bluuebunny/crossref_metadata_2025 huggingface dataset:
Categorical duality for Yetter-Drinfeld algebras
Authors: Neshveyev Sergey , Yamashita Makoto
Published: January 2023

We study tensor structures on

(\operatorname{Rep} G)

-module categories defined by actions of a compact quantum group

G

on unital C

^*

-algebras. We show that having a tensor product which defines the module structure is equivalent to enriching the action of

G

to the structure of a braided-commutative Yetter–Drinfeld algebra. This shows that the category of braided-commutative Yetter–Drinfeld

G

-C

^*

-algebras is equivalent to the category of generating unitary tensor functors from

\operatorname{Rep} G

into C

^*

-tensor categories. To illustrate this equivalence, we discuss coideals of quotient type in

C(G)

, Hopf–Galois extensions and noncommutative Poisson boundaries.
Algebraic Kasparov $K$-theory. I
Authors: Garkusha Grigory
Published: January 2023

This paper is to construct unstable, Morita stable and stable bivariant algebraic Kasparov

K

-theory spectra of

k

-algebras. These are shown to be homotopy invariant, excisive in each variable

K

-theories. We prove that the spectra represent universal unstable, Morita stable and stable bivariant homology theories respectively.
The Dirac operator with mass $m_0 \ge 0$: non-existence of zero modes and of threshold eigenvalues
Authors: Kalf Hubert , Okaji Takashi , Yamada Osanobu
Published: January 2023

A simple global condition on the potential is given which excludes zero modes of the massless Dirac operator. As far as local conditions at infinity are concerned, it is shown that at energy zero the Dirac equation without mass term has no non-trivial

L^2

-solutions at infinity for potentials which are either very slowly varying or decaying at most like

r^{-s}

with

s \in (0,1)

. When a mass term is present, it is proved that at the thresholds there are again no such solutions when the potential decays at most like

r^{-s}

with

s \in (0,2)

. In both situations the decay rate is optimal.
Long range diffusion reaction model on population dynamics
Authors: Abual-Rub M.S.
Published: January 2023

A model for long range di usion reaction on population dynamics has been considered, and conditions for the existence and uniqueness of solutions to the model in Lp;q norms has been obtained.
Combinatorics of the basic stratum of some simple Shimura varieties
Authors: Kret Arno
Published: January 2023

We express the cohomology of the basic Newton stratum of some unitary Shimura varieties associated to division algebras in terms of automorphic representations of the group in the Shimura datum.
Equivariant cycles and cancellation for motivic cohomology
Authors: Heller J. , Voineagu M. , Østvær P.A.
Published: January 2023

We introduce a Bredon motivic cohomology theory for smooth schemes defined over a field and equipped with an action by a finite group. These cohomology groups are defined for finite dimensional representations as the hypercohomology of complexes of equivariant correspondences in the equivariant Nisnevich topology. We generalize the theory of presheaves with transfers to the equivariant setting and prove a Cancellation Theorem.
Equivariant operational Chow rings of $T$-linear schemes
Authors: Gonzales Richard P.
Published: January 2023

We study

T

-linear schemes, a class of objects that includes spherical and Schubert varieties. We provide a localization theorem for the equivariant Chow cohomology of these schemes that does not depend on resolution of singularities. Furthermore, we give an explicit presentation of the equivariant Chow cohomology of possibly singular complete spherical varieties admitting a smooth equivariant envelope (e.g., group embeddings).
On Suslin matrices and their connection to spin groups
Authors: Chintala Vineeth Reddy
Published: January 2023

A representation of the Clifford algebra (of a hyperbolic module) is given using what are called Suslin matrices. This explicit construction is used to analyze the corresponding Spin groups. Moreover, some properties of Suslin matrices are better understood via this link to Clifford algebras.
On homological stability for configuration spaces on closed background manifolds
Authors: Cantero Federico , Palmer Martin
Published: January 2023

We introduce a new map between configuration spaces of points in a background manifold– the replication map – and prove that it is a homology isomorphism in a range with certain coefficients. This is particularly of interest when the background manifold is closed, in which case the classical stabilisation map does not exist. We then establish conditions on the manifold and on the coefficients under which homological stability holds for configuration spaces on closed manifolds. These conditions are sharp when the background manifold is a two-dimensional sphere, the classical counterexample in the field. For field coefficients this extends results of Church [Church, 2012] and Randal–Williams [Randal–Williams, 2013a] to the case of odd characteristic, and for

p

-local coefficients it improves results of Bendersky–Miller [Bendersky and Miller, 2014].
Rectification of algebras and modules
Authors: Hinich Vladimir
Published: January 2023

Let

\mathcal O

be a topological (colored) operad. The Lurie

\infty

-category of

\mathcal O

-algebras with values in (

\infty

-category of) complexes is compared to the

\infty

-category underlying the model category of (classical) dg

\mathcal O

-algebras. This can be interpreted as a “rectification” result for Lurie operad algebras. A similar result is obtained for modules over operad algebras, as well as for algebras over topological PROPs.
A combinatorial interpretation for Schreyer's tetragonal invariants
Authors: Castryck Wouter , Cools Filip
Published: January 2023

Schreyer has proved that the graded Betti numbers of a canonical tetragonal curve are determined by two integers

b_1

and

b_2

, associated to the curve through a certain geometric construction. In this article we prove that in the case of a smooth projective tetragonal curve on a toric surface, these integers have easy interpretations in terms of the Newton polygon of its defining Laurent polynomial. We can use this to prove an intrinsicness result on Newton polygons of small lattice width.
Moduli spaces for quilted surfaces and Poisson structures
Authors: Li-Bland David , Ševera Pavol
Published: January 2023

