bluuebunny/crossref_metadata_2025 huggingface dataset:
Pfaffian quartic surfaces and representations of Clifford algebras
Authors: Coskun Emre , Kulkarni Rajesh S. , Mustopa Yusuf
Published: January 2023

Given a general ternary form

f=f(x_1,x_2,x_3)

of degree 4 over an algebraically closed field of characteristic zero, we use the geometry of K3 surfaces and van den Bergh's correspondence between representations of the generalized Clifford algebra

C_f

associated to

f

and Ulrich bundles on the surface

X_f:={w^4=f(x_1,x_2,x_3)} \subseteq {P}^3

to construct a positive-dimensional family of 8-dimensional irreducible representations of

C_f.

The main part of our construction, which is of independent interest, uses recent work of Aprodu-Farkas on Green's Conjecture together with a result of Basili on complete intersection curves in

{P}^3

to produce simple Ulrich bundles of rank 2 on a smooth quartic surface

X \subseteq {P}^3

with determinant

O_X(3).

This implies that every smooth quartic surface in

{P}^3

is the zerolocus of a linear Pfaffian, strengthening a result of Beauville-Schreyer on general quartic surfaces.
Large parallel volumes of finite and compact sets in d-dimensional Euclidean space
Authors: Kampf J. , Kiderlen M.
Published: January 2023

The

r

-parallel volume

V(C_r)

of a compact subset

C

in

d

-di­men­sional Euclidean space is the volume of the set

C_r

of all points of Euclidean distance at most

r>0

from

C

. According to Steiner's formula,

V(C_r)

is a polynomial in

r

when

C

is convex. For finite sets

C

satisfying a certain geometric condition, a Laurent expansion of

V(C_r)

for large

r

is obtained. The dependence of the coefficients on the geometry of

C

is explicitly given by so-called intrinsic power volumes of

C

. In the planar case such an expansion holds for all finite sets

C

. Finally, when

C

is a compact set in arbitrary dimension, it is shown that the difference of large

r

-parallel volumes of

C

and of its convex hull behaves like

cr^{d-3}

, where

c

is an intrinsic power volume of

C

.
Framed sheaves over treefolds and symmetric obstruction theories
Authors: Oprea Dragos
Published: January 2023

We note that open moduli spaces of sheaves over local Calabi-Yau surface geometries framed along the divisor at infinity admit symmetric perfect obstruction theories. We calculate the corresponding Donaldson-Thomas weighted Euler characteristics (as well as the topological Euler characteristics). Furthermore, for blowup geometries, we discuss the contribution of exceptional curves.
The polytope of all triangulations of a point configuration
Authors: De Loera Jesús A. , Hosten Serkan , Santos Francisco , Sturmfels Bernd
Published: January 2023

We study the convex hull

P_{\cal A}

of the 0-1 incidence vectors of all triangulations of a point configuration

{\cal A}

. This was called the universal polytope in citeBIFIST. The affine span of

P_{\cal A}

is described in terms of the co-circuits of the oriented matroid of

{\cal A}

. Its intersection with the positive orthant is a quasi-integral polytope

Q_{\cal A}

whose integral hull equals

P_{\cal A}

. We present the smallest example where

Q_{\cal A}

and

P_{\cal A}

differ. The duality theory for regular triangulations in citeBIGEST is extended to cover all triangulations. We discuss potential applications to enumeration and optimization problems regarding all triangulations.
Erratum for “Galois algebras, Hasse principle and induction-restriction methods” [cf. Documenta Math. 16 (2011), 677–707]
Authors: Bayer-Fluckiger E. , Parimala R.
Published: January 2023

This is a correction to [BP 11] E. Bayer-Fluckiger, R. Parimala, Galois algebras, Hasse principle and induction–restriction methods, Documenta Math. 16 (2011), 677–707.
Nekovár duality over p-adic Lie extensions of global fields
Authors: Lim Meng Fai , Sharifi Romyar T.
Published: January 2023

