Non-Archimedean Geometry and Applications

The workshop focused on recent developments in non-Archimedean analytic geometry with various applications to other fields. The topics of the talks included foundational results on analytic spaces as well as applications to the local Langlands conjecture, birational geometry, p-adic cohomology theories, Shimura varieties and the non-Archimedean Simpson correspondence.

We report on the results and techniques of the author’s recent joint work with Richard Taylor, which analyzes in detail the bad reduction of certain Shimura varieties in order to prove the compatibility of local and global Langlands correspondences, obtaining as a consequence the local Langlands conjecture for GL(n) of a p-adic field. These Shimura varieties have natural models over p-adic integer rings, as moduli spaces for abelian varieties with additional structure. The starting point of the work with Taylor is the stratification of the special fiber of an integral model in minimal level, according to the isogeny type of the universal family of p-divisible groups attached to these abelian varieties Similar stratifications can conjecturally be constructed for any Shimura variety, and indeed are known to exist for most PEL types. We discuss a series of conjectures regarding the behavior of vanishing cycles along these strata, with the aim of extending Kottwitz’ conjectures on the cohomology of Shimura varieties to the case of bad reduction.KeywordsParabolic SubgroupSpecial FiberLevi FactorShimura VarietyDiscrete Series RepresentationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We are concerned with rigid analytic geometry in the general setting of Henselian fields K with separated analytic structure, whose theory was developed by Cluckers–Lipshitz–Robinson. It unifies earlier work and approaches of numerous mathematicians. Separated analytic structures admit reasonable relative quantifier elimination in a suitable analytic language. However, the rings of global analytic functions with two kinds of variables seem not to have good algebraic properties such as Noetherianity or excellence. Therefore, the usual global resolution of singularities from rigid analytic geometry is no longer at our disposal. Our main purpose is to give a definable version of the canonical desingularization algorithm (the hypersurface case) due to Bierstone–Milman so that both of these powerful tools are available in the realm of non-Archimedean analytic geometry at the same time. It will be carried out within a category of definable, strong analytic manifolds and maps, which is more flexible than that of affinoid varieties and maps. Strong analytic objects are those definable ones that remain analytic over all fields elementarily equivalent to K. This condition may be regarded as a kind of symmetry imposed on ordinary analytic objects. The strong analytic category makes it possible to apply a model-theoretic compactness argument in the absence of the ordinary topological compactness. On the other hand, our closedness theorem enables application of resolution of singularities to topological problems involving the topology induced by valuation. Eventually, these three results will be applied to such issues as the existence of definable retractions or extending continuous definable functions. The established results remain valid for strictly convergent analytic structures, whose classical examples are complete, rank one valued fields with the Tate algebras of strictly convergent power series. The earlier techniques and approaches to the purely topological versions of those issues cannot be carried over to the definable settings because, among others, non-Archimedean geometry over non-locally compact fields suffers from lack of definable Skolem functions.

We show that non-Archimedean analytic geometry can be viewed as relative algebraic geometry in the sense of Toën–Vaquié–Vezzosi over the category of non-Archimedean Banach spaces. For any closed symmetric monoidal quasi-abelian category we define a topology on certain subcategories of the category of (relative) affine schemes. In the case that the monoidal category is the category of abelian groups, the topology reduces to the ordinary Zariski topology. By examining this topology in the case that the monoidal category is the category of Banach spaces we recover the G-topology or the topology of admissible subsets on affinoids which is used in rigid or Berkovich analytic geometry. This gives a functor of points approach to non-Archimedean analytic geometry. We demonstrate that the category of Berkovich analytic spaces (and also rigid analytic spaces) embeds fully faithfully into the category of (relative) schemes in our version of relative algebraic geometry. We define a notion of quasi-coherent sheaf on analytic spaces which we use to characterize surjectivity of covers. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.

In this paper we formulate an abstract quotient theorem for convergent maps of PO-spaces (4.3), being inspired by the work [6] of I. F. Donin. This theorem, which is proved in detail in [5], gives a criterion for certain quotient groupoids over the category of germs of analytic spaces to have a semiuniversal deformation. With this result in hands, one can solve many local moduli problems in analytic geometry in a unified manner, see [3, 4] and [5]. Let us finally remark that we use the notions and notations introduced in [2] without any further comment.

Cohomology of Drinfeld Modular Varieties provides an introduction, in two volumes, both to this subject and to the Langlands correspondence for function fields. These varieties are the analogues for function fields of the Shimura varieties over number fields. The Langlands correspondence is a conjectured link between automorphic forms and Galois representations over a global field. By analogy with the number-theoretic case, one expects to establish the conjecture for function fields by studying the cohomology of Drinfeld modular varieties, which has been done by Drinfeld himself for the rank two case. This second volume is concerned with the Arthur-Selberg trace formula, and with the proof in some cases of the Rmamanujan-Petersson conjecture and the global Langlands conjecture for function fields. It is based on graduate courses taught by the author, who uses techniques which are extensions of those used to study Shimura varieties. Though the author considers only the simpler case of function rather than number fields, many important features of the number field case can be illustrated. Several appendices on background material keep the work reasonably self-contained. It is the first book on this subject and will be of much interest to all researchers in algebraic number theory and representation theory.