Let

G

be a Lie group endowed with a bi-invariant pseudo-Riemannian metric. Then the moduli space of flat connections on a principal

G

-bundle,

P\to \Sigma

, over a compact oriented surface with boundary,

\Sigma

, carries a Poisson structure. If we trivialize

P

over a finite number of points on

\partial \Sigma

then the moduli space carries a quasi-Poisson structure instead. Our first result is to describe this quasi-Poisson structure in terms of an intersection form on the fundamental groupoid of the surface, generalizing results of Massuyeau and Turaev citeMassuyeau:2012uw,Turaev:2007jh. Our second result is to extend this framework to quilted surfaces, i.e. surfaces where the structure group varies from region to region and a reduction (or relation) of structure occurs along the borders of the regions, extending results of the second author citeSevera:2011ug,Severa98,Severa:2005vla. We describe the Poisson structure on the moduli space for a quilted surface in terms of an operation on spin networks, i.e. graphs immersed in the surface which are endowed with some additional data on their edges and vertices. This extends the results of various authors citeGoldman:1986eh,Goldman:1984hr,Roche:2000ws,Andersen:1996ur.
Regularity of projection operators attached to worm domains
Authors: Barrett David E. , Ehsani Dariush , Peloso Marco M.
Published: January 2023

We construct a projection operator on an unbounded worm domain which maps subspaces of

W

^s to themselves. The subspaces are determined by a Fourier decomposition of

W

^s according to a rotational invariance of the worm domain.
On additive higher Chow groups of affine schemes
Authors: Krishna Amalendu , Park Jinhyun
Published: January 2023

We show that the multivariate additive higher Chow groups of a smooth affine

k

-scheme

\mathrm{Spec} (R)

essentially of finite type over a perfect field

k

of characteristic

\not = 2

form a differential graded module over the big de Rham–Witt complex

\mathbb W_{m}\Omega

^bullet_R. In the univariate case, we show that additive higher Chow groups of

\mathrm{Spec} (R)

form a Witt-complex over

R

. We use these structures to prove an étale descent for multivariate additive higher Chow groups.
On asymptotic bounds for the number of irreducible components of the moduli space of surfaces of general type. II.
Authors: Lönne Michael , Penegini Matteo
Published: January 2023

In this paper we investigate the asymptotic growth of the number of irreducible and connected components of the moduli space of surfaces of general type corresponding to certain families of surfaces isogenous to a higher product with group

(\mathbb Z/2\mathbb Z)^k

. We obtain a significantly higher growth than the one in our previous paper [LP14].
Free curves on varieties
Authors: Gounelas Frank
Published: January 2023

We study various generalisations of rationally connected varieties, allowing the connecting curves to be of higher genus. The main focus will be on free curves

f:C\to X

with large unobstructed deformation space as originally defined by Kollár, but we also give definitions and basic properties of varieties

X

covered by a family of curves of a fixed genus

g

so that through any two general points of

X

there passes the image of a curve in the family. We prove that the existence of a free curve of genus

g\geq1

implies the variety is rationally connected in characteristic zero and initiate a study of the problem in positive characteristic.
The harmonic analysis of lattice counting on real spherical spaces
Authors: Krötz Bernhard , Sayag Eitan , Schlichtkrull Henrik
Published: January 2023

By the collective name of

\it

lattice counting we refer to a setup introduced in [DRS] that aims to establish a relationship between arithmetic and randomness in the context of affine symmetric spaces. In this paper we extend the geometric setup from symmetric to real spherical spaces and continue to develop the approach with harmonic analysis which was initiated in [DRS].
Hyperplane mass partitions via relative equivariant obstruction theory
Authors: Blagojević Pavle V.M. , Frick Florian , Haase Albert , Ziegler Günter M.
Published: January 2023

The Grünbaum–Hadwiger–Ramos hyperplane mass partition problem was introduced by Grünbaum (1960) in a special case and in general form by Ramos (1996). It asks for the «admissible» triples

(d,j,k)

such that for any

j

masses in

\mathbb R^d

there are

k

hyperplanes that cut each of the masses into

2^k

equal parts. Ramos' conjecture is that the Avis–Ramos necessary lower bound condition

dk\ge j(2^{k}-1)

is also sufficient. We develop a «join scheme» for this problem, such that non-existence of an

{\mathfrak S_{k}^\pm}

-equivariant map between spheres

(S^{d})^{*k} \rightarrow S(W_{k}\oplus U_{k}^{\oplus j})

that extends a test map on the subspace of

(S^{d})^{*k}

where the hyperoctahedral group

\mathfrak S_{k}^\pm

acts non-freely, implies that

(d,j,k)

is admissible. For the sphere

(S^{d})^{*k}

we obtain a very efficient regular cell decomposition, whose cells get a combinatorial interpretation with respect to measures on a modified moment curve. This allows us to apply relative equivariant obstruction theory successfully, even in the case when the difference of dimensions of the spheres

(S^{d})^{*k}

and

S(W_{k}\oplus U_{k}^{\oplus j})

is greater than one. The evaluation of obstruction classes leads to counting problems for concatenated Gray codes. Thus we give a rigorous, unified treatment of the previously announced cases of the Grünbaum–Hadwiger–Ramos problem, as well as a number of new cases for Ramos' conjecture.
Gerstenhaber-schack and Hochschild cohomologies of Hopf algebras
Authors: Bichon Julien
Published: January 2023

