On the uniformization of L-valued frames

This note discusses the appropriate way of uniformizing the notion of an L-valued frame introduced by A. Pultr and S. Rodabaugh in [Lattice-valued frames, functor categories, and classes of sober spaces, Chapter 6 of Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Kluwer, 2003]. It covers the case of a completely distributive lattice L (which is, in a certain sense, the most general one) and studies the corresponding category of uniform L-valued frames

The densest lattices in PGL3(Q2)

We find the smallest possible covolume for lattices in PGL3(Q2), show that there are exactly two lattices with this covolume, and describe them explicitly. They are commensurable, and one of them appeared in Mumford's construction of his fake projective plane. We also discuss a new 2-adic uniformization of another fake projective plane.Comment: Minor error correcte

Liouville and Toda field theories on Riemann surfaces

We study the Liouville theory on a Riemann surface of genus g by means of their associated Drinfeld--Sokolov linear systems. We discuss the cohomological properties of the monodromies of these systems. We identify the space of solutions of the equations of motion which are single--valued and local and explicitly represent them in terms of Krichever--Novikov oscillators. Then we discuss the operator structure of the quantum theory, in particular we determine the quantum exchange algebras and find the quantum conditions for univalence and locality. We show that we can extend the above discussion to $sl_n$ Toda theories.Comment: 41 pages, LaTeX, SISSA-ISAS 27/93/E

On the p-adic uniformization of unitary Shimura curves

We prove $p$-adic uniformization for Shimura curves attached to the group of unitary similitudes of certain binary skew hermitian spaces $V$ with respect to an arbitrary CM field $K$ with maximal totally real subfield $F$. For a place $v|p$ of $F$ that is not split in $K$ and for which $V_v$ is anisotropic, let $\nu$ be an extension of $v$ to the reflex field $E$. We define an integral model of the corresponding Shimura curve over ${\rm Spec}\, O_{E, (\nu)}$ by means of a moduli problem for abelian schemes with suitable polarization and level structure prime to $p$. The formulation of the moduli problem involves a Kottwitz condition, an Eisenstein condition, and an adjusted invariant. The first two conditions are conditions on the Lie algebra of the abelian varieties; the last condition is a condition on the Riemann form of the polarization. The uniformization of the formal completion of this model along its special fiber is given in terms of the formal Drinfeld upper half plane for $F_v$. The proof relies on the construction of the contracting functor which relates a relative Rapoport-Zink space for strict formal $O_{F_v}$-modules with a Rapoport-Zink space of $p$-divisible groups which arise from the moduli problem, where the $O_{F_v}$-action is usually not strict when $F_v\ne \mathbb {Q}_p$. Our main tool is the theory of displays, in particular the Ahsendorf functor.Comment: 145 page

The basic locus of the unitary Shimura variety with parahoric level structure, and special cycles

In this paper, we study the basic locus in the fiber at $p$ of a certain unitary Shimura variety with a certain parahoric level structure. The basic locus $\widehat{\mathcal{M}^{ss}}$ is uniformized by a formal scheme $\mathcal{N}$ which is called Rapoport-Zink space. We show that the irreducible components of the induced reduced subscheme $\mathcal{N}_{red}$ of $\mathcal{N}$ are Deligne-Lusztig varieties and their intersection behavior is controlled by a certain Bruhat-Tits building. Also, we define special cycles in $\mathcal{N}$ and study their intersection multiplicities.Comment: 54 page

From non-commutative diagrams to anti-elementary classes

Anti-elementarity is a strong way of ensuring that a class of structures , in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the form L $\infty$$\lambda$. We prove that many naturally defined classes are anti-elementary, including the following: $\bullet$ the class of all lattices of finitely generated convex {\ell}-subgroups of members of any class of {\ell}-groups containing all Archimedean {\ell}-groups; $\bullet$ the class of all semilattices of finitely generated {\ell}-ideals of members of any nontrivial quasivariety of {\ell}-groups; $\bullet$ the class of all Stone duals of spectra of MV-algebras-this yields a negative solution for the MV-spectrum Problem; $\bullet$ the class of all semilattices of finitely generated two-sided ideals of rings; $\bullet$ the class of all semilattices of finitely generated submodules of modules; $\bullet$ the class of all monoids encoding the nonstable $K_0$-theory of von Neumann regular rings, respectively C*-algebras of real rank zero; $\bullet$ (assuming arbitrarily large Erd"os cardinals) the class of all coordinatizable sectionally complemented modular lattices with a large 4-frame. The main underlying principle is that under quite general conditions, for a functor $\Phi$ : A $\rightarrow$ B, if there exists a non-commutative diagram D of A, indexed by a common sort of poset called an almost join-semilattice, such that $\bullet$ $\Phi$ D^I is a commutative diagram for every set I, $\bullet$ $\Phi$ D is not isomorphic to $\Phi$ X for any commutative diagram X in A, then the range of $\Phi$ is anti-elementary.Comment: 49 pages. Journal of Mathematical Logic, World Scientific Publishing, In pres

T-duality Twists and Asymmetric Orbifolds

We study some aspects of asymmetric orbifolds of tori, with the orbifold group being some $\mathbb{Z}_N$ subgroup of the T-duality group and, in particular, provide a concrete understanding of certain phase factors that may accompany the T-duality operation on the stringy Hilbert space in toroidal compactification. We discuss how these T-duality twist phase factors are related to the symmetry and locality properties of the closed string vertex operator algebra, and clarify the role that they enact in the modular covariance of the orbifold theory, mainly using asymmetric orbifolds of tori which are root lattices as working examples.Comment: 67 pages. v2: references added and typos correcte