Tate duality is a Pontryagin duality between the

i

th Galois cohomology group of the absolute Galois group of a local field with coefficents in a finite module and the

(2-i)

th cohomology group of the Tate twist of the Pontryagin dual of the module. Poitou-Tate duality has a similar formulation, but the duality now takes place between Galois cohomology groups of a global field with restricted ramification and compactly-supported cohomology groups. Nekovár proved analogues of these in which the module in question is a finitely generated module

T

over a complete commutative local Noetherian ring

R

with a commuting Galois action, or a bounded complex thereof, and the Pontryagin dual is replaced with the Grothendieck dual

T^*

, which is a bounded complex of the same form. The cochain complexes computing the Galois cohomology groups of

T

and

T^*(1)

are then Grothendieck dual to each other in the derived category of finitely generated

R

-modules. Given a

p

-adic Lie extension of the ground field, we extend these to dualities between Galois cochain complexes of induced modules of

T

and

T^*(1)

in the derived category of finitely generated modules over the possibly noncommutative Iwasawa algebra with

R

-coefficients.
An equivariant main conjecture in Iwasawa theory and the Coates-Sinnott conjecture
Authors: Taleb Reza
Published: January 2023

We formulate and prove an Equivariant Main Conjecture (EMC) for

\it

all prime numbers

p

under the assumptions

\mu = 0

and the validity of the 2-adic Main Conjecture in Iwasawa theory citeWi. This equivariant version coincides with the version, which Ritter and Weiss formulated and proved for odd

p

under the assumption

\mu=0

in citeRW2. Our proof combines the approach of Ritter and Weiss with ideas and techniques used by Greither and Popescu in citeGP2 in a recent proof of an equivalent formulation of the above EMC under the same assumptions (

p

odd and

\mu=0

) as in citeRW2. As an application of the EMC we prove the Coates-Sinnott Conjecture, again assuming

\mu=0

and the 2-adic Main Conjecture.
Random walks in compact groups
Authors: Varjú P.P.
Published: January 2023

Let

X_1,X_2,\ldots

be independent identically distributed random elements of a compact group

G

. We discuss the speed of convergence of the law of the product

X_l\cdots X_1

to the Haar measure. We give poly-log estimates for certain finite groups and for compact semi-simple Lie groups. We improve earlier results of Solovay, Kitaev, Gamburd, Shahshahani and Dinai.
Torelli theorem for the Deligne–Hitchin moduli space. II
Authors: Biswas Indranil , Gómez Tomás L.
Published: January 2023

Let

X

and

X'

be compact Riemann surfaces of genus at least three. Let

G

and

G'

be nontrivial connected semisimple linear algebraic groups over

\mathbb C

. If some components

\mathcal M_{\mathrm{DH}}^d(X,G)

and

\mathcal M_{\mathrm{DH}}^{d'}(X',G')

of the associated Deligne–Hitchin moduli spaces are biholomorphic, then

X'

is isomorphic to

X

or to the conjugate Riemann surface

\overline X

.
On the multi-Koszul property for connected algebras.
Authors: Herscovich Estanislao
Published: January 2023

\small

In this article we introduce the notion of emphmulti-Koszul algebra for the case of a locally finite dimensional nonnegatively graded connected algebra, as a generalization of the notion of (generalized) Koszul algebras defined by R. Berger for homogeneous algebras. This notion also extends and generalizes the one recently introduced by the author and A. Rey, which was for the particular case of algebras further assumed to be finitely generated in degree 1 and with a finite dimensional space of relations. The idea of this new notion for this generality, which should be perhaps considered as a probably interesting common property for several of these algebras, was to find a grading independent description of some of the more appealing features shared by all generalized Koszul algebras. It includes several new interesting examples, textite.g. the super Yang-Mills algebras introduced by M. Movshev and A. Schwarz, which are not generalized Koszul or even multi-Koszul for the previous definition given by the author and Rey in any natural manner. On the other hand, we provide an equivalent description of the new definition in terms of the textrmTor (or textrmExt) groups, similar to the existing one for homogeneous algebras, and we show that several of the typical homological computations performed for the generalized Koszul algebras are also possible in this more general setting. In particular, we give a very explicit description of the