We show that the Gerstenhaber-Schack cohomology of a Hopf algebra determines its Hochschild cohomology, and in particular its Gerstenhaber-Schack cohomological dimension bounds its Hochschild cohomological dimension, with equality of the dimensions when the Hopf algebra is cosemisimple of Kac type. Together with some general considerations on free Yetter-Drinfeld modules over adjoint Hopf subalgebras and the monoidal invariance of Gerstenhaber-Schack cohomology, this is used to show that both Gerstenhaber-Schack and Hochschild cohomological dimensions of the coordinate algebra of the quantum permutation group are 3.
The Cohomology of Canonical Quotients of Free Groups and Lyndon Words
Authors: Efrat Ido
Published: January 2023

For a prime number

p

and a free profinite group

S

, let

S^{(n,p)}

be the

n

th term of its lower

p

-central filtration, and

S^{[n,p]}

the corresponding quotient. Using tools from the combinatorics of words, we construct a canonical basis of the cohomology group

H^2(S^{[n,p]}, \Bbb Z/p)

, which we call the Lyndon basis, and use it to obtain structural results on this group. We show a duality between the Lyndon basis and canonical generators of

S^{(n,p)}/S^{(n+1,p)}

. We prove that the cohomology group satisfies shuffle relations, which for small values of

n

fully describe it.
Vector-Valued Modular Forms and the Gauss Map
Authors: Dalla Piazza Francesco , Fiorentino Alessio , Grushevsky Samuel , Perna Sara , Salvati Manni Riccardo
Published: January 2023

We use the gradients of theta functions at odd two-torsion points – thought of as vector-valued modular forms – to construct holomorphic differential forms on the moduli space of principally polarized abelian varieties, and to characterize the locus of decomposable abelian varieties in terms of the Gauss images of two-torsion points.
Toeplitz Operators on Higher Cauchy-Riemann Spaces
Authors: Engliš Miroslav , Zhang Genkai
Published: January 2023

We develop a theory of Toeplitz, and to some extent Hankel, operators on the kernels of powers of the boundary d-bar operator, suggested by Boutet de Monvel and Guillemin, and on their analogues, somewhat better from the point of view of complex analysis, defined using instead the covariant Cauchy-Riemann operators of Peetre and the second author. For the former, Dixmier class membership of these Hankel operators is also discussed. Our main tool are the generalized Toeplitz operators (with pseudodifferential symbols), in particular there appears naturally the problem of finding parametrices of matrices of such operators in situations when the principal symbol fails to be elliptic.
On the automorphic induction for radical $p$-extensions and some other functorial operations $\pmod p$
Authors: Clozel Laurent
Published: January 2023

Soit

E/F

une extension de degré

d

de corps de nombres,

\Bbb A_E, \Bbb A_F

leurs anneaux d'adèles. Le principe de fonctorialité de Langlands postule entre autres une opération d'induction automorphe [1] construisant une représentation automorphe de

GL(nd, \Bbb A_F)

à partir d'une représentation cuspidale de

GL(n, \Bbb A_E)

. Quand l'extension

E/F

est résoluble, ceci a été démontré dans [1]. Si l'extension n'est pas galoisienne, le seul cas connu, dû à Jacquet et Piatetski-Shapiro [18], est celui où

n=1

et

d=3

. Nous montrons l'existence de l'induction automorphe quand

d=p

est un nombre premier et l'extension radicielle, mais pour des classes de cohomologie mod

p

. Les démonstrations reposent sur le travail récent de Treumann et Venkatesh [20].
Lax Colimits and Free Fibrations in $\infty$-Categories
Authors: Gepner David , Haugseng Rune , Nikolaus Thomas
Published: January 2023

We define and discuss lax and weighted colimits of diagrams in

\infty

-categories and show that the coCartesian fibration corresponding to a functor is given by its lax colimit. A key ingredient, of independent interest, is a simple characterization of the free Cartesian fibration on a functor of

\infty

-categories. As an application of these results, we prove that 2-representable functors are preserved under exponentiation, and also that the total space of a presentable Cartesian fibration between is presentable, generalizing a theorem of Makkai and Paré to the

\infty

-categories setting. Lastly, in an appendix, we observe that pseudofunctors between (2,1)-categories give rise to functors between

\infty

-categories via the Duskin nerve. setting and the Duskin nerve.
Milne's Correcting Factor and Derived De Rham Cohomology. II
Authors: Morin Baptiste
Published: January 2023

Milne's correcting factor, which appears in the Zeta-value at

s=n

of a smooth projective variety

X

over a finite field

\Bbb F_q

, is the Euler characteristic of the derived de Rham cohomology of

X/\Bbb{Z}

modulo the Hodge filtration

F^n

. In this note, we extend this result to arbitrary separated schemes of finite type over

\Bbb F_q

of dimension at most

d

, provided resolution of singularities for schemes of dimension at most

d

holds. More precisely, we show that Geisser's generalization of Milne's factor, whenever it is well defined, is the Euler characteristic of the

eh

-cohomology with compact support of the derived de Rham complex relative to

\Bbb Z

modulo

F^n

.
Torsion 1-cycles and the coniveau spectral sequence
Authors: Ma Shouhei
Published: January 2023

We relate the torsion part of the Abel-Jacobi kernel in the Griffiths group of 1-cycles to a birational invariant analogous to the degree 4 unramified cohomology and an invariant associated to the generalized Hodge conjecture in degree

2{\dim}(X)-3

. We also describe in terms of

\mathcal H

-cohomology the Griffiths group of 1-cycles and the group of torsion cycles algebraically equivalent to zero of arbitrary dimension.
Geometry of Positive Configurations in Affine Buildings
Authors: Le Ian , O'Dorney Evan
Published: January 2023

