scholarly journals On the Iwasawa theory of the Lubin–Tate moduli space

2013 ◽  
Vol 149 (5) ◽  
pp. 793-839 ◽  
Author(s):  
Jan Kohlhaase
Keyword(s):  
Moduli Space ◽  
General Linear Group ◽  
Hecke Algebra ◽  
Iwasawa Theory ◽  
The Other ◽  
Deformation Space ◽  
Ground Field ◽  
Formal Module ◽  
Deformation Problem ◽  
Formal Algebra

AbstractWe study the affine formal algebra$R$of the Lubin–Tate deformation space as a module over two different rings. One is the completed group ring of the automorphism group$\Gamma $of the formal module of the deformation problem, the other one is the spherical Hecke algebra of a general linear group. In the most basic case of height two and ground field$\mathbb {Q}_p$, our structure results include a flatness assertion for$R$over the spherical Hecke algebra and allow us to compute the continuous (co)homology of$\Gamma $with coefficients in $R$.

scholarly journals Wonderful compactifications of Bruhat-Tits buildings

2017 ◽  
Vol Volume 1 ◽  
Author(s):  
Bertrand Remy ◽  
Amaury Thuillier ◽  
Annette Werner
Keyword(s):  
Local Field ◽  
Semisimple Group ◽  
The Other ◽  
Analytic Space ◽  
Ground Field ◽  
Berkovich Analytic Space ◽  
Euclidean Building ◽  
The Relationship

Given a split semisimple group over a local field, we consider the maximal Satake-Berkovich compactification of the corresponding Euclidean building. We prove that it can be equivariantly identified with the compactification which we get by embedding the building in the Berkovich analytic space associated to the wonderful compactification of the group. The construction of this embedding map is achieved over a general non-archimedean complete ground field. The relationship between the structures at infinity, one coming from strata of the wonderful compactification and the other from Bruhat-Tits buildings, is also investigated.

ℤ3-actions on Horikawa surfaces

2021 ◽  
pp. 2150097
Author(s):  
Vicente Lorenzo
Keyword(s):  
Moduli Space ◽  
General Type ◽  
Admissible Pair ◽  
Algebraic Surfaces ◽  
The Other ◽  
Connected Components ◽  
Surfaces Of General Type ◽  
Canonical Models ◽  
Normal Surfaces ◽  
Stable Surfaces

Minimal algebraic surfaces of general type [Formula: see text] such that [Formula: see text] are called Horikawa surfaces. In this note, [Formula: see text]-actions on Horikawa surfaces are studied. The main result states that given an admissible pair [Formula: see text] such that [Formula: see text], all the connected components of Gieseker’s moduli space [Formula: see text] contain surfaces admitting a [Formula: see text]-action. On the other hand, the examples considered allow us to produce normal stable surfaces that do not admit a [Formula: see text]-Gorenstein smoothing. This is illustrated by constructing non-smoothable normal surfaces in the KSBA-compactification [Formula: see text] of Gieseker’s moduli space [Formula: see text] for every admissible pair [Formula: see text] such that [Formula: see text]. Furthermore, the surfaces constructed belong to connected components of [Formula: see text] without canonical models.

There exist conjugate simple braids whose associated permutations are not strongly conjugate

2007 ◽  
Vol 143 (3) ◽  
pp. 663-667 ◽  
Author(s):  
P. M. G. MANCHÓN
Keyword(s):  
Central Element ◽  
Hecke Algebra ◽  
The Other ◽  
Topological Dynamics ◽  
Ultra Summit Set

AbstractIf two permutations are strongly conjugate, then their corresponding positive permutation braids (also called simple braids) are conjugate. In this paper we exhibit two conjugate simple braids whose associated permutations are not strongly conjugate. In terms of the grey and black graphs with vertices in the ultra summit set defined in [1], this result can be reformulated by saying that there are ultra summit sets with simple braids (hence with canonical length k = 1) in which the grey graph is not connected. Birman, Gebhardt and González Meneses have given similar examples, but with k ≥ 2 [1]. Recall that the set of simple braids on n strings is a basis of the Hecke algebra Hn. If two simple braids on n strings are conjugate, the associated permutations are centrally conjugate, which means that the coefficients of any central element of Hn corresponding to these simple braids are equal. Working on topological dynamics, Hall and Carvalho [8] have discovered two braids on 12 strings (one and its reverse) which are not conjugate. Since one braid is the reverse of the other, their corresponding permutations are centrally conjugate. For n ≤ 6 we checked that central conjugacy implies conjugacy of the corresponding simple braids.