A_{\infty}

-algebra structure of the Yoneda algebra of a multi-Koszul algebra, which has a similar pattern as for the case of generalized Koszul algebras. We also show that a finitely generated multi-Koszul algebra with a finite dimensional space of relations is a

K_{2}

algebra in the sense of T. Cassidy and B. Shelton.
Unitary cycles on Shimura curves and the Shimura lift. I
Authors: Sankaran Siddarth
Published: January 2023

This paper concerns two families of divisors, which we call the 'orthogonal' and 'unitary' special cycles, defined on integral models of Shimura curves. The orthogonal family was studied extensively by Kudla-Rapoport-Yang, who showed that they are closely related to the Fourier coefficients of modular forms of weight 3/2, while the unitary divisors are analogues of cycles appearing in more recent work of Kudla-Rapoport on unitary Shimura varieties. Our main result relates these two families by (a formal version of) the Shimura lift.
Minimization of the energy of the nonrelativistic one-electron Pauli-Fierz model over quasifree states
Authors: Bach Volker , Breteaux Sébastien
Published: January 2023

In this article the existence of a minimizer for the energy for the nonrelativistic one-electron Pauli-Fierz model within the class of quasifree states is established. To this end it is shown that the minimum of the energy on quasifree states coincides with the minimum of the energy on pure quasifree states, where existence and uniqueness of a minimizer holds. Infrared and ultraviolet cutoffs are assumed, along with sufficiently small coupling constant and momentum of the dressed electron. A perturbative expression of the minimum of the energy on quasifree states for a small momentum of the dressed electron and small coupling constant is given. We also express the Lagrange equation for the minimizer in terms of the generalized one particle density matrix of the pure quasifree state.
Non-supramenable groups acting on locally compact spaces
Authors: Kellerhals Julian , Monod Nicolas , Rørdam Mikael
Published: January 2023

Supramenability of groups is characterised in terms of invariant measures on locally compact spaces. This opens the door to constructing interesting crossed product

C^*

-algebras for non-supramenable groups. In particular, stable Kirchberg algebras in the UCT class are constructed using crossed products for both amenable and non-amenable groups.
Counting polynomials for linear codes, hyperplane arrangements, and matroids
Authors: Plesken Wilhelm , Bächler Thomas
Published: January 2023

Thomas-decomposition of a polynomial systems and the resulting counting polynomials are applied to the theory of linear codes, hyperplane arrangements, and vector matroids to reinterpret known polynomials such as characteristic polynomials and weight enumerator, to introduce a new polynomial counting the matrices defining the same matroid, and to introduce the concept of essential flats revealing a structure which allows to rewrite the rank generating polynomial as a sum of products of univariate polynomials. Our concepts make no essential distinction between finite and infinite fields.
A global quantum duality principle for subgroups and homogeneous spaces
Authors: Ciccoli Nicola , Gavarini Fabio
Published: January 2023

For a complex or real algebraic group

G

, with

\mathfrak{g} := \mathrm{Lie}(G)

, quantizations of
global
type are suitable Hopf algebras

F_q[G]

or

U_q(\mathfrak{g})

over

{C}\big[q,q^{-1}\big]

. Any such quantization yields a structure of Poisson group on

G

, and one of Lie bialgebra on

\mathfrak{g}

: correspondingly, one has dual Poisson groups

G^*

and a dual Lie bialgebra

\mathfrak{g}^*

. In this context, we introduce suitable notions of
quantum subgroup
and, correspondingly, of
quantum homogeneous space
, in three versions:
weak
,
proper
and
strict
(also called
flat
in the literature). The last two notions only apply to those subgroups which are coisotropic, and those homogeneous spaces which are Poisson quotients; the first one instead has no restrictions whatsoever.
Trace class operators, regulators, and assembly maps in $K$-theory
Authors: Cortiñas Guillermo , Tartaglia Gisela
Published: January 2023