Positive configurations of points in the affine building were introduced in [the author, Geom. Topol. 20, No. 3, 1673–1735 (2016; Zbl 1348.30023)] as the basic object needed to define higher laminations. We start by giving a self-contained, elementary definition of positive configurations of points in the affine building and their basic properties. Then we study the geometry of these configurations. The canonical functions on triples of flags that were defined by V. Fock and A. Goncharov [Publ. Math., Inst. Hautes Étud. Sci. 103, 1–211 (2006; Zbl 1099.14025)] have a tropicalization that gives functions on triples of points in the affine Grassmannian. One expects that these functions, though of algebro-geometric origin, have a simple description in terms of the metric structure on the corresponding affine building. We give a several conjectures describing the tropicalized canonical functions in terms of the geometry of affine buildings, and give proofs of some of them. The statements involve minimal networks and have some resemblance to the max-flow/min-cut theorem, which also plays a role in the proofs in unexpected ways. The conjectures can be reduced to purely algebraic statements about valuations of lattices that we argue are interesting in their own right. One can view these conjectures as the first examples of intersection pairings between higher laminations. They fit within the framework of the Duality Conjectures of [Fock and Goncharov, loc. cit.].
Differential Embedding Problems over Complex Function Fields
Authors: Bachmayr Annette , Harbater David , Hartmann Julia , Wibmer Michael
Published: January 2023

We introduce the notion of differential torsors, which allows the adaptation of constructions from algebraic geometry to differential Galois theory. Using these differential torsors, we set up a general framework for applying patching techniques in differential Galois theory over fields of characteristic zero. We show that patching holds over function fields over the complex numbers. As the main application, we prove the solvability of all differential embedding problems over complex function fields, thereby providing new insight on the structure of the absolute differential Galois group, i.e., the fundamental group of the underlying Tannakian category.
Presentations of subshifts and their topological conjugacy invariants
Authors: Matsumoto Kengo
Published: January 2023

We introduce the notions of symbolic matrix system and

\lambda

-graph system that are presentations of subshifts. They are generalized notions of symbolic matrix and

\lambda

-graph for sofic subshifts to general subshifts. We then formulate strong shift equivalence and shift equivalence between symbolic matrix systems and show that two subshifts are topologically conjugate if and only if the associated canonical symbolic matrix systems are strong shift equivalent. We construct several kinds of shift equivalence invariants for symbolic matrix systems. They are the dimension groups, the Bowen-Franks groups and the nonzero spectrum that are generalizations of the corresponding notions for nonnegative matrices. The K-groups for symbolic matrix systems are introduced. They are also shift equivalence invariants and stronger than the Bowen-Franks groups but weaker than the dimension triples. These kinds of shift equivalence invariants naturally induce topological conjugacy invariants for subshifts.
HNN Extensions of Quasi-Lattice Ordered Groups and their Operator Algebras
Authors: Raeburn Iain , Tolich Ilija , an Huef Astrid
Published: January 2023

The Baumslag-Solitar group is an example of an HNN extension. J. Spielberg [J. Lond. Math. Soc., II. Ser. 86, No. 3, 728–754 (2012; Zbl 1264.46040)] showed that it has a natural positive cone, and that it is then a quasi-lattice ordered group in the sense of A. Nica [J. Oper. Theory 27, No. 1, 17–52 (1992; Zbl 0809.46058)]. We give conditions for an HNN extension of a quasi-lattice ordered group

(G,P)

to be quasi-lattice ordered. In that case, if

(G,P)

is amenable as a quasi-lattice ordered group, then so is the HNN extension.
Differential Operators and Families of Automorphic Forms on Unitary Groups of Arbitrary Signature
Authors: Eischen Ellen , Fintzen Jessica , Mantovan Elena , Varma Ila
Published: January 2023

In the 1970's, Serre exploited congruences between

q

-expansion coefficients of Eisenstein series to produce

p

-adic families of Eisenstein series and, in turn,

p

-adic zeta functions. Partly through integration with more recent machinery, including Katz's approach to

p

-adic differential operators, his strategy has influenced four decades of developments. Prior papers employing Katz's and Serre's ideas exploiting differential operators and congruences to produce families of automorphic forms rely crucially on

q

-expansions of automorphic forms. The overarching goal of the present paper is to adapt the strategy to automorphic forms on unitary groups, which lack

q

-expansions when the signature is of the form

(a, b)

,

a\ne b

. In particular, this paper completely removes the restrictions on the signature present in prior work. As intermediate steps, we achieve two key objectives. First, partly by carefully analyzing the action of the Young symmetrizer on Serre-Tate expansions, we explicitly describe the action of differential operators on the Serre-Tate expansions of automorphic forms on unitary groups of arbitrary signature. As a direct consequence, for each unitary group, we obtain congruences and families analogous to those studied by Katz and Serre. Second, via a novel lifting argument, we construct a

p

-adic measure taking values in the space of

p

-adic automorphic forms on unitary groups of any prescribed signature. We relate the values of this measure to an explicit

p

-adic family of Eisenstein series. One application of our results is to the recently completed construction of

p

-adic

L

-functions for unitary groups by the first named author, Harris, Li, and Skinner.
A Local-Global Principle for Symplectic $\mathrm K_2$
Authors: Lavrenov Andrei
Published: January 2023

