Hodge theory into broader and deeper contexts.