Let

G

be a group and let

KH

be homotopy algebraic

K

-theory. We prove that if

G

satisfies the rational

KH

isomorphism conjecture for the group algebra

L^1[G]

with coefficients in the algebra of trace-class operators in Hilbert space, then it also satisfies the

K

-theoretic Novikov conjecture for the group algebra over the integers, and the rational injectivity part of the Farrell-Jones conjecture with coefficients in any number field.
A conductor formula for completed group algebras
Authors: Nickel Andreas
Published: January 2023

Let

\mathfrak o

be the ring of integers in a finite extension of

\mathbb Q_p

. If

G

is a finite group and

\Gamma

is a maximal

\mathfrak o

-order containing the group ring

\mathfrak o[G]

, Jacobinski's conductor formula gives a complete description of the central conductor of

\Gamma

into

\mathfrak o [G]

in terms of characters of

G

. We prove a similar result for completed group algebras

\mathfrak o [[G]]

, where

G

is a

p

-adic Lie group of dimension 1. We will also discuss several consequences of this result.
Torsion in the crystalline cohomology of singular varieties
Authors: Bhatt Bhargav
Published: January 2023

This paper discusses some examples showing that the crystalline cohomology of even very mildly singular projective varieties tends to be quite large. In particular, any singular projective variety with at worst ordinary double points has infinitely generated crystalline cohomology in at least two cohomological degrees. These calculations rely critically on comparisons between crystalline and derived de Rham cohomology.
Characterisation of the Berkovich spectrum of the Banach algebra of bounded continuous functions
Authors: Mihara Tomoki
Published: January 2023

For a complete valuation field

k

and a topological space

X

, we prove the universality of the underlying topological space of the Berkovich spectrum of the Banach

k

-algebra

\mathrm{C}_{\mathrm{bd}}(X,k)

of bounded continuous

k

-valued functions on

X

. This result yields three applications: a partial solution to an analogue of Kaplansky conjecture for the automatic continuity problem over a local field, comparison of two ground field extensions of

\mathrm{C}_{\mathrm{bd}}(X,k)

, and non-Archimedean Gel'fand theory.
Hyperbolic manifolds of small volume
Authors: Belolipetsky Mikhail , Emery Vincent
Published: January 2023

We conjecture that for every dimension

n \neq 3

there exists a noncompact hyperbolic

n

-manifold whose volume is smaller than the volume of any compact hyperbolic

n

-manifold. For dimensions

n \le 4

and

n = 6

this conjecture follows from the known results. In this paper we show that the conjecture is true for arithmetic hyperbolic

n

-manifolds of dimension

n\ge 30

.
Fano threefolds with 2-torus action
Authors: Süß Hendrik
Published: January 2023

Following the work of Altmann and Hausen we give a combinatorial description for smooth Fano threefolds admitting a 2-torus action. We show that a whole variety of properties and invariants can be read off from this description. As an application we prove and disprove the existence of Kähler-Einstein metrics for some of these Fano threefolds, calculate their Cox rings and some of their toric canonical degenerations.
Cluster equivalence and graded derived equivalence.
Authors: Amiot Claire , Oppermann Steffen
Published: January 2023

In this paper we introduce a new approach for organizing algebras of global dimension at most 2. We introduce the notion of cluster equivalence for these algebras, based on whether their generalized cluster categories are equivalent. We are particularly interested in the question how much information about an algebra is preserved in its generalized cluster category, or, in other words, how closely two algebras are related if they have equivalent generalized cluster categories. Our approach makes use of the cluster-tilting objects in the generalized cluster categories: We first observe that cluster-tilting objects in generalized cluster categories are in natural bijection with cluster-tilting subcategories of derived categories, and then prove a recognition theorem for the latter. Using this recognition theorem we give a precise criterion when two cluster equivalent algebras are derived equivalent. For a given algebra we further describe all the derived equivalent algebras which have the same canonical cluster tilting object in their generalized cluster category. Finally we show that in general, if two algebras are cluster equivalent, then (under certain conditions) the algebras can be graded in such a way that the categories of graded modules are derived equivalent. To this end we introduce mutation of graded quivers with potential, and show that this notion reflects mutation in derived categories.
Classification of normal toric varieties over a valuation ring of rank one
Authors: Gubler Walter , Soto Alejandro
Published: January 2023

