Rigid Analytic Spaces Research Articles

⌢𝒟-modules on rigid analytic spaces I

Abstract We introduce a sheaf of infinite order differential operators 𝒟 ⌢ {\overset{\frown}{\mathcal{D}}} on smooth rigid analytic spaces that is a rigid analytic quantisation of the cotangent bundle. We show that the sections of this sheaf over sufficiently small affinoid varieties are Fréchet–Stein algebras, and use this to define co-admissible sheaves of 𝒟 ⌢ {\overset{\frown}{\mathcal{D}}} -modules. We prove analogues of Cartan’s Theorems A and B for co-admissible 𝒟 ⌢ {\overset{\frown}{\mathcal{D}}} -modules.

T-modules and Pila–Wilkie estimates

We present in this work an upper bound estimate of the number of torsion points contained into the transcendent part of an algebraic subvariety of an abelian and uniformizable T-module in function of their degree, taking an analogous shape as in [PW]. We give in particular a version of the Implicit Function Theorem for analytic sets provided by a non-archimedean topology, we present our definition of analytic space and we introduce the notion of dimension on it, showing that it is coherent with the classic notion of dimension on rigid analytic spaces. We then use such results to prove a theorem of density of regular points (which we define in this new context). We finally give some results which allow to apply our main theorem to our strategy to prove Manin–Mumford conjecture in this setting.

On the reconstruction problem in mirror symmetry

Let π:M→B be a Lagrangian torus fibration with singularities such that the fibers are of Maslov index zero, and unobstructed. The paper constructs a rigid analytic space M0∨ over the Novikov field which is a deformation of the semi-flat complex structure of the dual torus fibration over the smooth locus B0⊂B of π. Transition functions of M0∨ are obtained via A∞ homomorphisms which capture the wall-crossing phenomenon of moduli spaces of holomorphic disks.

On a conjecture of Rapoport and Zink

In their book, Rapoport and Zink constructed rigid analytic period spaces \({\mathcal {F}}^{wa}\) for Fontaine’s filtered isocrystals, and period morphisms from PEL moduli spaces of p-divisible groups to some of these period spaces. They conjectured the existence of an étale bijective morphism \({\mathcal {F}}^{a}\to {\mathcal {F}}^{wa}\) of rigid analytic spaces and of a universal local system of ℚ p -vector spaces on \({\mathcal {F}}^{a}\). Such a local system would give rise to a tower of étale covering spaces \(\breve {{\mathcal {E}}}_{{\widetilde {K}}}\) of \({\mathcal {F}}^{a}\), equipped with a Hecke-action, and an action of the automorphism group J(ℚ p ) of the isocrystal with extra structure.For Hodge-Tate weights n−1 and n we construct in this article an intrinsic Berkovich open subspace \({\mathcal {F}}^{0}\) of \({\mathcal {F}}^{wa}\) and the universal local system on \({\mathcal {F}}^{0}\). We show that only in exceptional cases \({\mathcal {F}}^{0}\) equals all of \({\mathcal {F}}^{wa}\) and when the Shimura group is \(\operatorname {GL}_{n}\) we determine all these cases. We conjecture that the rigid-analytic space associated with \({\mathcal {F}}^{0}\) is the maximal possible \({\mathcal {F}}^{a}\), and that \({\mathcal {F}}^{0}\) is connected. We give evidence for these conjectures. For those period spaces possessing PEL period morphisms, we show that \({\mathcal {F}}^{0}\) equals the image of the period morphism. Then our local system is the rational Tate module of the universal p-divisible group and carries a J(ℚ p )-linearization. We construct the tower \(\breve {{\mathcal {E}}}_{{\widetilde {K}}}\) of étale covering spaces, and we show that it is canonically isomorphic in a Hecke and J(ℚ p )-equivariant way to the tower constructed by Rapoport and Zink using the universal p-divisible group.

P-adic modular forms of non-integral weight over Shimura curves

AbstractIn this work, we set up a theory of p-adic modular forms over Shimura curves over totally real fields which allows us to consider also non-integral weights. In particular, we define an analogue of the sheaves of kth invariant differentials over the Shimura curves we are interested in, for any p-adic character. In this way, we are able to introduce the notion of overconvergent modular form of any p-adic weight. Moreover, our sheaves can be put in p-adic families over a suitable rigid analytic space, that parametrizes the weights. Finally, we define Hecke operators, including the U operator, that acts compactly on the space of overconvergent modular forms. We also construct the eigencurve.

