Inscribed Banach–Colmez Spaces

• Inscribed Banach–Colmez spaces are refined p-adic analytic objects that integrate classical Banach–Colmez structures with enriched tangent bundles and differential frameworks.
• They embed within modern geometric approaches—using v-sheaves, diamonds, and stacks—to enable advanced slope filtration and analytical decomposition.
• Their applications span period maps, moduli of local shtukas, and cohomological smoothness in p-adic Hodge theory, yielding new insights in arithmetic geometry.

Inscribed Banach–Colmez spaces are a refined class of analytic objects at the intersection of p-adic functional analysis, algebraic geometry, and arithmetic geometry. They arise from embedding classical Banach–Colmez spaces (p-adic Banach spaces equipped with a $\mathbb{C}_p$-algebra of analytic functions) within larger categorical or geometric frameworks such as $v$-sheaves, diamonds, and stacks, and are distinguished by their enriched tangent bundles and differential structures. These structures are central to new formulations of period maps, moduli spaces of local shtukas, and differential-topological invariants in p-adic geometry and representation theory.

1. Foundations: Banach–Colmez Spaces and Analytic Enrichment

A Banach–Colmez space, as first formalized by Colmez and Fontaine, is a p-adic Banach space $E$ equipped with the additional data of a $\mathbb{C}_p$-algebra of analytic functions. The defining property is that $E$ admits a presentation

$$0 \longrightarrow V \longrightarrow E \longrightarrow L \longrightarrow 0$$

where $V$ is a finite-dimensional $\mathbb{Q}_p$-vector space ("height") and $L$ is a finite-dimensional $\mathbb{C}_p$-vector space ("dimension") (Plût, 2016). The invariants

$$\dim\,E = \dim_{\mathbb{C}_p} L, \qquad \mathrm{ht}\,E = \dim_{\mathbb{Q}_p} V$$

are intrinsic and additive on short exact sequences, and these objects naturally encode extensions appearing in p-adic Hodge theory.

The core analytic structure is provided by the algebra of analytic functions, often realized via convergent power series, which enables a Gelfand transform and the definition of spectral norms. In the inscribed context, Banach–Colmez spaces are embedded as internal tangent bundles or fibers in moduli-theoretic or stack-theoretic frameworks, yielding a "differential" enhancement of their classical analytic geometry.

2. The Inscribed Framework: $v$-Sheaves, Diamonds, and Differential Enrichment

The inscribed variant, as developed in recent work (Howe, 15 Aug 2025), generalizes the theory of diamonds and $v$-sheaves by considering stacks with internal tangent bundles modeled on relative Banach–Colmez spaces. For a given smooth rigid space $Z$ over a $p$-adic field, the tangent bundle $T_Z$ is inscribed via an exact tensor functor from $p$-adic representations to twistor bundles on an appropriately thickened Fargues–Fontaine curve.

In this setting, the fiber functor

$\omega: \mathrm{Rep}\,G \to \mathrm{Vect}(Z)$

need not preserve a chosen filtration, but locally (in the étale topology) one may select a filtration-preserving connection and compute the infinitesimal variation using the difference

$$\nabla - \nabla': V \longrightarrow V \otimes \Omega_Z,$$

yielding the Kodaira–Spencer map

$$\kappa_{\omega_\nabla^{Fil}}: T_Z \longrightarrow \omega(\mathfrak g) / Fil^0 \omega(\mathfrak g)$$

with $\mathfrak g = \mathrm{Lie}\,G$, capturing geometric infinitesimal deformations in the inscribed context. This formalism is functorial and tensor-compatible; the resulting inscribed structures are only visible after differentiating within the $v$-sheaf or diamond formalism.

3. Structural Characterization and Slope Filtration

An essential feature of the Banach–Colmez category is the presence of a canonical Harder–Narasimhan (HN) slope filtration on oblique spaces (those which interpolate between algebraic and analytic). For an oblique inscribed Banach–Colmez space $X$, there exists a unique decreasing filtration by positive rationals: $$\cdots \supset \mathrm{Fil}^{\alpha} X \supset \mathrm{Fil}^{\beta} X \supset \cdots \quad (\alpha, \beta \in \mathbb{Q}^+)$$ with graded pieces

$$\mathrm{Gr}^{\alpha} X = \mathrm{Fil}^{\alpha}X\,/\,\bigcup_{\beta > \alpha}\mathrm{Fil}^{\beta}X \simeq E_{d,h}^{n_\alpha(X)},$$

where $\alpha = d/h$ and $E_{d,h} = \{x \in B^+ : \varphi^h(x) = p^d x\}$ (Plût, 2016). The slope filtration decomposes inscribed Banach–Colmez spaces into semistable analytic strata, directly generalizing Kedlaya’s slope filtration for differential modules over the Robba ring, and facilitates fine geometric analysis in moduli problems.

