bluuebunny/crossref_metadata_2025 huggingface dataset:
Algebraic groups of type $D_4$, triality, and composition algebras.
Authors: Chernousov Vladimir , Elduque Alberto , Knus Max-Albert
Published: January 2023

Conjugacy classes of outer automorphisms of order 3 of simple algebraic groups of classical type

D_4

are classified over arbitrary fields. There are two main types of conjugacy classes. For one type the fixed algebraic groups are simple of type

G_2

; for the other type they are simple of type

A_2

when the characteristic is different from 3 and are not smooth when the characteristic is 3. A large part of the paper is dedicated to the exceptional case of characteristic 3. A key ingredient of the classification of conjugacy classes of trialitarian automorphisms is the fact that the fixed groups are automorphism groups of certain composition algebras.
The variety of polar simplices
Authors: Ranestad Kristian , Schreyer Frank-Olaf
Published: January 2023

A collection of n distinct hyperplanes

L_i = {l_i=0} \subset {\mathbf P}^{n-1}

, the

(n-1)

-dimensional projective space over an algebraically closed field of characteristic not equal to 2, is a polar simplex of a smooth quadric

Q^{n-2}={q=0}

, if each

L_i

is the polar hyperplane of the point

p_i = \bigcap_{j \ne i} L_j

, equivalently, if

q= l_1^2+\ldots+l_n^2

for suitable choices of the linear forms

l_i

. In this paper we study the closure

VPS(Q,n) \subset \mathrm{Hilb}_{n}(\check{\mathbf P}^{n-1})

of the variety of sums of powers presenting

Q

from a global viewpoint:

VPS(Q,n)

is a smooth Fano variety of index 2 and Picard number 1 when

n<6

, and

VPS(Q,n)

is singular when

n\geq 6

.
Projective varieties with bad semi-stable reduction at 3 only
Authors: Abrashkin Victor
Published: January 2023

Suppose

F=W(k)[1/p]

where

W(k)

is the ring of Witt vectors with coefficients in algebraically closed field

k

of characteristic

p\ne 2

. We construct integral theory of

p

-adic semi-stable representations of the absolute Galois group of

F

with Hodge-Tate weights from

[0,p)

. This modification of Breuil's theory results in the following application in the spirit of the Shafarevich Conjecture. If

Y

is a projective algebraic variety over

\mathbb Q

with good reduction modulo all primes

l\ne 3

and semi-stable reduction modulo 3 then for the Hodge numbers of

Y_C=Y\otimes _{\mathbb Q}\ C

, one has

h^2(Y_C)=h^{1,1}(Y_C)

.
A geometric quantization of the Kostant–Sekiguchi correspondence for scalar type unitary highest weight representations
Authors: Möllers Jan
Published: January 2023

For any Hermitian Lie group

G

of tube type we give a geometric quantization procedure of certain

K_{\mathbb C}

-orbits in

\mathfrak p_{\mathbb C}^*

to obtain all scalar type highest weight representations. Here

K_{\mathbb C}

is the complexification of a maximal compact subgroup

K\subseteq G

with corresponding Cartan decomposition

\mathfrak g=\mathfrak k+\mathfrak p

of the Lie algebra of

G

. We explicitly realize every such representation

\pi

on a Fock space consisting of square integrable holomorphic functions on its associated variety

\mathrm{Ass}(\pi)\subseteq\mathfrak p_{\mathbb C}^*

. The associated variety

\mathrm{Ass}(\pi)

is the closure of a single nilpotent

K_{\mathbb C}

-orbit

\mathcal O^{K_{\mathbb C}}\subseteq\mathfrak p_{\mathbb C}^*

which corresponds by the Kostant–Sekiguchi correspondence to a nilpotent coadjoint

G

-orbit

\mathcal O^G\subseteq\mathfrak g^*

. The known Schrödinger model of

\pi

is a realization on

L^2(\mathcal O)

, where

\mathcal O\subseteq\mathcal O^G

is a Lagrangian submanifold. We construct an intertwining operator from the Schrödinger model to the new Fock model, the generalized Segal–Bargmann transform, which gives a geometric quantization of the Kostant–Sekiguchi correspondence (a notion invented by Hilgert, Kobayashi, Ørsted and the author). The main tool in our construction are multivariable

I

- and

K

-Bessel functions on Jordan algebras which appear in the measure of

\mathcal O^{K_{\mathbb C}}

, as reproducing kernel of the Fock space and as integral kernel of the Segal–Bargmann transform. As a corollary to our construction we also obtain the integral kernel of the unitary inversion operator in the Schrödinger model in terms of a multivariable

J

-Bessel function as well as explicit Whittaker vectors.
On the cyclic homology of ringed spaces and schemes
Authors: Keller Bernhard
Published: January 2023

We prove that the cyclic homology of a scheme with an ample line bundle coincides with the cyclic homology of its category of algebraic vector bundles. As a byproduct of the proof, we obtain a new construction of the Chern character of a perfect complex on a ringed space.
Extending self-maps to projective space over finite fields
Authors: Poonen Bjorn
Published: January 2023

Using the closed point sieve, we extend to finite fields the following theorem proved by A. Bhatnagar and L. Szpiro over infinite fields: if

X

is a closed subscheme of

{P}^n

over a field, and

\phi \colon X \to X

satisfies

\phi^* \mathscr{O}_X(1) \simeq \mathscr{O}_X(d)

for some

d \ge 2

, then there exists

r \ge 1

such that

\phi^r

extends to a morphism

{P}^n \to {P}^n

.
The virtual Haken conjecture (with an appendix by Ian Agol, Daniel Groves and Jason Manning).
Authors: Agol Ian
Published: January 2023