On arithmetic families of filtered -modules and crystalline representations

AbstractWe consider stacks of filtered$\varphi $-modules over rigid analytic spaces and adic spaces. We show that these modules parameterize$p$-adic Galois representations of the absolute Galois group of a$p$-adic field with varying coefficients over an open substack containing all classical points. Further, we study a period morphism (defined by Pappas and Rapoport) from a stack parameterizing integral data, and determine the image of this morphism.

Tropical analytic geometry, Newton polygons, and tropical intersections

In this paper we use the connections between tropical algebraic geometry and rigid-analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in several variables: if f1,…,fn are n convergent power series in n variables with coefficients in a non-Archimedean field K, we give a formula for the valuations and multiplicities of the common zeros of f1,…,fn. We use rigid-analytic methods to show that stable complete intersections of tropical hypersurfaces compute algebraic multiplicities even when the intersection is not tropically proper. These results are naturally formulated and proved using the theory of tropicalizations of rigid-analytic spaces, as introduced by Einsiedler, Kapranov, and Lind (2006) [14] and Gubler (2007) [20]. We have written this paper to be as readable as possible both to tropical and arithmetic geometers.

Period spaces for Hodge structures in equal characteristic

We develop the analog in equal positive characteristic of Fontaine’s theory for crystalline Galois representations of a p-adic eld. In particular we describe the analog of Fontaine’s functor which assigns to a crystalline Galois representation an isocrystal with a Hodge ltration. In equal characteristic the role of isocrystals and Hodge ltrations is played by z-isocrystals and Hodge-Pink structures. The latter were invented by Pink. Our rst main result in this article is the analog of the Colmez-Fontaine Theorem that \weakly implies admissible. Next we construct period spaces for Hodge-Pink structures on a xed z-isocrystal. These period spaces are analogs of the Rapoport-Zink period spaces for Fontaine’s ltered isocrystals in mixed characteristic and likewise are rigid analytic spaces. For our period spaces we prove the analog of a conjecture of Rapoport and Zink stating the existence of a \universal local system on a Berkovich open subspace of the period space. As a consequence of \weakly implies admissible this Berkovich open subspace contains every classical rigid analytic point of the period space. As the principal tool to demonstrate these results we use the analog of Kedlaya’s Slope Filtration Theorem which we also formulate and prove here.

Cut-by-curves criterion for the overconvergence of p-adic differential equations

In this paper, we prove a ‘cut-by-curves criterion’ for the overconvergence of integrable connections on certain smooth rigid analytic spaces and certain smooth varieties over p-adic fields.

On period spaces for p-divisible groups

In their book Rapoport and Zink constructed rigid analytic period spaces for Fontaine's filtered isocrystals, and period morphisms from moduli spaces of p-divisible groups to some of these period spaces. We determine the image of these period morphisms, thereby contributing to a question of Grothendieck. We give examples showing that only in rare cases the image is all of the Rapoport–Zink period space. To cite this article: U. Hartl, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Deformation theory of rigid-analytic spaces

In this paper, we study a deformation theory of rigid analytic spaces. We develop a theory of cotangent complexes for rigid geometry which fits in with our deformations. We then use the complexes to give a cohomological description of infinitesimal deformation of rigid analytic spaces. Moreover, we will prove an existence of a formal versal family for a proper rigid analytic space.

Uniform properties of rigid subanalytic sets

In the context of rigid analytic spaces over a non-Archimedean valued field, a rigid subanalytic set is a Boolean combination of images of rigid analytic maps. We give an analytic quantifier elimination theorem for (complete) algebraically closed valued fields that is independent of the field; in particular, the analytic quantifier elimination is independent of the valued field’s characteristic, residue field and value group, in close analogy to the algebraic case. This provides uniformity results about rigid subanalytic sets. We obtain uniform versions of smooth stratification for subanalytic sets and the Łojasiewicz inequalities, as well as a unfiorm description of the closure of a rigid semianalytic set.