scholarly journals Indecomposable orthogonal invariants of several matrices over a field of positive characteristic

2020 ◽  
pp. 1-11
Author(s):  
Artem Lopatin
Keyword(s):  
Characteristic Polynomial ◽  
Orthogonal Group ◽  
General Linear Group ◽  
Positive Characteristic ◽  
The Other ◽  
Symmetric Matrices ◽  
Maximal Degree ◽  
Infinite Field ◽  
The Given ◽  
Field Of Positive Characteristic

We consider the algebra of invariants of [Formula: see text]-tuples of [Formula: see text] matrices under the action of the orthogonal group by simultaneous conjugation over an infinite field of characteristic [Formula: see text] different from two. It is well known that this algebra is generated by the coefficients of the characteristic polynomial of all products of generic and transpose generic [Formula: see text] matrices. We establish that in case [Formula: see text] the maximal degree of indecomposable invariants tends to infinity as [Formula: see text] tends to infinity. In other words, there does not exist a constant [Formula: see text] such that it only depends on [Formula: see text] and the considered algebra of invariants is generated by elements of degree less than [Formula: see text] for any [Formula: see text]. This result is well-known in case of the action of the general linear group. On the other hand, for the rest of [Formula: see text] the given phenomenon does not hold. We investigate the same problem for the cases of symmetric and skew-symmetric matrices.

scholarly journals IWASAWA THEORY FOR THE SYMMETRIC SQUARE OF A CM MODULAR FORM AT INERT PRIMES

2011 ◽  
Vol 54 (2) ◽  
pp. 241-259
Author(s):  
ANTONIO LEI
Keyword(s):  
Modular Form ◽  
Complex Multiplication ◽  
Iwasawa Theory ◽  
The Other ◽  
Selmer Groups ◽  
Supersingular Primes ◽  
Supersingular Prime ◽  
Symmetric Square ◽  
The One ◽  
Invent Math

AbstractLet f be a modular form with complex multiplication (CM) and p an odd prime that is inert in the CM field. We construct two p-adic L-functions for the symmetric square of f, one of which has the same interpolating properties as the one constructed by Delbourgo and Dabrowski (A. Dabrowski and D. Delbourgo, S-adic L-functions attached to the symmetric square of a newform, Proc. Lond. Math. Soc. 74(3) (1997), 559–611), whereas the other one has a similar interpolating properties but corresponds to a different eigenvalue of the Frobenius. The symmetry between these two p-adic L-functions allows us to define the plus and minus p-adic L-functions à la Pollack (R. Pollack, on the p-adic L-function of a modular form at a supersingular prime, Duke Math. J. 118(3) (2003), 523–558). We also define the plus and minus p-Selmer groups analogous to the ones defined by Kobayashi (S. Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152(1) (2003), 1–36). We explain how to relate these two sets of objects via a main conjecture.

scholarly journals Derived categories of quintic del Pezzo fibrations

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Fei Xie
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Derived Categories ◽  
Derived Category ◽  
The Other ◽  
Del Pezzo Surfaces ◽  
Linear Section ◽  
Projective Duality ◽  
Del Pezzo ◽  
Space Approach

AbstractWe provide a semiorthogonal decomposition for the derived category of fibrations of quintic del Pezzo surfaces with rational Gorenstein singularities. There are three components, two of which are equivalent to the derived categories of the base and the remaining non-trivial component is equivalent to the derived category of a flat and finite of degree 5 scheme over the base. We introduce two methods for the construction of the decomposition. One is the moduli space approach following the work of Kuznetsov on the sextic del Pezzo fibrations and the components are given by the derived categories of fine relative moduli spaces. The other approach is that one can realize the fibration as a linear section of a Grassmannian bundle and apply homological projective duality.