We prove that cubulated hyperbolic groups are virtually special. The proof relies on results of Haglund and Wise which also imply that they are linear groups, and quasi-convex subgroups are separable. A consequence is that closed hyperbolic 3-manifolds have finite-sheeted Haken covers, which resolves the virtual Haken question of Waldhausen and Thurston's virtual fibering question. An appendix to this paper by Agol, Groves, and Manning proves a generalization of the main result of citeAGM08.
Essential Whittaker functions for $\mathrm{GL}(n)$
Authors: Matringe Nadir
Published: January 2023

We give a constructive proof of the existence of the essential Whittaker function of a generic representation of

GL(n,F)

, for

F

a non-archimedean local field, using mirabolic restriction techniques.
The zeta function of a finite category
Authors: Noguchi Kazunori
Published: January 2023

We define the zeta function of a finite category. We prove a theorem that states a relationship between the zeta function of a finite category and the Euler characteristic of finite categories, called the series Euler characteristic [BL08]. Moreover, it is shown that for a covering of finite categories,

P:E\to B

, the zeta function of

E

is that of

B

to the power of the number of sheets in the covering. This is a categorical analogue of the unproved conjecture of Dedekind for algebraic number fields and the Dedekind zeta functions.
Ramified Satake isomorphisms for strongly parabolic characters
Authors: Kamgarpour Masoud , Schedler Travis
Published: January 2023

For certain characters of the compact maximal torus of a reductive

p

-adic group, which we call strongly parabolic characters, we prove Satake-type isomorphisms. Our results generalize those of Satake, Howe, Bushnell and Kutzko, and Roche.
Stability of the tangent bundle of the wonderful compactification of an adjoint group
Authors: Biswas Indranil , Kannan S.Senthamarai
Published: January 2023

Let

G

be a complex linear algebraic group which is simple of adjoint type. Let

\overline G

be the wonderful compactification of

G

. We prove that the tangent bundle of

\overline G

is stable with respect to every polarization on

\overline G

.
Algebraic cycles and fibrations
Authors: Vial Charles
Published: January 2023

Let

f:X \rightarrow B

be a projective surjective morphism between quasi-projective varieties. The goal of this paper is the study of the Chow groups of

X

in terms of the Chow groups of

B

and of the fibres of

f

. One of the applications concerns quadric bundles. When

X

and

B

are smooth projective and when

f

is a flat quadric fibration, we show that the Chow motive of

X

is «built» from the motives of varieties of dimension less than the dimension of

B

.
Invariants of upper motives
Authors: Haution Olivier
Published: January 2023

Let

H

be a homology theory for algebraic varieties over a field

k

. To a complete

k

-variety

X

, one naturally attaches an ideal

H_{X}(k)

of the coefficient ring

H(k)

. We show that, when

X

is regular, this ideal depends only on the upper Chow motive of

X

. This generalises the classical results asserting that this ideal is a birational invariant of smooth varieties for particular choices of

H

, such as the Chow group. When

H

is the Grothendieck group of coherent sheaves, we obtain a lower bound on the canonical dimension of varieties. When

H

is the algebraic cobordism, we give a new proof of a theorem of Levine and Morel. Finally we discuss some splitting properties of geometrically unirational field extensions of small transcendence degree.
Densities of the Raney distributions
Authors: Mlotkowski Wojciech , Penson Karol A. , Życzkowski Karol
Published: January 2023

We prove that if

p\ge1

and

0< r\le p

then the sequence

\binom{mp+r}m\frac{r}{mp+r}

is positive definite. More precisely, it is the moment sequence of a probability measure

\mu(p,r)

with compact support contained in

[0,+\infty)

. This family of measures encompasses the multiplicative free powers of the Marchenko-Pastur distribution as well as the Wigner's semicircle distribution centered at

x=2

. We show that if

p>1

is a rational number and

0<r\le p

then

\mu(p,r)

is absolutely continuous and its density

W_{p,r}(x)

can be expressed in terms of the generalized hypergeometric functions. In some cases, including the multiplicative free square and the multiplicative free square root of the Marchenko-Pastur measure,

W_{p,r}(x)

turns out to be an elementary function.
Eisenstein series and the trace formula for $\mathrm{GL}(2)$ over a function field
Authors: Flicker Yuval Z.
Published: January 2023

We write out and prove the trace formula for a convolution operator on the space of cusp forms on

GL(2)

over the function field

F

of a smooth projective absolutely irreducible curve over a finite field. The proof – which follows Drinfeld – is complete and all terms in the formula are explicitly computed. The structure of the homogeneous space

\operatorname{GL}(2,F)\backslash \operatorname{GL}(2,\mathbb A)

is studied in section 2 by means of locally free sheaves of

\mathcal O_X

-modules. Section 3 deals with the regularization and computation of the geometric terms, over conjugacy classes. Section 4 develops the theory of intertwining operators and Eisenstein Series, and the trace formula is proven in section 5.
$E_n$-regularity implies $E_{n-1}$-regularity
Authors: Tabuada Gonçalo
Published: January 2023

Vorst and Dayton-Weibel proved that

K_n

-regularity implies

K_{n-1}

-regularity. In this article we generalize this result from (commutative) rings to differential graded categories and from algebraic

K

-theory to any functor which is Morita invariant, continuous, and localizing. Moreover, we show that regularity is preserved under taking desuspensions, fibers of morphisms, direct factors, and arbitrary direct sums. As an application, we prove that the above implication also holds for schemes. Along the way, we extend Bass' fundamental theorem to this broader setting and establish a Nisnevich descent result which is of independent interest.
An application of Hermitian K-theory: sums-of-squares formulas
Authors: Xie Heng
Published: January 2023