Normal toric varieties over a field or a discrete valuation ring are classified by rational polyhedral fans. We generalize this classification to normal toric varieties over an arbitrary valuation ring of rank one. The proof is based on a generalization of Sumihiro's theorem to this non-noetherian setting. These toric varieties play an important role for tropicalizations.
Operational $K$-theory
Authors: Anderson Dave , Payne Sam
Published: January 2023

We study the operational bivariant theory associated to the covariant theory of Grothendieck groups of coherent sheaves, and prove that it has many geometric properties analogous to those of operational Chow theory. This operational

K

-theory agrees with Grothendieck groups of vector bundles on smooth varieties, admits a natural map from the Grothen­dieck group of perfect complexes on general varieties, satisfies descent for Chow envelopes, and is

{A}^1

-homotopy invariant.

Furthermore, we show that the operational

K

-theory of a complete linear variety is dual to the Grothendieck group of coherent sheaves. As an application, we show that the

K

-theory of perfect complexes on any complete toric threefold surjects onto this group. Finally we identify the equivariant operational

K

-theory of an arbitrary toric variety with the ring of integral piecewise exponential functions on the associated fan.
On the construction of the Kan loop group
Authors: Waldhausen Friedhelm
Published: January 2023

A re-make, not of the construction, but of its description.
Deriving Auslander's formula
Authors: Krause Henning
Published: January 2023

Auslander's formula shows that any abelian category

\mathsf C

is equivalent to the category of coherent functors on

\mathsf C

modulo the Serre subcategory of all effaceable functors. We establish a derived version of this equivalence. This amounts to showing that the homotopy category of injective objects of some appropriate Grothendieck abelian category (the category of ind-objects of

\mathsf C

) is compactly generated and that the full subcategory of compact objects is equivalent to the bounded derived category of

\mathsf C

. The same approach shows for an arbitrary Grothendieck abelian category that its derived category and the homotopy category of injective objects are well-generated triangulated categories. For sufficiently large cardinals

\alpha

we identify their

\alpha

-compact objects and compare them.
On two higher Chow groups of schemes over a finite field
Authors: Kondo Satoshi , Yasuda Seidai
Published: January 2023

Given a separated scheme

X

of finite type over a finite field, its higher Chow groups

CH_{-1}(X, 1)

and

CH_{-2}(X, 3)

are computed explicitly.
Reductions of Galois representations for slopes in $(1,2)$
Authors: Bhattacharya Shalini , Ghate Eknath
Published: January 2023

We describe the semisimplifications of the mod

p

reductions of certain crystalline two-dimensional local Galois representations of slopes in

(1,2)

and all weights. The proof uses the compatibility between the

p

-adic and mod

p

Local Langlands Correspondences for

\operatorname{GL}_2(\mathbb Q_p)

. We also give a complete description of the submodules generated by the second highest monomial in the mod

p

symmetric power representations of

{\operatorname{GL}}_2(\mathbb F_p)

.
On the algebraicity of special $L$-values of Hermitian modular forms.
Authors: Bouganis Thanasis
Published: January 2023

In this work we prove some results on the algebraicity of special

L

-values attached to Hermitian modular forms. Our work is based on techniques developed by Goro Shimura in his book «Arithmeticity in the Theory of Automorphic Forms», and our results are in some cases complementary to results obtained previously by Michael Harris on the same question.
Classification of categorical subspaces of locally noetherian schemes
Authors: Kanda Ryo
Published: January 2023

We classify the prelocalizing subcategories of the category of quasi-coherent sheaves on a locally noetherian scheme. In order to give the classification, we introduce the notion of a local filter of subobjects of the structure sheaf. The essential part of the argument is given as results on a Grothendieck category with certain properties. We also classify the localizing subcategories, the closed subcategories, and the bilocalizing subcategories in terms of filters.
Triple Massey products over global fields
Authors: Minác Ján , Tân Nguyên Duy
Published: January 2023