We prove that an element of the relative symplectic Steinberg group

g\in\mathrm{StSp}_{2n}(R[t],\,tR[t])

is trivial if and only if its image under any maximal localisation homomorphism is trivial.
Random matrices and $K$-theory for exact $C^*$-algebras
Authors: Haagerup U. , Thorbjørnsen S.
Published: January 2023

In this paper we find asymptotic upper and lower bounds for the spectrum of random operators of the form

S^*S=\Big(\sum_{i=1}^ra_i\otimes Y_i^{(n)}\Big)^* \Big(\sum_{i=1}^ra_i\otimes Y_i^{(n)}\Big),

where

a_1,\ldots,a_r

are elements of an exact

C^*

-algebra and

Y_1^{(n)},\ldots,Y_r^{(n)}

are complex Gaussian random

n\times n

matrices, with independent entries. Our result can be considered as a generalization of results of Geman (1981) and Silverstein (1985) on the asymptotic behavior of the largest and smallest eigenvalue of a random matrix of Wishart type. The result is used to give new proofs of:
The Complexity of a Flat Groupoid
Authors: Romagny Matthieu , Rydh David , Zalamansky Gabriel
Published: January 2023

Grothendieck proved that any finite epimorphism of noetherian schemes factors into a finite sequence of effective epimorphisms. We define the complexity of a flat groupoid

R\rightrightarrows X

with finite stabilizer to be the length of the canonical sequence of the finite map

R\rightarrow X\times_{X/R} X

, where

X/R

is the Keel-Mori geometric quotient. For groupoids of complexity at most 1, we prove a theorem of descent along the quotient

X\rightarrow X/R

and a theorem on the existence of the quotient of a groupoid by a normal subgroupoid. We expect that the complexity could play an important role in the finer study of quotients by groupoids.
On the Quotients of the Maximal Unramified 2-Extension of a Number Field
Authors: Maire Christian
Published: January 2023

Let

\mathrm{K}

be a totally imaginary number field. Denote by

G_{\mathrm{K}}^{ur}(2)

the Galois group of the maximal unramified pro-2 extension of

\mathrm{K}

. By using cup-products in étale cohomology of

\mathrm{Spec}\mathcal O_{\mathrm{K}}

we study situations where

G_{\mathrm{K}}^{ur}(2)

has no quotient of cohomological dimension 2. For example, in the family of imaginary quadratic fields

\mathrm{K}

, the group

G_{\mathrm{K}}^{ur}(2)

almost never has a quotient of cohomological dimension 2 and of maximal 2-rank. We also give a relation between this question and that of the 4-rank of the class group of

\mathrm{K}

, showing in particular that when ordered by absolute value of the discriminant, more than 99% of imaginary quadratic fields satisfy an alternative (but equivalent) form of the unramified Fontaine–Mazur conjecture (at

p = 2

).
Arithmetic Families of $(\varphi,\Gamma)$-Modules and Locally Analytic Representations of $GL_2(Q_p)$
Authors: Gaisin Ildar , Rodrigues Jacinto Joaquín
Published: January 2023

Let

A

be a

\bold{Q}_p

-affinoid algebra, in the sense of Tate. We develop a theory of locally convex

A

-modules parallel to the treatment in the case of a field by Schneider and Teitelbaum. We prove that there is an integration map linking a category of locally analytic representations in

A

-modules and separately continuous relative distribution modules. There is a suitable theory of locally analytic cohomology for these objects and a version of Shapiro's Lemma, generalizing results of Kohlhaase. As an application we propose a

p

-adic Langlands correspondence in families: For a regular trianguline

(\varphi,\Gamma)

-module of dimension

2

over the relative Robba ring

\mathcal R_A

we construct a locally analytic

GL_2(Q_p)

-representation in

A

-modules.
Cylindrical Wigner Measures
Authors: Falconi Marco
Published: January 2023

In this paper we study the semiclassical behavior of quantum states acting on the

\mathrm{C}^\ast

-algebra of canonical commutation relations, from a general perspective. The aim is to provide a unified and flexible approach to the semiclassical analysis of bosonic systems. We also give a detailed overview of possible applications of this approach to mathematical problems of both axiomatic relativistic quantum field theories and nonrelativistic many body systems. If the theory has infinitely many degrees of freedom, the set of Wigner measures, i.e. the classical counterpart of the set of quantum states, coincides with the set of all cylindrical measures acting on the algebraic dual of the space of test functions for the field, and this reveals a very rich semiclassical structure compared to the finite-dimensional case. We characterize the cylindrical Wigner measures and the a priori properties they inherit from the corresponding quantum states.
Model Topoi and Motivic Homotopy Theory
Authors: Raptis Georgios , Strunk Florian
Published: January 2023

Given a small simplicial category

\mathcal{C}

whose underlying ordinary category is equipped with a Grothendieck topology

\tau

, we construct a model structure on the category of simplicially enriched presheaves on

\mathcal{C}

where the weak equivalences are the local weak equivalences of the underlying (non-enriched) simplicial presheaves. We show that this model category is a

t

-complete model topos and describe the Grothendieck topology

[\tau]

on the homotopy category of

\mathcal{C}

that corresponds to this model topos. After we first review a proof showing that the motivic homotopy theory is not a model topos, we specialize this construction to the category of smooth schemes of finite type, which is simplicially enriched using the standard algebraic cosimplicial object, and compare the result with the motivic homotopy theory. We also collect some partial positive results on the exactness properties of the motivic localization functor.
Kähler Geometry on Hurwitz Spaces
Authors: Naumann Philipp
Published: January 2023