By using Hermitian

K

-theory, we improve D. Dugger and D. Isaksen's condition (some powers of 2 dividing some binomial coefficients) for the existence of sums-of-squares formulas.
A note on the global Langlands conjecture.
Authors: Lapid Erez M.
Published: January 2023

The theory of base change is used to give some new examples of the Global Langlands Conjecture. The Galois representations involved have solvable image and are not monomial, although some multiple of them in the Grothendieck group is monomial. Thus, it gives nothing new about Artin's Conjecture itself. An application is given to a question which arises in studying multiplicities of cuspidal representations of

SL_n

. We explain how the (conjectural) adjoint lifting can prove GLC for a family of representations containing the tetrahedral 2-dimensional ones.
On the integrality of modular symbols and Kato's Euler system for elliptic curves
Authors: Wuthrich Christian
Published: January 2023

Let

E/\mathbb Q

be an elliptic curve. We investigate the denominator of the modular symbols attached to

E

. We show that one can change the curve in its isogeny class to make these denominators coprime to any given odd prime of semi-stable reduction. This has applications to the integrality of Kato's Euler system and the main conjecture in Iwasawa theory for elliptic curves.
The Picard group of the compactified universal Jacobian
Authors: Melo Margarida , Viviani Filippo
Published: January 2023

We determine explicitly the Picard groups of the universal Jacobian stack and of its compactification over the stack of stable curves. Along the way, we prove some results concerning the gerbe structure of the universal Jacobian stack over its rigidification by the natural action of the multiplicative group and relate this with the existence of generalized Poincaré line bundles. We also compare our results with Kouvidakis-Fontanari computations of the divisor class group of the universal (compactified) Jacobian scheme.
Semistable modules over Lie algebroids in positive characteristic
Authors: Langer Adrian
Published: January 2023

We study Lie algebroids in positive characteristic and moduli spaces of their modules. In particular, we show a Langton's type theorem for the corresponding moduli spaces. We relate Langton's construction to Simpson's construction of gr-semistable Griffiths transverse filtration. We use it to prove a recent conjecture of Lan-Sheng-Zuo that semistable systems of Hodge sheaves on liftable varieties in positive characteristic are strongly semistable.
The local cohomology of the Jacobian ring
Authors: Sernesi Edoardo
Published: January 2023

We study the 0-th local cohomology module

H^0_{\mathbf{m}}(R(f))

of the jacobian ring

R(f)

of a singular reduced complex projective hypersurface

X

, by relating it to the sheaf of logarithmic vector fields along

X

. We investigate the analogies between

H^0_{\mathbf{m}}(R(f))

and the well known properties of the jacobian ring of a nonsingular hypersurface. In particular we study self-duality, Hodge theoretic and Torelli type questions for

H^0_{\mathbf{m}}(R(f))

.
Big de Rham-Witt cohomology: basic results
Authors: Chatzistamatiou Andre
Published: January 2023

Let

X

be a smooth projective

R

-scheme, where

R

is a smooth

\mathbb Z

-algebra. As constructed by Hesselholt, we have the absolute big de Rham–Witt complex

\mathbb W\Omega^*_X

of

X

at our disposal. There is also a relative version

\mathbb W\Omega^*_{X/R}

with

\mathbb W(R)

-linear differential. In this paper we study the hypercohomology of the relative (big) de Rham-Witt complex after truncation with finite truncation sets

S

. We show that it is a projective

\mathbb W_S(R)

-module, provided that the de Rham cohomology is a flat

R

-module. In addition, we establish a Poincaré duality theorem. explicit description of the relative de Rham–Witt complex of a smooth

\lambda

-ring, which may be of independent interest.
Partial classification of the Baumslag-Solitar group von Neumann algebras
Authors: Meesschaert Niels , Vaes Stefaan
Published: January 2023

We prove that the rational number

|n/m|

is an invariant of the group von Neumann algebra of the Baumslag–Solitar group

\operatorname{BS}(n,m)

. More precisely, if

L(\operatorname{BS}(n,m))

is isomorphic with

L(\operatorname{BS}(n',m'))

, then

|n'/m'| = |n/m|^{\pm 1}

. We obtain this result by associating to abelian, but not maximal abelian, subalgebras of a II

_1

factor, an equivalence relation that can be of type III. In particular, we associate to

L(\operatorname{BS}(n,m))

a canonical equivalence relation of type III

_{|n/m|}

.
Correction to the paper “Classical motivic polylogarithm according to Beilinson and Deligne”
Authors: Wildeshaus Jörg , Huber Annette
Published: January 2023

The Hodge theoretic appendix of our paper contains results (A.2.10, A.2.11) about the explicit shape of the groups of one-extensions in the category of Hodge structures. Regrettably, they are incorrect in general. Here, we give the right formulae. Since they coincide with the old ones for Hodge structures of strictly negative weights (e.g., Tate twists

A(n), n \ge 1)

, this correction has no effect on the main text of our paper.
On Galois representations of weight one
Authors: Wiese Gabor
Published: January 2023

A two-dimensional Galois representation into the Hecke algebra of Katz modular forms of weight one over a finite field of characteristic

p

is constructed and is shown to be unramified at

p

in most cases.
Special values of anticyclotomic Rankin–Selberg $L$-functions
Authors: Hsieh Ming-Lun
Published: January 2023

In this article, we construct a class of anticyclotomic

p

-adic Rankin–Selberg

L

-functions for Hilbert modular forms, generalizing the construction of Brakočevic, Bertolini, Darmon and Prasanna in the elliptic case. Moreover, building on works of Hida, we give a necessary and sufficient criterion for the vanishing of the Iwasawa