Let

K

be a global field which contains a primitive

p

-th root of unity, where

p

is a prime number. M. J. Hopkins and K. G. Wickelgren showed that for

p=2

, any triple Massey product over

K

with respect to

\mathbb F_p

, contains 0 whenever it is defined. We show that this is true for all primes

p

.
On a conjecture of Izhboldin on similarity of quadratic forms
Authors: Hoffmann Detlev W.
Published: January 2023

In his paper

\it

Motivic equivalence of quadratic forms, Izhboldin modifies a conjecture of Lam and asks whether two quadratic forms, each of which isomorphic to the product of an Albert form and a

k

-fold Pfister form, are similar provided they are equivalent modulo

I^{k+3}

. We relate this conjecture to another conjecture on the dimensions of anisotropic forms in

I^{k+3}

. As a consequence, we obtain that Izhboldin's conjecture is true for

k\leq 1

.
On $F$-crystalline representations
Authors: Cais Bryden , Liu Tong
Published: January 2023

We extend the theory of Kisin modules and crystalline representations to allow more general coefficient fields and lifts of Frobenius. In particular, for a finite and totally ramified extension

F/\mathbb Q_p

, and an arbitrary finite extension

K/F

, we construct a general class of infinite and totally wildly ramified extensions

K_\infty/K

so that the functor

V\mapsto V|_{G_{K_\infty}}

is fully-faithfull on the category of

F

-crystalline representations

V

. We also establish a new classification of

F

-Barsotti–Tate groups via Kisin modules of height 1 which allows more general lifts of Frobenius.
Quadratic and symmetric bilinear forms on modules with unique base over a semiring
Authors: Izhakian Zur , Knebusch Manfred , Rowen Louis
Published: January 2023

We study quadratic forms on free modules with unique base, the situation that arises in tropical algebra, and prove the analog of Witt's Cancelation Theorem. Also, the tensor product of an indecomposable bilinear module

(U, \gamma)

with an indecomposable quadratic module

(V,q)

is indecomposable, with the exception of one case, where two indecomposable components arise.
Outer automorphisms of algebraic groups and a Skolem-Noether theorem for Albert algebras
Authors: Garibaldi Skip , Petersson Holger P.
Published: January 2023

The question of existence of outer automorphisms of a simple algebraic group

G

arises naturally both when working with the Galois cohomology of

G

and as an example of the algebro-geometric problem of determining which connected components of

Aut(G)

have rational points. The existence question remains open only for four types of groups, and we settle one of the remaining cases, type ^3

D

_4. The key to the proof is a Skolem-Noether theorem for cubic étale subalgebras of Albert algebras which is of independent interest. Necessary and sufficient conditions for a simply connected group of outer type

A

to admit outer automorphisms of order 2 are also given.
A characterization of semiprojectivity for subhomogeneous $C^*$-algebras
Authors: Enders Dominic
Published: January 2023

We study semiprojective, subhomogeneous

C^*

-algebras and give a detailed description of their structure. In particular, we find two characterizations of semiprojectivity for subhomogeneous mbox

C^*

-algebras: one in terms of their primitive ideal spaces and one by means of special direct limit structures over one-dimensional NCCW complexes. These results are obtained by working out several new permanence results for semiprojectivity, including a complete description of its behavior with respect to extensions by homogeneous mbox

C^*

-algebras.
A new discriminant algebra construction
Authors: Biesel Owen , Gioia Alberto
Published: January 2023

A discriminant algebra operation sends a commutative ring

R

and an

R

-algebra

A

of rank

n

to an

R

-algebra

\Delta

_A/R of rank 2 with the same discriminant bilinear form. Constructions of discriminant algebra operations have been put forward by Rost, Deligne, and Loos. We present a simpler and more explicit construction that does not break down into cases based on the parity of

n

. We then prove properties of this construction, and compute some examples explicitly.
Categorified trace for module tensor categories over braided tensor categories
Authors: Henriques André , Penneys David , Tener James
Published: January 2023