The classical Hurwitz space

\mathcal{H}^{n,b}

is a fine moduli space for simple branched coverings of the Riemann sphere

\mathbb{P}^1

by compact hyperbolic Riemann surfaces. In the article we study a generalized Weil-Petersson metric on the Hurwitz space, which was introduced in [R. Axelsson et al., Manuscr. Math. 147, No. 1–2, 63–79 (2015; Zbl 1319.32012)]. For this purpose, Horikawa's deformation theory of holomorphic maps is refined in the presence of hermitian metrics in order to single out distinguished representatives. Our main result is a curvature formula for a subbundle of the tangent bundle on the Hurwitz space obtained as a direct image. This covers the case of the curvature of the fibers of the natural map

\mathcal{H}^{n,b} \to \mathcal{M}_g

.
On a Construction Due to Khoshkam and Skandalis
Authors: Sundar S.
Published: January 2023

In this paper, we consider the Wiener-Hopf algebra, denoted

\mathcal{W}(A,P,G,\alpha)

, associated to an action of a discrete subsemigroup

P

of a group

G

on a

C^\ast

-algebra

A

. We show that

\mathcal{W}(A,P,G,\alpha)

can be represented as a groupoid crossed product. As an application, we show that when

P=\mathbb{F}_{n}^{+}

, the free semigroup on

n

generators, the

K

-theory of

\mathcal{W}(A,P,G,\alpha)

and the

K

-theory of

A

coincides.
Green Functions and Higher Deligne-Lusztig Characters
Authors: Chen Zhe
Published: January 2023

We give a generalisation of the character formula of Deligne-Lusztig representations from the finite field case to the truncated formal power series case. Motivated by this generalisation, we give a definition of Green functions for these local rings, and prove some basic properties along the lines of the finite field case, like a summation formula. As an application we show that the higher Deligne-Lusztig characters and Gérardin's characters agree at regular semisimple elements.
Topological Cyclic Homology Via the Norm
Authors: Angeltveit Vigleik , Blumberg Andrew J. , Gerhardt Teena , Hill Michael A. , Lawson Tyler , Mandell Michael A.
Published: January 2023

We describe a construction of the cyclotomic structure on topological Hochschild homology (

THH

) of a ring spectrum using the Hill-Hopkins-Ravenel multiplicative norm. Our analysis takes place entirely in the category of equivariant orthogonal spectra, avoiding use of the Bökstedt coherence machinery. We are also able to define two relative versions of topological cyclic homology (

TC

) and

TR

-theory: one starting with a ring

C_n

-spectrum and one starting with an algebra over a cyclotomic commutative ring spectrum

A

. We describe spectral sequences computing the relative theory over

A

in terms of

TR

over the sphere spectrum and vice versa. Furthermore, our construction permits a straightforward definition of the Adams operations on

TR

and

TC

.
Julia Sets for Polynomial Diffeomorphisms of $\mathbb{C}^2$ are not Semianalytic
Authors: Bedford Eric , Kim Kyounghee
Published: January 2023

For any polynomial diffeomorphism

f

of

\mathbb{C}^2

with positive entropy, neither the Julia set of

f

nor of its inverse

f^{-1}

is semianalytic.
Special Subvarieties in Mumford-Tate Varieties
Authors: Mohajer Abolfazl , Müller-Stach Stefan , Zuo Kang
Published: January 2023

Let

X=\Gamma \backslash D

be a Mumford-Tate variety, i.e., a quotient of a Mumford-Tate domain

D=G(\mathcal{R})/V

by a discrete subgroup

\Gamma

. Mumford-Tate varieties are generalizations of Shimura varieties. We define the notion of a special subvariety

Y \subset X

(of Shimura type), and formulate necessary criteria for

Y

to be special. Our method consists in looking at finitely many compactified special curves

C_i

in

Y

, and testing whether the inclusion

\bigcup_i C_i \subset Y

satisfies certain properties. One of them is the so-called relative proportionality condition. In this paper, we give a new formulation of this numerical criterion in the case of Mumford-Tate varieties

X

. In this way, we give necessary and sufficient criteria for a subvariety

Y

of

X

to be a special subvariety of Shimura type in the sense of the André-Oort conjecture. We discuss in detail the important case where

X=A_g

, the moduli space of principally polarized abelian varieties.
On Free Resolutions of Iwasawa Modules
Authors: Nichifor Alexandra , Palvannan Bharathwaj
Published: January 2023

Let

\Lambda

(isomorphic to

\mathbb{Z}_p[[T]]

) denote the usual Iwasawa algebra and

G

denote the Galois group of a finite Galois extension

L/K

of totally real fields. When the non-primitive Iwasawa module over the cyclotomic

\mathbb{Z}_p

-extension has a free resolution of length one over the group ring

\Lambda[G]

, we prove that the validity of the non-commutative Iwasawa main conjecture allows us to find a representative for the non-primitive

p

-adic

L

-function (which is an element of a

K_1

-group) in a maximal

\Lambda

-order. This integrality result involves a study of the Dieudonné determinant. Using a cohomolgoical criterion of Greenberg, we also deduce the precise conditions under which the non-primitive Iwasawa module has a free resolution of length one. As one application of the last result, we consider an elliptic curve over

\mathbb{Q}

with a cyclic isogeny of degree

p^2

. We relate the characteristic ideal in the ring

\Lambda

of the Pontryagin dual of its non-primitive Selmer group to two characteristic ideals, viewed as elements of group rings over

\Lambda

, associated to two non-primitive classical Iwasawa modules.
On the Distribution of Satake Parameters for Siegel Modular Forms
Authors: Knightly Andrew , Li Charles
Published: January 2023