Direct immersogeometric fluid flow and heat transfer analysis of objects represented by point clouds

It has been known for many years that the stabilization of the Arthur-Selberg trace formula would, or perhaps we should write will, have important consequences for the Langlands functoriality program as well as for the study of the Galois representations on the l-adic cohomology of Shimura varieties. At present, full stabilization is still only known for SL(2) and U(3) and their inner forms [LL,R]. The automorphic and arithmetic consequences of stabilization for U(3) form the subject of the influential volume [LR]. Under somewhat restrictive hypotheses, one can sometimes derive the expected corollaries of the stable trace formula. Examples of such pseudo-stabilization include Kottwitz' analysis in [K2] of the zeta functions of certain Shimura varieties attached to twisted forms of unitary groups over totally real fields, and the proof in [L1] of stable cyclic base change of automorphic representations which are locally Steinberg at at least two places. These conditional results have been used successfully to provide non-trivial examples of compatible systems of l-adic representations attached to certain classes of automorphic representations of GL(n) [C3], and of non-trivial classes of cohomology of S-arithmetic groups [BLS, L1]. Conditional results also suffice for important local applications, such as the local Langlands conjecture for GL(n) [HT, He]. The present article develops a technique for obtaining conditional base change and functorial transfer. Let Un be a unitary group over a number field F attached to a quadratic extension E/F. The technique applies to quadratic base change from Un to GL(n)E, and to transfer between inner forms of unitary groups. Roughly speaking, if p is an automorphic representation of U which is locally supercuspidal at two places of F split in E, then the expected consequences of the stable trace formula hold for p; in particular p admits a base change to a cuspidal automorphic representation of GL(n)E (Theorem 2.2.2). Slightly more general results are available when F is totally real and E is totally imaginary, and when p is of cohomological type. Automorphic descent from GL(n)E to Un can be proved under analogous hypotheses (Theorem 2.4.1, Theorem 3.1.2). Finally, we prove transfer between distinct inner forms of unitary groups (Jacquet-Langlands transfer) under quite general local hypotheses (Theorem 2.1.2 and, in a more precise form, Theorem 3.1.6 and Proposition 3.1.7). As in [L1], all results are obtained from the simple version of the Arthur-Selberg trace formula, in which non-elliptic and non-cuspidal terms are absent.

Une suite spectrale de Hochschild–Serre pour l'uniformisation de Rapoport–Zink

The coordination geometry of silver complexes of the green fluorescence protein chromophore is highly diverse and dependent on the size and shape of the counteranion in the complexes. Only complexes that have vacancies at the silver atom prove to be heterogeneous catalysts in the “green” synthesis of 1,4‐benzodioxines. The results reveal an interesting strategy of exploiting crystal packing forces to enhance catalytic efficiency by just changing the counteranion.

Achieving control over coordination geometries in lanthanide complexes remains a challenge to the coordination chemist. This is particularly the case in the field of molecule-based magnetism, where barriers for magnetic relaxation processes as well as tunneling pathways are strongly influenced by the lanthanide coordination geometry. Addressing the challenge of design of 4f-element coordination environments, the ubiquitous Ln(hfac)3 moieties have been shown to be applicable as Lewis acids coordinating transition metal acetylacetonates facially leading to simple, chiral lanthanide–transition metal heterodinuclear complexes. The broad scope of this approach is illustrated by the synthesis of a range of such complexes LnM: LnM(hfac)3(μ2-acac-O,O,O′)3 (Ln = La, Pr, Gd; M = Cr, Fe, Ga), with approximate three-fold symmetry. The complexes have been crystallographically characterized and exhibit polymorphism for some combinations of 4f and 3d metal centers. However, an isostructural set of systems spanning several lanthanides which exhibit spontaneous resolution in the orthorhombic Sohncke space group P212121 is presented here. The electronic structure and ensuing magnetic properties have been studied by EPR spectroscopy and magnetometry. The GdFe, PrFe, and PrCr complexes exhibit ferromagnetic coupling, while GdCr exhibits antiferromagnetic coupling. GdGa exhibits slow relaxation of the magnetization in applied static fields.