Given a braided pivotal category

\mathcal C

and a pivotal module tensor category

\mathcal M

, we define a functor

\operatorname{Tr}_\mathcal C:\mathcal M \to \mathcal C

, called the associated categorified trace. By a result of Bezrukavnikov, Finkelberg and Ostrik, the functor

\operatorname{Tr}_{\mathcal C}

comes equipped with natural isomorphisms

\tau_{x,y}:\operatorname{Tr}_{\mathcal C}(x \otimes y) \to \operatorname{Tr}_{\mathcal C}(y \otimes x)

, which we call the traciators. This situation lends itself to a diagramatic calculus of 'strings on cylinders', where the traciator corresponds to wrapping a string around the back of a cylinder. We show that

\operatorname{Tr}_{\mathcal C}

in fact has a much richer graphical calculus in which the tubes are allowed to branch and braid. Given algebra objects

A

and

B

, we prove that

\operatorname{Tr}_{\mathcal C}(A)

and

\operatorname{Tr}_{\mathcal C}(A \otimes B)

are again algebra objects. Moreover, provided certain mild assumptions are satisfied,

\operatorname{Tr}_{\mathcal C}(A)

and

\operatorname{Tr}_{\mathcal C}(A \otimes B)

are semisimple whenever

A

and

B

are semisimple.
Special modules over positively based algebras
Authors: Kildetoft Tobias , Mazorchuk Volodymyr
Published: January 2023

We use the Perron-Frobenius Theorem to define, study and, in some sense, classify special simple modules over arbitrary finite dimensional positively based algebras. For group algebras of finite Weyl groups with respect to the Kazhdan-Lusztig basis, this agrees with Lusztig's notion of a special module introduced in [Lu1].
Twistor spaces with a pencil of fundamental divisors
Authors: Kreußler B.
Published: January 2023

In this paper simply connected twistor spaces

Z

containing a pencil of fundamental divisors are studied. The Riemannian base for such spaces is diffeomorphic to the connected sum

n\mathbb{CP}^2

. We obtain for

n\ge 5

a complete description of the set of curves intersecting the fundamental line bundle

K^-\frac12

negatively. For this purpose we introduce a combinatorial structure, called
blow-up graph
. We show that for generic

S\in\mid-\frac12 K\mid

the algebraic dimension can be computed by the formula

a(Z)=1+\kappa^{-1}(S)

. A detailed study of the anti Kodaira dimension

\kappa^{-1}(S)

of rational surfaces permits to read off the algebraic dimension from the blow-up graphs. This gives a characterisation of Moishezon twistor spaces by the structure of the corresponding blow-up graphs. We study the behaviour of these graphs under small deformations. The results are applied to prove the main existence result, which states that every blow-up graph belongs to a fundamental divisor of a twistor space. We show, furthermore, that a twistor space with

\dim\mid-\frac12 K\mid=3

is a LeBrun space [LeB2]. We characterise such spaces also by the property to contain a smooth rational non-real curve

C

with

C.(-\frac12 K)=2-n

.
Rigidity and Frobenius Structure
Authors: Crew Richard
Published: January 2023

We show that an irreducible ordinary differential equation on the projective line has a Frobenius structure for a power of some prime

p

if it is rigid in the sense of Katz and satisfies some other reasonable (and necessary) conditions relative to the prime

p

.
Automorphisms of a Symmetric Product of a Curve (with an Appendix by Najmuddin Fakhruddin)
Authors: Biswas Indranil , Gómez Tomás L.
Published: January 2023

Let

X

be an irreducible smooth projective curve of genus

g>2

defined over an algebraically closed field of characteristic different from two. We prove that the natural homomorphism from the automorphisms of