We prove a harmonically weighted equidistribution result for the

p

-th Satake parameters of the family of automorphic cuspidal representations of

\operatorname{PGSp}(2n)

of fixed weight

\mathtt{k}

and prime-to-

p

level

N\to \infty

. The main tool is a new asymptotic Petersson formula for

\operatorname{PGSp}(2n)

in the level aspect.
Quasi-Polynomiality of Monotone Orbifold Hurwitz Numbers and Grothendieck's Dessins d'Enfants
Authors: Kramer Reinier , Lewański Danilo , Shadrin Sergey
Published: January 2023

We prove quasi-polynomiality for monotone and strictly monotone orbifold Hurwitz numbers. The second enumerative problem is also known as enumeration of a special kind of Grothendieck's dessins d'enfants or

r

-hypermaps. These statements answer positively two conjectures proposed by Do-Karev and Do-Manescu. We also apply the same method to the usual orbifold Hurwitz numbers and obtain a new proof of the quasi-polynomiality in this case. In the second part of the paper we show that the property of quasi-polynomiality is equivalent in all these three cases to the property that the

n

-point generating function has a natural representation on the

n

-th cartesian powers of a certain algebraic curve. These representations are necessary conditions for the Chekhov-Eynard-Orantin topological recursion.
Two interesting oriented matroids
Authors: Richter-Gebert Jürgen
Published: January 2023

Oriented matroids are a combinatorial model for configurations in real vector spaces. A central role in the theory is played by the realizability problem: Given an oriented matroid, find an associated vector configuration.
Functoriality of Automorphic $\mathrm{L}$-Invariants and Applications
Authors: Gehrmann Lennart
Published: January 2023

We study the behaviour of automorphic

\mathrm{L}

-invariants associated to cuspidal representations of

\mathrm{GL}(2)

of cohomological weight 0 under abelian base change and Jacquet-Langlands lifts to totally definite quaternion algebras. Under a standard non-vanishing hypothesis on automorphic

\mathrm{L}

-functions and some technical restrictions on the automorphic representation and the base field we get a simple proof of the equality of automorphic and arithmetic

\mathrm{L}

-invariants. This together with Spieß' results on

p

-adic

\mathrm{L}

-functions yields a new proof of the exceptional zero conjecture for modular elliptic curves – at least, up to sign.
Spinors and the Tangent Groupoid
Authors: Higson Nigel , Yi Zelin
Published: January 2023

The purpose of this article is to study Ezra Getzler's approach to the Atiyah-Singer index theorem from the perspective of Alain Connes' tangent groupoid. We shall construct a "rescaled" spinor bundle on the tangent groupoid, define a convolution operation on its smooth, compactly supported sections, and explain how the algebra so-obtained incorporates Getzler's symbol calculus.
Branched Projective Structures on a Riemann Surface and Logarithmic Connections
Authors: Biswas Indranil , Dumitrescu Sorin , Gupta Subhojoy
Published: January 2023

We study the set

\mathcal{P}_S

consisting of all branched holomorphic projective structures on a compact Riemann surface

X

of genus

g \geq 1

and with a fixed branching divisor

S := \sum_{i=1}^d n_i\cdot x_i

, where

x_i \in X

. Under the hypothesis that

n_i,=1

, for all

i

, with

d

a positive even integer such that

d \neq 2g-2

, we show that

\mathcal{P}_S

coincides with a subset of the set of all logarithmic connections with singular locus

S

, satisfying certain geometric conditions, on the rank two holomorphic jet bundle

J^1(Q)

, where

Q

is a fixed holomorphic line bundle on

X

such that

Q^{\otimes 2}= TX \otimes \mathcal{O}_X(S)

. The space of all logarithmic connections of the above type is an affine space over the vector space

H^0(X, K^{\otimes 2}_X \otimes\mathcal{O}_X(S))

of dimension

3g-3+d

. We conclude that

\mathcal{P}_S

is a subset of this affine space that has codimenison

d

at a generic point.
Annihilating Wild Kernels
Authors: Nickel Andreas
Published: January 2023

Let

L/K

be a finite Galois extension of number fields with Galois group

G

. Let

p

be an odd prime and

r>1

be an integer. Assuming a conjecture of Schneider, we formulate a conjecture that relates special values of equivariant Artin

L

-series at

s=r

to the compact support cohomology of the étale

p

-adic sheaf

\mathbb{Z}_p(r)

. We show that our conjecture is essentially equivalent to the

p

-part of the equivariant Tamagawa number conjecture for the pair

(h^0(\text{Spec}(L))(r),\mathbb{Z}[G])

. We derive from this explicit constraints on the Galois module structure of Banaszak's

p

-adic wild kernels.
Subtle Characteristic Classes and Hermitian Forms
Authors: Tanania Fabio
Published: January 2023

Following
A. Smirnov
and
A. Vishik
["Subtle characteristic classes", Preprint, ], we compute the motivic cohomology ring of the Nisnevich classifying space of the unitary group associated to the standard split hermitian form of a quadratic extension. This provides us with subtle characteristic classes which take value in the motivic cohomology of the Čech simplicial scheme associated to a hermitian form. Comparing these new classes with subtle Stiefel-Whitney classes arising in the orthogonal case, we obtain relations among the latter ones holding in the motivic cohomology of the Čech simplicial scheme associated to a quadratic form divisible by a 1-fold Pfister form. Moreover, we present a description of the motive of the torsor corresponding to a hermitian form in terms of its subtle characteristic classes.
The Abel-Jacobi map for cubic threefold and periods of Fano threefolds of degree $14$
Authors: Iliev A. , Markushevich D.
Published: January 2023

