Mostrando recursos 1 - 13 de 13

  1. Relative trace formulas for unitary hyperbolic spaces

    Tsuzuki, Masao
    We develop relative trace formulas of unitary hyperbolic spaces for split rank $1$ unitary groups over totally real number fields.

  2. Classifying spaces of degenerating mixed Hodge structures, IV: The fundamental diagram

    Kato, Kazuya; Nakayama, Chikara; Usui, Sampei
    We complete the construction of the fundamental diagram of various partial compactifications of the moduli spaces of mixed Hodge structures with polarized graded quotients. The diagram includes the space of nilpotent orbits, the space of $\mathrm{SL}(2)$ -orbits, and the space of Borel–Serre orbits. We give amplifications of this fundamental diagram and amplify the relations of these spaces. We describe how this work is useful in understanding asymptotic behaviors of Beilinson regulators and of local height pairings in degeneration. We discuss mild degenerations in which regulators converge.

  3. Canonical Kähler metrics and arithmetics: Generalizing Faltings heights

    Odaka, Yuji
    We extend the Faltings modular heights of Abelian varieties to general arithmetic varieties, show direct relations with the Kähler–Einstein geometry, the minimal model program, and Bost–Zhang’s heights and give some applications. Along the way, we propose the “arithmetic Yau–Tian–Donaldson conjecture” (the equivalence of a purely arithmetic property of a variety and its metrical property) and partially confirm it.

  4. Coefficient estimates of analytic endomorphisms of the unit disk fixing a point with applications to concave functions

    Ohno, Rintaro; Sugawa, Toshiyuki
    In this article, we discuss the coefficient regions of analytic self-maps of the unit disk with a prescribed fixed point. As an application, we solve the Fekete–Szegő problem for normalized concave functions with a pole in the unit disk.

  5. On a relation between the self-linking number and the braid index of closed braids in open books

    Ito, Tetsuya
    We prove a generalization of the Jones–Kawamuro conjecture that relates the self-linking number and the braid index of closed braids, for planar open books with certain additional conditions and modifications. We show that our result is optimal in some sense by giving several examples that do not satisfy a naive generalization of the Jones–Kawamuro conjecture.

  6. The étale cohomology of the general linear group over a finite field and the Dickson algebra

    Tezuka, Michishige; Yagita, Nobuaki
    Let $p\neq\ell$ be primes. We study the étale cohomology $H^{*}_{\text{\'{e}t}}(\mathrm{BGL}_{n}(\mathbb{F}_{p^{s}});\mathbb{Z}/{\ell})$ by using the stratification methods from Molina-Rojas and Vistoli. To compute this cohomology, we use the Dickson algebra and the Drinfeld space.

  7. On the distinguished spectrum of $\operatorname{Sp}_{2n}$ with respect to $\operatorname{Sp}_{n}\times\operatorname{Sp}_{n}$

    Lapid, Erez Moshe; Offen, Omer
    Given a reductive group $G$ and a reductive subgroup $H$ , both defined over a number field $F$ , we introduce the notion of the $H$ -distinguished automorphic spectrum of $G$ and analyze it for the pairs $(\operatorname{GL}_{2n},\operatorname{Sp}_{n})$ and $(\operatorname{Sp}_{2n},\operatorname{Sp}_{n}\times\operatorname{Sp}_{n})$ . In the first case we give a complete description by using results of Jacquet and Rallis as well as Offen and Yamana. In the second case we give an upper bound, generalizing vanishing results of Ash, Ginzburg, and Rallis, and a lower bound, extending results of Ginzburg, Rallis, and Soudry.

  8. The cyclotomic Iwasawa main conjecture for Hilbert cusp forms with complex multiplication

    Hara, Takashi; Ochiai, Tadashi
    We deduce the cyclotomic Iwasawa main conjecture for Hilbert modular cusp forms with complex multiplication from the multivariable main conjecture for CM number fields. To this end, we study in detail the behavior of the $p$ -adic $L$ -functions and the Selmer groups attached to CM number fields under specialization procedures.

  9. The moduli of representations of degree $2$

    Nakamoto, Kazunori
    There are six types of $2$ -dimensional representations in general. For any groups and any monoids, we can construct the moduli of $2$ -dimensional representations for each type: the moduli of absolutely irreducible representations, representations with Borel mold, representations with semisimple mold, representations with unipotent mold, representations with unipotent mold over ${\Bbb{F}}_{2}$ , and representations with scalar mold. We can also construct them for any associative algebras.

  10. Lattice multipolygons

    Higashitani, Akihiro; Masuda, Mikiya
    We discuss generalizations of some results on lattice polygons to certain piecewise linear loops which may have a self-intersection but have vertices in the lattice $\mathbb{Z}^{2}$ . We first prove a formula on the rotation number of a unimodular sequence in $\mathbb{Z}^{2}$ . This formula implies the generalized twelve-point theorem of Poonen and Rodriguez-Villegas. We then introduce the notion of lattice multipolygons, which is a generalization of lattice polygons, state the generalized Pick’s formula, and discuss the classification of Ehrhart polynomials of lattice multipolygons and also of several natural subfamilies of lattice multipolygons.

  11. On the geometry of the Lehn–Lehn–Sorger–van Straten eightfold

    Shinder, Evgeny; Soldatenkov, Andrey
    In this article we make a few remarks about the geometry of the holomorphic symplectic manifold $Z$ constructed by Lehn, Lehn, Sorger, and van Straten as a two-step contraction of the variety of twisted cubic curves on a cubic fourfold $Y\subset\mathbb{P}^{5}$ . We show that $Z$ is birational to a component of the moduli space of stable sheaves in the Calabi–Yau subcategory of the derived category of $Y$ . Using this description we deduce that the twisted cubics contained in a hyperplane section $Y_{H}=Y\cap H$ of $Y$ give rise to a Lagrangian subvariety $Z_{H}\subset Z$ . For a generic choice...

  12. Regular functions on spherical nilpotent orbits in complex symmetric pairs: Classical non-Hermitian cases

    Bravi, Paolo; Chirivî, Rocco; Gandini, Jacopo
    Given a classical semisimple complex algebraic group $G$ and a symmetric pair $(G,K)$ of non-Hermitian type, we study the closures of the spherical nilpotent $K$ -orbits in the isotropy representation of $K$ . For all such orbit closures, we study the normality, and we describe the $K$ -module structure of the ring of regular functions of the normalizations.

  13. The approximate pseudorandom walk accompanied by the pseudostochastic process corresponding to a higher-order heat-type equation

    Nakajima, Tadashi; Sato, Sadao
    As is well known, a standard random walk is approximate to the stochastic process corresponding to the heat equation. Lachal constructed the approximate pseudorandom walk which is accompanied by the pseudostochastic process corresponding to an even-order heat-type equation. We have two purposes for this article. The first is to construct the approximate pseudorandom walk which is accompanied by the pseudostochastic process corresponding to an odd-order heat-type equation. The other is to propose a construction method for the approximate pseudorandom walk which is accompanied by the pseudostochastic process corresponding to an even-order heat-type equation. This method is different from that of...

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