\mu

-invariant of this

p

-adic

L

-function vanishes and prove a result on the non-vanishing modulo

p

of central Rankin–Selberg

L

-values with anticyclotomic twists. These results have future applications to Iwasawa main conjecture for Rankin–Selberg convolution and Iwasawa theory for Heegner cycles.
Twisted $C^*$-algebras associated to finitely aligned higher-rank graphs
Authors: Sims Aidan , Whitehead Benjamin , Whittaker Michael F.
Published: January 2023

We introduce twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs and give a comprehensive treatment of their fundamental structural properties. We establish versions of the usual uniqueness theorems and the classification of gauge-invariant ideals. We show that all twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs are nuclear and satisfy the UCT, and that for twists that lift to real-valued cocycles, the

K

-theory of a twisted relative Cuntz-Krieger algebra is independent of the twist. In the final section, we identify a sufficient condition for simplicity of twisted Cuntz-Krieger algebras associated to higher-rank graphs which are not aperiodic. Our results indicate that this question is significantly more complicated than in the untwisted setting.
Local heights on Abelian varieties and rigid analytic uniformization
Authors: Werner Annette
Published: January 2023

We express classical and

p

-adic local height pairings on an abelian variety with split semistable reduction in terms of the corresponding pairings on the abelian part of the Raynaud extension (which has good reduction). Here we use an approach to height pairings via splittings of biextensions which is due to Mazur and Tate. We conclude with a formula comparing Schneider's

p

-adic height pairing to the

p

-adic height pairing in the semistable ordinary reduction case defined by Mazur and Tate.
Formality of derived intersections
Authors: Grivaux Julien
Published: January 2023

We study derived intersections of smooth analytic cycles, and provide in some cases necessary and sufficient conditions for this intersection be formal. In particular, if

X

is a complex submanifold of a complex manifold

Y

, we prove that

X

can be quantized if and only if the derived intersection of

X^2

and

\Delta_Y

is formal in

\mathrm{D}^{\mathrm{b}}\bigl (X^2 \bigr)

.
A remark on the gradient map
Authors: Biliotti L. , Ghigi A. , Heinzner P.
Published: January 2023

For a Hamiltonian action of a compact group

U

of isometries on a compact Kähler manifold

Z

and a compatible subgroup

G

of

U^{\mathbb C}

, we prove that for any closed

G

-invariant subset

Y\subset Z

the image of the gradient map

\mu_{\mathfrak p}(Y)

is independent of the choice of the invariant Kähler form

\omega

in its cohomology class

[\omega]

.
The pullbacks of principal coactions
Authors: Hajac Piotr M. , Wagner Elmar
Published: January 2023

We prove that the class of principal coactions is closed under one-surjective pullbacks in an appropriate category of algebras equipped with left and right coactions. This allows us to handle cases of

C^*

-algebras lacking two different non-trivial ideals. It also allows us to go beyond the category of comodule algebras. As an example of the former, we carry out an index computation for noncommutative line bundles over the standard Podle´s sphere using the Mayer-Vietoris-type arguments afforded by a one-surjective pullback presentation of the

C^*

-algebra of this quantum sphere. To instantiate the latter, we define a family of coalgebraic noncommutative deformations of the

\mathrm{U}(1)

-principal bundle

\mathrm{S}^7\rightarrow{C}\mathrm{P}^3

.
Motivic homology of semiabelian varieties
Authors: Sugiyama Rin
Published: January 2023

We generalize some classical results on the Chow group of an abelian variety to semiabelian varieties and to motivic (co)homology, using a result of Ancona–Enright-Ward–Huber [1] on a decomposition of the motive of a semiabelian variety in Voevodsky's category of motives.
Existence of stationary pulses for nonlocal reaction-diffusion equations
Authors: Volpert Vitaly , Vougalter Vitali
Published: January 2023

A nonlocal reaction-diffusion equation and a system of equations from population dynamics are considered on the whole axis. Existence of solutions in the form of stationary pulses is proved by a perturbation method. It is based on spectral properties of the linearized operators and on the implicit function theorem.
The rational homotopy of the $K(2)$-local sphere and the chromatic splitting conjecture for the prime 3 and level 2
Authors: Goerss Paul G. , Henn Hans-Werner , Mahowald Mark
Published: January 2023

We calculate the rational homotopy of the

K(2)

-local sphere

L_{K(2)}S^0

at the prime 3 and confirm Hopkins' chromatic splitting conjecture for

p=3

and

n=2

.
Asymptotics of spherical superfunctions on rank one Riemannian symmetric superspaces
Authors: Alldridge Alexander , Palzer Wolfgang
Published: January 2023

We compute the Harish-Chandra

c

-function for a generic class of rank-one purely non-compact Riemannian symmetric superspaces

X=G/K

in terms of Euler

\\Gamma

functions, proving that it is meromorphic. Compared to the even case, the poles of the

c

-function are shifted into the right half-space. We derive the full asymptotic Harish-Chandra series expansion of the spherical superfunctions on

X

. In the case where the multiplicity of the simple root is an even negative number, they have a closed expression as Jacobi polynomials for an unusual choice of parameters.
Martingale inequalities and operator space structures on $L_p$
Authors: Pisier Gilles
Published: January 2023

We describe a new operator space structure on

L_p

when

p

is an even integer and compare it with the one introduced in our previous work using complex interpolation. For the new structure, the Khintchine inequalities and Burkholder's martingale inequalities have a very natural form:the span of the Rademacher functions is completely isomorphic to the operator Hilbert space

OH

, and the square function of a martingale difference sequence

d_n

is

\Sigma d_n\otimes \bar d_n

. Various inequalities from harmonic analysis are also considered in the same operator valued framework. Moreover, the new operator space structure also makes sense for non-commutative