Novel macrocyclic bis(disulfide)tetramine ligands and several Cu(II) and Ni(II) complexes of them with additional ligands have been synthesized by the oxidative coupling of linear tetradentate N2S2 tetramines with iodine. Facile demetalation of the Ni(II) oxidation products affords the free 20-membered macrocycles meso-9 and rac-9 and the 22-membered macrocycle 16, all of which are potentially octadentate N4S4 ligands. X-ray structure analyses reveal distinctly different conformations for the two isomers of 9; meso-9 shows a stepped conformation in profile with the disulfide groups corresponding to the rise of the step, whereas rac-9 exhibits a V conformation with the disulfide groups near the vertex of the V. No metal complexes of rac-9 have been isolated. Crystallographic studies of three Cu(II) complexes reveal that depending upon the size of the macrocyclic ligand and the nature of the additional ligands (I-, NCO-, and CH3CN), the Cu(II) coordination geometry shows considerable variation (plasticity), with substantial changes in the Cu(II)-disulfide bonding. Thus, a diiodide salt contains six-coordinate Cu(II) to which all four bridging disulfide sulfur atoms form strong equatorial bonds. In contrast, isocyanato complexes of the 20- and 22-membered macrocycles exhibit trigonal-bipyramidal Cu(II) and distorted cis-octahedral Cu(II) geometries, respectively, having only one and no short equatorially bound sulfur atoms. The coordination geometry of the latter complex can also be described as four-coordinate seesaw with two semicoordinated S(disulfide) ligands. Disulfide-->Cu(II) ligand-to-metal charge transfer absorptions of both isocyanato-containing Cu(II) species appear too weak to observe, probably because of poor overlap of the sulfur orbitals with the Cu(II) d-vacancy. The dual disulfide-bridged Ni(II) units of the crystallographically characterized octahedral Ni(II) complex of meso-9 with axial iodide and acetonitrile ligands promote substantial antiferromagnetic coupling (J = -13.0(2) cm-1).

The title complexes, [M(C5O5)(C12H8N2)2], with M = Co(II), Ni(II) and Cu(II), all lie across twofold rotation axes, around which two 1,10-phenanthroline ligands are arranged in a chiral propeller manner. The Co(II) and Ni(II) complexes are isostructural, with octahedral coordination geometry, while the local geometry of the Cu(II) complex is severely distorted from octahedral.

In this work we develop a formalism of vanishing cycles for non-Archimedean analytic spaces which is an analog of that for complex analytic spaces from [SGA7], Exp. XIV. As an application we prove that in the equicharacteristic case the stalks of the vanishing cycles sheaves of a scheme X at a closed point x E X, depend only on the formal completion Spf(Ox,,) of X at x. In particular, any continuous homomorphism OX,x -+ y,y induces a homorphism from the stalks of the vanishing cycles sheaves of X at x to those of Y at y. Furthermore, we prove that, given Ox,x and O y, , there exists n > 1 such that, for any pair of continuous homomorphisms OX,x -Oy,y that coincide modulo the n-th power of the maximal ideal of Oy,y, the induced homomorphisms between the stalks of the vanishing cycles sheaves coincide. These facts generalize a result of G. Laumon from [Lau] (see Remark 7.6). Throughout the paper we fix a non-Archimedean field k (whose valuation is not assumed to be nontrivial). In ?1 we study etale Galois sheaves on k-analytic spaces. To define the vanishing cycles functor and to work with it, we use the language of pro-analytic spaces, i.e., pro-objects of the category of analytic spaces ([SGA4], Exp. I). Examples of such objects are the germs of analytic spaces as in [Ber2], ?3.4. Another example is considered in ?3. In ?4 we define the vanishing cycles functor and establish its basic properties. In ?5 we show that the vanishing cycles sheaves are trivial for smooth morphisms. In ?6 we prove a comparison theorem for vanishing cycles. This theorem is more general than its analog over C from [SGA7], Exp. XIV, and its proof does not use Hironaka's theorem on resolution of singularities. In ?7 we apply the comparison theorem to prove the properties of the vanishing cycles sheaves of schemes formulated above. It is worthwhile to note that this application is obtained by considering non-Archimedean analytic geometry over fields with trivial valuation. Like [Ber3], this work arose from a suggestion of P. Deligne to apply the etale cohomology theory from [Ber2] to the study of the vanishing cycles sheaves of schemes. I am very grateful to him for useful discussions on the subject. I also

In this chapter we start some basic facts on analytic and complex geometry (divisors, blowing-up, big line bundles), we prove the theorem of Siegel-Remmert-Thimm, that the field of meromorphic functions on a connected compact complex manifold is an algebraic field of transcendence degree less than the dimension of the manifold. Then we study in more detail Moishezon manifolds and their relation to projective manifolds. In particular we prove that a Moishezon manifold is projective if and only if it carries a Kähler metric. We end the section 2.2 by giving the solution of the Grauert-Riemenschneider conjecture as application of the holomorphic Morse inequalities from Theorem 1.7.1.KeywordsLine BundleComplex ManifoldAhler ManifoldHolomorphic Line BundleCompact Complex ManifoldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.