scholarly journals PERTURBATION OF THE GROUND VARIETIES OF c=1 STRING THEORY

1993 ◽  
Vol 08 (33) ◽  
pp. 3187-3199 ◽  
Author(s):  
DEBASHIS GHOSHAL ◽  
PORUS LAKDAWALA ◽  
SUNIL MUKHI
Keyword(s):  
String Theory ◽  
Moduli Space ◽  
The Self ◽  
The Other ◽  
Cosmological Perturbation ◽  
First Order ◽  
Singular Varieties ◽  
Universal Deformations ◽  
Other Hand

We discuss the effect of perturbations on the ground rings of c=1 string theory at the various compactification radii defining the AN points of the moduli space. We argue that perturbations by plus-type moduli define ground varieties which are equivalent to the unperturbed ones under redefinitions of the coordinates and hence cannot smoothen the singularity. Perturbations by the minus-type moduli, on the other hand, lead to semi-universal deformations of the singular varieties that can smoothen the singularity under certain conditions. To first order, the cosmological perturbation by itself can remove the singularity only at the self-dual (A1) point.

scholarly journals Simple Lie Algebras of Low Dimension Over GF (2)

2006 ◽  
Vol 9 ◽  
pp. 174-192 ◽  
Author(s):  
Michael Vaughan-Lee
Keyword(s):  
Lie Algebras ◽  
The Other ◽  
Ground Field ◽  
Dimensional Algebra ◽  
Simple Lie Algebras ◽  
Low Dimension ◽  
Restricted Lie Algebras ◽  
Central Simple

AbstractWe classify the simple Lie algebras of dimension at most 9 over GF(2). There is one of dimension 3 and one of dimension 6, there are two of dimension 7 and two of dimension 8, and there is one of dimension 9. The two simple Lie algebras of dimension 8 are restricted Lie algebras. If we extend the ground field to GF(4), then the six-dimensional algebra is no longer simple, and if we extend the ground field to GF(8) then the nine-dimensional algebra is no longer simple. But the other algebras are all central simple.

scholarly journals Heterotic orbifolds, reduced rank and SO(2n + 1) characters

2020 ◽  
Vol 35 (24) ◽  
pp. 2050132
Author(s):  
Hervé Partouche ◽  
Balthazar de Vaulchier
Keyword(s):  
Moduli Space ◽  
Degrees Of Freedom ◽  
Heterotic String ◽  
The Other ◽  
Internal Space ◽  
Reduced Rank ◽  
Gauge Symmetries ◽  
Discrete Torsion ◽  
Dimensional Minkowski Space ◽  
Heterotic Orbifolds

The moduli space of the maximally supersymmetric heterotic string in [Formula: see text]-dimensional Minkowski space contains various components characterized by the rank of the gauge symmetries of the vacua they parametrize. We develop an approach for describing in a unified way continuous Wilson lines which parametrize a component of the moduli space, together with discrete deformations responsible for the switch from one component to the other. Applied to a component that contains vacua with [Formula: see text] gauge-symmetry factors, our approach yields a description of all backgrounds of the component in terms of free-orbifold models. The orbifold generators turn out to act symmetrically or asymmetrically on the internal space, with or without discrete torsion. Our derivations use extensively affine characters of [Formula: see text]. As a by-product, we find a peculiar orbifold description of the heterotic string in 10 dimensions, where all gauge degrees of freedom arise as twisted states, while the untwisted sector reduces to the gravitational degrees of freedom.

scholarly journals Moduli space of J-holomorphic subvarieties

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Weiyi Zhang
Keyword(s):  
Algebraic Geometry ◽  
Linear System ◽  
Moduli Space ◽  
Complex Structure ◽  
The Other ◽  
Almost Complex Structure ◽  
Rational Surfaces ◽  
Uniqueness Results ◽  
Almost Complex ◽  
Higher Genus

AbstractWe study the moduli space of J-holomorphic subvarieties in a 4-dimensional symplectic manifold. For an arbitrary tamed almost complex structure, we show that the moduli space of a sphere class is formed by a family of linear system structures as in algebraic geometry. Among the applications, we show various uniqueness results of J-holomorphic subvarieties, e.g. for the fiber and exceptional classes in irrational ruled surfaces. On the other hand, non-uniqueness and other exotic phenomena of subvarieties in complex rational surfaces are explored. In particular, connected subvarieties in an exceptional class with higher genus components are constructed. The moduli space of tori is also discussed, and leads to an extension of the elliptic curve theory.

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