L_p

-spaces associated to a trace with analogous results. When

p\to \infty

and the trace is normalized, this gives us a tool to study the correspondence

E\mapsto \underline{E}

defined as follows: if

E\subset B(H)

is a completely isometric emdedding then

\underline{E}

is defined so that

\underline{E}\subset CB(OH)

is also one.
Spectral triples and finite summability on Cuntz–Krieger algebras
Authors: Goffeng Magnus , Mesland Bram
Published: January 2023

We produce a variety of odd bounded Fredholm modules and odd spectral triples on Cuntz–Krieger algebras by means of realizing these algebras as “the algebra of functions on a non-commutative space” coming from a sub shift of finite type. We show that any odd

K

-homology class can be represented by such an odd bounded Fredholm module or odd spectral triple. The odd bounded Fredholm modules that are constructed are finitely summable. The spectral triples are

\theta

-summable, although their phases will already on the level of analytic

K

-cycles be finitely summable bounded Fredholm modules. Using the unbounded Kasparov product, we exhibit a family of generalized spectral triples, related to work of Bellissard–Pearson, possessing mildly unbounded commutators, whilst still giving well defined

K

-homology classes.
Topology and purity for torsors
Authors: Antieau Benjamin , Williams Ben
Published: January 2023

We study the homotopy theory of the classifying space of the complex projective linear groups to prove that purity fails for

\operatorname{PGL}_p

-torsors on regular noetherian schemes when

p

is a prime. Extending our previous work when

p=2

, we obtain a negative answer to a question of Colliot-Thélène and Sansuc, for all

\operatorname{PGL}_p

. We also give a new example of the failure of purity for the cohomological filtration on the Witt group, which is the first example of this kind of a variety over an algebraically closed field.
Equivariant Fredholm modules for the full quantum flag manifold of $\mathrm{SU}_q(3)$
Authors: Voigt Christian , Yuncken Robert
Published: January 2023

We introduce

C^*

-algebras associated to the foliation structure of a quantum flag manifold. We use these to construct

{SL}_q(3,{C})

-equivariant Fredholm modules for the full quantum flag manifold

X_q = {SU}_q(3)/T

of

{SU}_q(3)

, based on an analytical version of the Bernstein-Gelfand-Gelfand complex. As a consequence we deduce that the flag manifold

X_q

satisfies Poincaré duality in equivariant

KK

-theory. Moreover, we show that the Baum-Connes conjecture with trivial coefficients holds for the discrete quantum group dual to

{SU}_q(3)

.
Invariant blocks under coprime actions
Authors: Malle Gunter , Navarro Gabriel , Späth Britta
Published: January 2023

If a finite group

A

acts coprimely as automorphisms on a finite group

G

, then the

A

-invariant Brauer

p

-blocks of

G

are exactly those that contain

A

-invariant irreducible characters.
Logarithmic bundles of multi-degree arrangements in $\Bbb P^n$
Authors: Angelini Elena
Published: January 2023

Let

D = {D_{1}, \ldots, D_{\ell}}

be a multi-degree arrangement with normal crossings on the complex projective space

{P}^n

, with degrees

d_{1}, \ldots, d_{\ell}

; let

\Omega_{{P}^n}^1(\log D)

be the logarithmic bundle attached to it. First we prove a Torelli type theorem when

D

has a sufficiently large number of components by recovering them as unstable smooth irreducible degree-

d_{i}

hypersurfaces of

\Omega_{{P}^n}^1(\log D)

. Then, when

n = 2

, by describing the moduli spaces containing

\Omega_{{P}^2}^1(\log D)

, we show that arrangements of a line and a conic, or of two lines and a conic, are not Torelli. Moreover we prove that the logarithmic bundle of three lines and a conic is related with the one of a cubic. Finally we analyze the conic-case.
The Eisenstein ideal and Jacquet-Langlands isogeny over function fields
Authors: Papikian Mihran , Wei Fu-Tsun
Published: January 2023

Let

\mathfrak p

and

\mathfrak q

be two distinct prime ideals of

\mathbb{F}_q[T]

. We use the Eisenstein ideal of the Hecke algebra of the Drinfeld modular curve

X_0(\mathfrak p\mathfrak q)

to compare the rational torsion subgroup of the Jacobian

J_0(\mathfrak p\mathfrak q)

with its subgroup generated by the cuspidal divisors, and to produce explicit examples of Jacquet-Langlands isogenies. Our results are stronger than what is currently known about the analogues of these problems over

\mathbb Q

.
Motivic equivalence of quadratic forms
Authors: Izhboldin Oleg T.
Published: January 2023

Let

X_\phi

and

X_\psi

be projective quadrics corresponding to quadratic forms

\phi

and

\psi

over a field

F

. If

X_\phi

is isomorphic to

X_\psi

in the category of Chow motives, we say that

\phi

and

\psi

are motivic isomorphic and write

\phi\stackrel{m}\sim\psi

. We show that in the case of odd-dimensional forms the condition

\phi\stackrel{m}\sim\psi

is equivalent to the similarity of

\phi

and

\psi

. After this, we discuss the case of even-dimensional forms. In particular, we construct examples of generalized Albert forms

q_1

and

q_2

such that

q_1\stackrel{m}\sim q_2

and

q_1\not\sim q_2

.
On a canonical class of Green currents for the unit sections of abelian schemes
Authors: Maillot V. , Rössler D.
Published: January 2023

We show that on any abelian scheme over a complex quasi-projective smooth variety, there is a Green current for the zero-section, which is axiomatically determined up to

\partial

and

\bar\partial

-exact differential forms. On an elliptic curve, this current specialises to a Siegel function. We prove generalisations of classical properties of Siegel functions, like distribution relations and reciprocity laws. Furthermore, as an application of a refined version of the arithmetic Riemann-Roch theorem, we show that the above current, when restricted to a torsion section, is the realisation in analytic Deligne cohomology of an element of the (Quillen)