4. Cohomological Smoothness and Moduli of Local Shtukas

The inscribed tangent bundles play a foundational role in the paper of moduli spaces of mixed characteristic local shtukas with one leg. For the moduli space $M_{b,[\mu]}^\tau \to \mathrm{Spd}(\mathbb{C}_p)$ with fixed determinant, the inscribed tangent space at a point $z$ is

$$T_{M_{b,[\mu]}^\tau} = BC(\mathcal{A}_{\mathrm{max}}^\circ),$$

where $BC(-)$ denotes the Banach–Colmez space of global sections of a vector bundle constructed via isocrystal data on the Fargues–Fontaine curve (Howe, 15 Aug 2025). A crucial geometric criterion—absence of the zero Harder–Narasimhan slope in $\mathcal{A}_{\mathrm{max}}^\circ$—characterizes the locus where the Banach–Colmez tangent spaces are connected (i.e., lack extra infinitesimal automorphisms).

The Fargues–Scholze Jacobian criterion applies: if the relevant tangent bundle admits only strictly positive slopes, then the structure morphism

$$M_{b,[\mu]}^{\tau,\mathrm{non}\text{-}\mathrm{vsp}} \to \mathrm{Spd}(\mathbb{C}_p)$$

is cohomologically smooth. This result generalizes Ivanov–Weinstein’s theorem on basic EL Rapoport–Zink spaces to the infinite level, non-basic case. It provides a conceptual framework for the finiteness properties of $\ell$-adic cohomology in these moduli spaces, linking geometric, analytic, and arithmetic structures.

5. Enriched Purity, Presentability, and Inscription

A categorical perspective on inscribed Banach–Colmez spaces emerges from the theory of enriched purity and presentability in Banach spaces (Rosický, 2022). When the category $Ban$ of Banach spaces is enriched over complete metric spaces, finite-dimensional Banach spaces are finitely presentable for the enriched hom-functor $Ban(A,-): Ban \to CMet$, allowing for $\epsilon$-approximate factorization in directed colimits. Pure morphisms—those admitting approximate lifting over finite-dimensional objects—are proved to be precisely ideals in the Banach space sense.

In this context, inscribed Banach–Colmez spaces may be interpreted as those admitting ideal (pure) embeddings within larger analytic structures. Moreover, classes of Banach spaces approximately injective with respect to morphisms with finite-dimensional domains and separable codomains (such as Lindenstrauss and Gurarii spaces) fall within this framework, pointing to model-theoretic characterizations of inscribed analytic objects in functional analysis.

6. Interconnections: Period Maps, Fourier Transform, and Arithmetic Geometry

Inscribed Banach–Colmez spaces provide the appropriate environment for constructing natural refinements of Hodge and Hodge–Tate period maps, as well as their derivatives and lattices over infinite level global and local Shimura varieties (Howe, 15 Aug 2025). The refined Liu–Zhu period map, whose derivative yields the geometric Sen morphism or canonical Higgs field, realizes the differential structure which is only apparent in the inscribed context.

Further, the $\ell$-adic Fourier transform on Banach–Colmez spaces (Anschütz et al., 2021) builds analytic bridges between representation theory and arithmetic geometry by generalizing classical Fourier transforms from finite-dimensional vector spaces to "inscribed" Picard stacks in $E$-vector spaces, using derived categorical methods and stability under duality and compactness.

7. Applications and Further Directions

The theory of inscribed Banach–Colmez spaces is central to the modern paper of $p$-adic period domains, moduli of local shtukas, higher-dimensional $v$-sheaves with internal tangent bundles, and the differential-topological analysis of diamonds. Applications range from new proofs of the weakly admissible implies admissible theorem and structure theory for p-adic representations to the geometric Langlands program in the $p$-adic setting.

A plausible implication is that further development of inscription techniques in analytic geometric frameworks will advance the classification of period morphisms, provide finer invariants in $p$-adic Hodge theory, and yield new tools for derived and stack-theoretic approaches to arithmetic moduli problems. The interplay between enrichment, purity, and analytic representation spaces suggests additional connections to non-Archimedean geometry, functional analysis, and derived algebraic geometry.