The p-adic monodromy-weight conjecture for p-adically uniformized varieties

A p-adically uniformized variety is a smooth projective variety whose associated rigid analytic space admits a uniformization by Drinfeld's p-adic symmetric domain. For such a variety we prove the monodromy-weight conjecture, which asserts that two independently defined filtrations on the log-crystalline cohomology of the special fiber in fact coincide. The proof proceeds by reducing the conjecture to a combinatorial statement about harmonic cochains on the Bruhat–Tits building, which was verified in our previous work.

Rigid analytic Picard theorems

We prove a geometric logarithmic derivative lemma for rigid analytic mappings to algebraic varieties in characteristic zero. We use the lemma to give a new and simpler proof (at least in characteristic zero) of Berkovich's little Picard theorem, which says there are no nonconstant rigid analytic maps from the affine line to nonsingular projective curves of positive genus, and of Cherry's result that there are no nonconstant rigid analytic maps from the affine line to Abelian varieties. Furthermore, we use the lemma to prove new theorems of little and big Picard type for dominant mappings, in close analogy with Griffiths and King. For the little Picard type theorem, we prove that if X is a smooth projective variety with a simple normal crossings divisor D such that ( X, D ) has nonnegative logarithmic Kodaira dimension, then there are no dominant rigid analytic maps f from A m to X \ D . For the big Picard type theorem, we prove that if Y is a nonsingular rigid analytic space, E is an effective simple normal crossings divisor on Y , and if X is a smooth projective variety with a simple normal crossings divisor D such that ( X, D ) is of log-general type, then any dominant rigid analytic map f : Y \ E →. X \ D extends to an analytic map from Y to X .

C2 building and projective space

AbstractWe study the stability map from the rigid analytic space of semistable points in P3 to convex sets in the building of Sp2 over a local field and construct a pure affinoid covering of the space of stable points.

Semi-stable models for rigid-analytic spaces

Let R be a complete discrete valuation ring with field of fractions K and let X K be a smooth, quasi-compact rigid-analytic space over Sp K. We show that there exists a finite separable field extension K' of K, a rigid-analytic space X' K' over Sp K' having a strictly semi-stable formal model over the ring of integers of K', and an etale, surjective morphism f : X' K' →X K of rigid-analytic spaces over Sp K. This is different from the alteration result of A.J. de Jong [dJ] who does not obtain that f is etale. To achieve this property we have to work locally on X K , i.e. our f is not proper and hence not an alteration.

The structure of analytic τ-sheaves

Let R be a complete discrete valuation F q -algebra whose residue field is algebraic over F q , and let K denote its fraction field. In this paper, we study the structure of τ-sheaves M without good reduction on the curve C ̃ K≔C× Spec K , seen as a rigid analytic space. One motivation is the Tate uniformization theorem for t-motives of Drinfeld modules, which we want to extend to general τ-sheaves. On the other hand, we are interested in the action of inertia on a generic Tate module T ℓ( M) of M. For a given τ-sheaf M on C ̃ K , we prove the existence of a maximal model M max for M on C ̃ R , an R-model of C ̃ K , and, over a finite separable extension R′ of R, of nondegenerate models M for M. We prove the following ‘semistability’ theorem: there exists a finite extension K′ of K, a nonempty open subscheme C′⊂ C, and a filtration 0⊂ M ̃ 1⊂⋯⊂ M ̃ | C ̃ K′′ of sub- τ-sheaves of M ̃ | C ̃ ′ K′ on C K′ , such that its subquotients are τ-sheaves with a good model. As a consequence, the action of the inertia group I K on the Tate modules T ℓ( M ̃ ) associated to M ̃ is potentially unipotent for almost all closed points ℓ of C.

P-Adic Automorphic Functions for the Unitary Group in Three Variables

Let G be a special unitary group in three variables over a local field and, a co-compact subgroup of G. We construct, -automorphic functions on the rigid analytic space of points in the projective plane stable under the action of tori in G.

Rigid analytic spaces with overconvergent structure sheaf

We introduce a category of 'rigid spaces with overconvergent structure sheaf' which we call dagger spaces --- this is the correct category in which de Rham cohomology in rigid analysis should be studied. We compare it with the (usual) category of rigid spaces, give Serre and Poincare duality theorems and explain the relation with Berthelot's rigid cohomology.

Analytic functions on Zariski open sets, and local cohomology

Abstract We give a new proof of a theorem of Siu and Trautmann on the vanishing of cohomology with support on complex or rigid analytic spaces. We indicate how our methods might be used to prove various comparison theorems in algebraic and analytic geometry.