K_1

group of the base, the corresponding denominator being given by the denominator of a Bernoulli number. This generalises the second Kronecker limit formula and the denominator 12 computed by Kubert, Lang and Robert in the case of Siegel units. Finally, we prove an analog in Arakelov theory of a Chern class formula of Bloch and Beauville, where the canonical current plays a key role.
Hessian ideals of a homogeneous polynomial and generalized Tjurina algebras
Authors: Dimca Alexandru , Sticlaru Gabriel
Published: January 2023

Using the minors in Hessian matrices, we introduce new graded algebras associated to a homogeneous polynomial. When the associated projective hypersurface has isolated singularities, these algebras are related to some new local algebras associated to isolated hypersurface singularities, which generalize their Tjurina algebras. One consequence of our results is a new very rapid way to determine the number of weighted homogeneous singularities of such a hypersurface.
Drinfeld centers for bicategories
Authors: Meir Ehud , Szymik Markus
Published: January 2023

We generalize Drinfeld's notion of the center of a tensor category to bicategories. In this generality, we present a spectral sequence to compute the basic invariants of Drinfeld centers: the abelian monoid of isomorphism classes of objects, and the abelian automorphism group of its identity object. There is an associated obstruction theory that explains the difference between the Drinfeld center and the center of the classifying category. For examples, we discuss bicategories of groups and bands, rings and bimodules, as well as fusion categories.
On the motive of a commutative algebraic group
Authors: Ancona Giuseppe , Enright-Ward Stephen , Huber Annette
Published: January 2023

We prove a canonical Künneth decomposition for the motive of a commutative group scheme over a field. Moreover, we show that this decomposition behaves under the group law just as in cohomology. We also deduce applications of the decomposition to the existence of a weight filtration, computation of any Weil cohomology theory and study of 1-motives.
Multiplicative structures on algebraic $K$-theory
Authors: Barwick Clark
Published: January 2023

The algebraic

K

-theory of Waldhausen

\infty

-categories is the functor corepresented by the unit object for a natural symmetric monoidal structure. We therefore regard it as the stable homotopy theory of homotopy theories. In particular, it respects all algebraic structures, and as a result, we obtain the Deligne Conjecture for this form of

K

-theory.
Stability of matter for the Hartree-Fock functional of the relativistic electron-positron field
Authors: Bach Volker , Barbaroux Jean-Marie , Helffer Bernard , Siedentop Heinz
Published: January 2023

We investigate stability of matter of the Hartree-Fock functional of the relativistic electron-positron field – in suitable second quantization – interacting via a second quantized Coulomb field and a classical magnetic field. We are able to show that stability holds for a range of nuclear charges

Z_1,..,Z_K\leq Z

and fine structure constants

\alpha

that include the physical value of

\alpha

and elements up to holmium (

Z=67

).
Equivariant embeddings of commutative linear algebraic groups of corank one
Authors: Arzhantsev Ivan , Kotenkova Polina
Published: January 2023

Let

\mathbb K

be an algebraically closed field of characteristic zero,

\mathbb G_m=(\mathbb K\setminus{0},\times)

be its multiplicative group, and

\mathbb G_a=(\mathbb K,+)

be its additive group. Consider a commutative linear algebraic group

\mathbb G=(\mathbb G_m)^r\times\mathbb G_a

. We study equivariant

\mathbb G

-embeddings, i.e. normal

\mathbb G

-varieties

X

containing

\mathbb G

as an open orbit. We prove that

X

is a toric variety and all such actions of

\mathbb G

on

X

correspond to Demazure roots of the fan of

X

. In these terms, the orbit structure of a

\mathbb G

-variety

X

is described.
Syzygies of 5-gonal canonical curves
Authors: Bopp Christian
Published: January 2023

We show that for 5-gonal curves of odd genus

g\ge 13

and even genus

g\ge 28

the

\lceil \frac{g-1}2\rceil

-th syzygy module of the curve is not determined by the syzygies of the scroll swept out by the special pencil of degree 5.
$\mathcal L$-invariant for Siegel-Hilbert forms
Authors: Rosso Giovanni
Published: January 2023

We prove a formula for the Greenberg–Benois

\mathcal L

-invariant of the spin, standard and adjoint Galois representations associated with Siegel–Hilbert modular forms. In order to simplify the calculation, we give a new definition of the

\mathcal L

-invariant for a Galois representation

V

of a number field

F\neq \mathbb Q

; we also check that it is compatible with Benois' definition for

\operatorname{Ind}_{F}^{\mathbb Q}(V)

.
Enumerating exceptional collections of line bundles on some surfaces of general type
Authors: Coughlan Stephen
Published: January 2023

We use constructions of surfaces as abelian covers to write down exceptional collections of line bundles of maximal length for every surface

X

in certain families of surfaces of general type with

p_{g}=0

and

K_{X}^{2}=3,4,5,6,8

. We also compute the algebra of derived endomorphisms for an appropriately chosen exceptional collection, and the Hochschild cohomology of the corresponding quasiphantom category. As a consequence, we see that the subcategory generated by the exceptional collection does not vary in the family of surfaces. Finally, we describe the semigroup of effective divisors on each surface, answering a question of Alexeev.
Singularities, double points, controlled topology and chain duality
Authors: Ranicki Andrew
Published: January 2023

A manifold is a Poincaré duality space without singularities. McCrory obtained a homological criterion of a global nature for deciding if a polyhedral Poincaré duality space is a homology manifold, i.e. if the singularities are homologically inessential. A homeomorphism of manifolds is a degree 1 map without double points. In this paper combinatorially controlled topology and chain complex methods are used to provide a homological criterion of a global nature for deciding if a degree 1 map of polyhedral homology manifolds has acyclic point inverses, i.e. if the double points are homologically inessential.
Product systems over Ore monoids
Authors: Albandik Suliman , Meyer Ralf
Published: January 2023