The Abe–Jacobi maps of the families of elliptic quintics and rational quartics lying on a smooth cubic threefold are studied. It is proved that their generic fiber is the 5-dimensional projective space for quintics, and a smooth 3-dimensional variety birational to the cubic itself for quartics. The paper is a continuation of the recent work of Markushevich–Tikhomirov, who showed that the first Abel–Jacobi map factors through the moduli component of stable rank 2 vector bundles on the cubic threefold with Chern numbers

c_1=0, c_2=2

obtained by Serre's construction from elliptic quintics, and that the factorizing map from the moduli space to the intermediate Jacobian is étale. The above result implies that the degree of the étale map is 1, hence the moduli component of vector bundles is birational to the intermediate Jacobian. As an application, it is shown that the generic fiber of the period map of Fano varieties of degree 14 is birational to the intermediate Jacobian of the associated cubic threefold.
On Projective Kähler Manifolds of Partially Positive Curvature and Rational Connectedness
Authors: Heier Gordon , Wong Bun
Published: January 2023

In a previous paper, we proved that a projective Kähler manifold of positive total scalar curvature is uniruled. At the other end of the spectrum, it is a well-known theorem of Campana and Kollár-Miyaoka-Mori that a projective Kähler manifold of positive Ricci curvature is rationally connected. In the present work, we investigate the intermediate notion of

k

-positive Ricci curvature and prove that for a projective

n

-dimensional Kähler manifold of

k

-positive Ricci curvature the MRC fibration has generic fibers of dimension at least

n-k+1

. We also establish an analogous result for projective Kähler manifolds of semi-positive holomorphic sectional curvature based on an invariant which records the largest codimension of maximal subspaces in the tangent spaces on which the holomorphic sectional curvature vanishes. In particular, the latter result confirms a conjecture of S.-T. Yau in the projective case.
Double Shuffle Relations for Refined Symmetric Multiple Zeta Values
Authors: Hirose Minoru
Published: January 2023

Symmetric multiple zeta values (SMZVs) are elements in the ring of all multiple zeta values modulo the ideal generated by

\zeta(2)

introduced by Kaneko–Zagier as counterparts of finite multiple zeta values. It is known that symmetric multiple zeta values satisfy double shuffle relations and duality relations. In this paper, we construct certain lifts of SMZVs which live in the ring generated by all multiple zeta values and

2\pi i

as certain iterated integrals on

\mathbb{P}^1\setminus\{0,1,\infty\}

along a certain closed path. We call these lifted values refined symmetric multiple zeta values (RSMZVs). We show double shuffle relations and duality relations for RSMZVs. These relations are refinements of the double shuffle relations and the duality relations of SMZVs. Furthermore, we compare RSMZVs to other variants of lifts of SMZVs. Especially, we prove that RSMZVs coincide with Bachmann–Takeyama–Tasaka's

\xi

-values [
H. Bachmann
et al., Compos. Math. 154, No. 12, 2701–2721 (2018; Zbl 1429.11161)].
Frobenius and Spherical Codomains and Neighbourhoods
Authors: Hochenegger Andreas , Meachan Ciaran
Published: January 2023

Given an exact functor between triangulated categories which admits both adjoints and whose cotwist is either zero or an autoequivalence, we show how to associate a unique full triangulated subcategory of the codomain on which the functor becomes either Frobenius or spherical, respectively. We illustrate our construction with examples coming from projective bundles and smooth blowups. This work generalises results about spherical subcategories obtained by Martin Kalck, David Ploog and the first author.
Involutions of Azumaya Algebras
Authors: First Uriya A. , Williams Ben
Published: January 2023

We consider the general circumstance of an Azumaya algebra

A

of degree

n

over a locally ringed topos

(\mathbf{X}, \mathcal O_\mathbf{X})

where the latter carries a (possibly trivial) involution, denoted

\lambda

. This generalizes the usual notion of involutions of Azumaya algebras over schemes with involution, which in turn generalizes the notion of involutions of central simple algebras. We provide a criterion to determine whether two Azumaya algebras with involutions extending

\lambda

are locally isomorphic, describe the equivalence classes obtained by this relation, and settle the question of when an Azumaya algebra

A

is Brauer equivalent to an algebra carrying an involution extending

\lambda

, by giving a cohomological condition. We remark that these results are novel even in the case of schemes, since we allow ramified, non-trivial involutions of the base object. We observe that, if the cohomological condition is satisfied, then

A

is Brauer equivalent to an Azumaya algebra of degree

2n

carrying an involution. By comparison with the case of topological spaces, we show that the integer

2n

is minimal, even in the case of a nonsingular affine variety

X

with a fixed-point free involution. As an incidental step, we show that if

R

is a commutative ring with involution for which the fixed ring

S

is local, then either

R

is local or

R/S

is a quadratic étale extension of rings.
The integral cohomology algebras of ordered configuration spaces of spheres
Authors: Feichtner Eva Maria , Ziegler Günter M.
Published: January 2023

We compute the cohomology algebras of spaces of ordered point configurations on spheres,

F(S^k,n)

, with integer coefficients. For

k=2

we describe a product structure that splits

F(S^2,n)

into well-studied spaces. For

k>2

we analyze the spectral sequence associated to a classical fiber map on the configuration space. In both cases we obtain a complete and explicit description of the integer cohomology algebra of

F(S^k,n)

in terms of generators, relations and linear bases. There is 2-torsion occuring if and only if

k

is even. We explain this phenomenon by relating it to the Euler classes of spheres. Our rather classical methods uncover combinatorial structures at the core of the problem.