We interpret the Cuntz–Pimsner covariance condition as a nondegeneracy condition for representations of product systems. We show that Cuntz–Pimsner algebras over Ore monoids are constructed through inductive limits and section algebras of Fell bundles over groups. We construct a groupoid model for the Cuntz–Pimsner algebra coming from an action of an Ore monoid on a space by topological correspondences. We characterise when this groupoid is effective or locally contracting and describe its invariant subsets and invariant measures.
Milne's correcting factor and derived de Rham cohomology
Authors: Morin Baptiste
Published: January 2023

Milne's correcting factor is a numerical invariant playing an important role in formulas for special values of zeta functions of varieties over finite fields. We show that Milne's factor is simply the Euler characteristic of the derived de Rham complex (relative to Z) modulo the Hodge filtration.
Tame class field theory for singular varieties over algebraically closed fields
Authors: Geisser Thomas , Schmidt Alexander
Published: January 2023

Let

X

be a separated scheme of finite type over an algebraically closed field

k

and let

m

be a natural number. By an explicit geometric construction using torsors we construct a pairing between the first mod

m

Suslin homology and the first mod

m

tame étale cohomology of

X

. We show that the induced homomorphism from the mod

m

Suslin homology to the abelianized tame fundamental group of

X

mod

m

is surjective. It is an isomorphism of finite abelian groups if

(m, char(k)) = 1

, and for general

m

if resolution of singularities holds over

k

.
Lifting from two elliptic modular forms to Siegel modular forms of half-integral weight of even degree
Authors: Hayashida Shuichi
Published: January 2023

The aim of this paper is to construct lifts from two elliptic modular forms to Siegel modular forms of half-integral weight of even degree under the assumption that the constructed Siegel modular form is not identically zero. The key ingredient of the proof is a new Maass relation for Siegel modular forms of half-integral weight and any degree.
Cohomology of polynomial functiors on free groups
Authors: Djament Aurélien , Pirashvili Teimuraz , Vespa Christine
Published: January 2023

We show that extension groups between two polynomial functors on free groups are the same in the category of all functors and in a subcategory of polynomial functors of bounded degree. The proof relies on functorial properties of the group ring of free groups and its filtration by powers of the augmentation ideal. We give some applications, in particular in term of homological dimension.
Equivariant Poincaré series and topology of valuations
Authors: Campillo A. , Delgado F. , Gusein-Zade S.M.
Published: January 2023

The equivariant with respect to a finite group action Poincaré series of a collection of

r

valuations was defined earlier as a power series in

r

variables with the coefficients from a modification of the Burnside ring of the group. Here we show that (modulo simple exceptions) the equivariant Poincaré series determines the equivariant topology of the collection of valuations.
The isomorphism problem for semigroup $C^*$-algebras of right-angled Artin monoids
Authors: Eilers Søren , Li Xin , Ruiz Efren
Published: January 2023

Semigroup C*-algebras for right-angled Artin monoids were introduced and studied by Crisp and Laca. In the paper at hand, we are able to present the complete answer to their question of when such C*-algebras are isomorphic. The answer to this question is presented both in terms of properties of the graph defining the Artin monoids as well as in terms of classification by K-theory, and is obtained using recent results from classification of non-simple C*-algebras. Moreover, we are able to answer another natural question: Which of these semigroup C*-algebras for right-angled Artin monoids are isomorphic to graph algebras? We give a complete answer, and note the consequence that many of the C*-algebras under study are semiprojective.
Some endoscopic properties of the essentially tame Jacquet-Langlands correspondence
Authors: Tam Kam-Fai
Published: January 2023

Let

F

be a non-Archimedean local field of characteristic 0 and

G

be an inner form of the general linear group

G^*=GL

_n over

F

. We show that the rectifying character appearing in the essentially tame Jacquet-Langlands correspondence of Bushnell and Henniart for

G

and

G^*

can be factorized into a product of some special characters, called zeta-data in this paper, in the theory of endoscopy of Langlands and Shelstad. As a consequence, the essentially tame local Langlands correspondence for

G

can be described using admissible embeddings of L-tori.
Remarks on $L^p$-boundedness of wave operators for Schrödinger operators with threshold singularities
Authors: Yajima K.
Published: January 2023

We consider the continuity property in Lebesgue spaces

L^{p}(\mathbb R^{m})

of the wave operators

W_\pm

of scattering theory for Schrödinger operators

H=-\Delta + V

on

\mathbb R^m

,

|V(x)|\le C\langle x \rangle^{-\delta}

for some

\delta>2

when

H

is of exceptional type, i.e.

\mathcal N=\{u \in \langle x \rangle^{s} L^{2}(\mathbb R^{m}) \colon (1+ (-\Delta)^{-1}V)u=0 \}\not=\{0\}

for some

1/2<s<\delta-1/2

. It has recently been proved by Goldberg and Green for

m\ge 5

that

W_\pm

are in general bounded in

L^{p}(\mathbb R^{m})

for

1\le p<m/2

, for

1\le p<m

if all

\phi \in \mathcal N

satisfy

\int_{\mathbb R^{m}} V\phi dx=0

and, for

1\le p<\infty

if

\int_{\mathbb R^{m}} x_{i} V\phi dx=0, i=1, \dots, m

in addition. We make the results for

p>m/2

more precise and prove in particular that these conditions are also necessary for the stated properties of

W_\pm

. We also prove that, for

m=3, W_\pm

are bounded in

L^{p}(\mathbb R^{3})

for

1<p<3

and that the same holds for

1<p<\infty

if and only if all

\phi\in \mathcal N

satisfy

\int_{\mathbb R^{3}}V\phi dx=0

and

\int_{\mathbb R^{3}} x_{i} V\phi dx=0

,

i=1, 2, 3

, simultaneously.
Hyperelliptic Schottky problem and stable modular forms
Authors: Codogni Giulio
Published: January 2023

It is well known that, fixed an even, unimodular, positive definite quadratic form, one can construct a modular form in each genus; this form is called the theta series associated to the quadratic form. Varying the quadratic form, one obtains the ring of stable modular forms. We show that the differences of theta series associated to specific pairs of quadratic forms vanish on the locus of hyperelliptic Jacobians in each genus. In our examples, the quadratic forms have rank 24, 32 and 48. The proof relies on a geometric result about the boundary of the Satake compactification of the hyperelliptic locus. We also study the monoid formed by the moduli space of all principally polarised abelian varieties, the operation being the product of abelian varieties. We use this construction to show that the ideal of stable modular forms vanishing on the hyperelliptic locus in each genus is generated by differences of theta series.
Some results on Bessel functionals for GSp(4)
Authors: Roberts Brooks , Schmidt Ralf
Published: January 2023

We prove that every irreducible, admissible representation

\pi

of

\operatorname{GSp}(4,F)

, where

F

is a non-archimedean local field of characteristic zero, admits a Bessel functional, provided

\pi

is not one-dimensional. If

\pi

is not supercuspidal, we explicitly determine the set of all Bessel functionals admitted by

\pi

, and prove that Bessel functionals of a fixed type are unique. If

\pi

is supercuspidal, we do the same for all split Bessel functionals.
The local monodromy as a generalized algebraic correspondence (with an appendix by Spencer Bloch)
Authors: Consani Caterina
Published: January 2023

For an algebraic, normal-crossings degeneration over a local field the local monodromy operator and its powers naturally define Galois equivariant classes in the

\ell

-adic (middle dimensional) cohomology groups of some precise strata of the special fiber of a normal-crossings model associated to the fiber product degeneration. The paper addresses the question whether these classes are algebraic. It is shown that the answer is positive for any degeneration whose special fiber has (locally) at worst triple points singularities. These algebraic cycles are responsible for and they explain geometrically the presence of poles of local Euler L-factors at integers on the left of the left-central point.
On zeta elements for $\Bbb G_m$
Authors: Burns David , Kurihara Masato , Sano Takamichi
Published: January 2023

In this paper, we present a unifying approach to the general theory of abelian Stark conjectures. To do so we define natural notions of ‘zeta element’, of ‘Weil-étale cohomology complexes’ and of ‘integral Selmer groups’ for the multiplicative group

\mathbb G_m

over finite abelian extensions of number fields. We then conjecture a precise connection between zeta elements and Weil-étale cohomology complexes, we show this conjecture is equivalent to a special case of the equivariant Tamagawa number conjecture and we give an unconditional proof of the analogous statement for global function fields. We also show that the conjecture entails much detailed information about the arithmetic properties of generalized Stark elements including a new family of integral congruence relations between Rubin–Stark elements (that refines recent conjectures of Mazur and Rubin and of the third author) and explicit formulas in terms of these elements for the higher Fitting ideals of the integral Selmer groups of

G_{m}

, thereby obtaining a clear and very general approach to the theory of abelian Stark conjectures. As first applications of this approach, we derive, amongst other things, a proof of (a refinement of) a conjecture of Darmon concerning cyclotomic units, a proof of (a refinement of) Gross’s ‘Conjecture for Tori’ in the case that the base field is Q, explicit conjectural formulas for both annihilating elements and, in certain cases, the higher Fitting ideals (and hence explicit structures) of ideal class groups and a strong refinement of many previous results concerning abelian Stark conjectures.
$p$-adic L-functions of automorphic forms and exceptional zeros
Authors: Deppe Holger
Published: January 2023

We construct

p

-adic L-functions for automorphic representations of

\mathrm{GL}

_2 of a number field

F

, and show that the corresponding

p

-adic L-function of a modular elliptic curve

E

over

F

has an extra zero at the central point for each prime above

p

at which

E

has split multiplicative reduction, a part of the exceptional zero conjecture.
$\mathrm C^*$-algebras of Boolean inverse monoids -- traces and invariant means
Authors: Starling Charles
Published: January 2023

To a Boolean inverse monoid

S

we associate a universal C*-algebra

C^*_{B}(S)

and show that it is equal to Exel's tight C*-algebra of

S

. We then show that any invariant mean on

S

(in the sense of Kudryavtseva, Lawson, Lenz and Resende) gives rise to a trace on

C^*_{B}(S)

, and vice-versa, under a condition on

S

equivalent to the underlying groupoid being Hausdorff. Under certain mild conditions, the space of traces of

C^*_{B}(S)

is shown to be isomorphic to the space of invariant means of

S

. We then use many known results about traces of C*-algebras to draw conclusions about invariant means on Boolean inverse monoids; in particular we quote a result of Blackadar to show that any metrizable Choquet simplex arises as the space of invariant means for some AF inverse monoid

S

.
On geometric aspects of diffuse groups. With an appendix by Nathan Dunfield.
Authors: Kionke Steffen , Dunfield Jean Raimbault
Published: January 2023

Bowditch introduced the notion of diffuse groups as a geometric variation of the unique product property. We elaborate on various examples and non-examples, keeping the geometric point of view from Bowditch's paper. In particular, we discuss fundamental groups of flat and hyperbolic manifolds. Appendix refappendix settles an open question by providing an example of a group which is diffuse but not left-orderable.