X

to the automorphisms of the symmetric product

\mathrm{Sym}^d(X)

is an isomorphism if

d>2g-2

. In an appendix, Fakhruddin proves that the isomorphism class of the symmetric product of a curve determines the isomorphism class of the curve.
Iwasawa theory and explicit reciprocity law. A remake of an article of P. Colmez
Authors: Perrin-Riou Bernadette
Published: January 2023

Let

V

be a crystalline

p

-adic representation of the absolute Galois group of

\mathbb Q_p

. The author has built the Iwasawa theory of such a representation in Invent. Math (1994) and conjectured a reciprocity law which has been proved by P. Colmez. In this text, we write the initial construction with simplification and the proof of P. Colmez in a different language. This point of view will allow us to study the universal norms in the geometric cohomology classes associated to

V

by Bloch and Kato in a forthcoming article.
Homological Stability of Automorphism Groups of Quadratic Modules and Manifolds
Authors: Friedrich Nina
Published: January 2023

We prove homological stability for both general linear groups of modules over a ring with finite stable rank and unitary groups of quadratic modules over a ring with finite unitary stable rank. In particular, we do not assume the modules and quadratic modules to be well-behaved in any sense: for example, the quadratic form may be singular. This extends results by van der Kallen and Mirzaii-van der Kallen respectively. Combining these results with the machinery introduced by Galatius-Randal-Williams to prove homological stability for moduli spaces of simply-connected manifolds of dimension

2n \geq 6

, we get an extension of their result to the case of virtually polycyclic fundamental groups.
Deformation Theory with Homotopy Algebra Structures on Tensor Products
Authors: Robert-Nicoud Daniel
Published: January 2023

In order to solve two problems in deformation theory, we establish natural structures of homotopy Lie algebras and of homotopy associative algebras on tensor products of algebras of different types and on mapping spaces between coalgebras and algebras. When considering tensor products, such algebraic structures extend the Lie algebra or associative algebra structures that can be obtained by means of the Manin products of operads. These new homotopy algebra structures are proven to be compatible with the concepts of homotopy theory:

\infty

-morphisms and the Homotopy Transfer Theorem. We give a conceptual interpretation of their Maurer-Cartan elements. In the end, this allows us to construct the deformation complex for morphisms of algebras over an operad and to represent the deformation

\infty

-groupoid for differential graded Lie algebras.
Relative Property (T) for Nilpotent Subgroups
Authors: Chatterji Indira , Morris Dave Witte , Shah Riddhi
Published: January 2023

We show that relative Property (T) for the abelianization of a nilpotent normal subgroup implies relative Property (T) for the subgroup itself. This and other results are a consequence of a theorem of independent interest, which states that if

H

is a closed subgroup of a locally compact group

G

, and

A

is a closed subgroup of the center of

H

, such that

A

is normal in

G

, and

(G/A, H/A)

has relative Property (T), then

(G, H^{(1)})

has relative Property (T), where

H^{(1)}

is the closure of the commutator subgroup of

H

. In fact, the assumption that

A

is in the center of

H

can be replaced with the weaker assumption that

A

is abelian and every

H

-invariant finite measure on the unitary dual of

A

is supported on the set of fixed points.
On Endoscopic $p$-Adic Automorphic Forms for $\mathrm{SL}_2$
Authors: Ludwig Judith
Published: January 2023

We show the existence of some non-classical cohomological

p

-adic automorphic eigenforms for

\mathrm{SL}_2

using endoscopy and the geometry of eigenvarieties. These forms seem to account for some non-automorphic members of classical global

L

-packets.
Quasi-Homogeneity of the Moduli Space of Stable Maps to Homogeneous Spaces
Authors: Bärligea Christoph Mark
Published: January 2023

Let

G

be a connected, simply connected, simple, complex, linear algebraic group. Let

P

be an arbitrary parabolic subgroup of

G

. Let

X=G/P

be the

G

-homogeneous projective space attached to this situation. Let

d\in H_2(X)

be a degree. Let

\overline{M}_{0,3}(X,d)

be the (coarse) moduli space of three pointed genus zero stable maps to

X

of degree

d

. We prove under reasonable assumptions on

d

that

\overline{M}_{0,3}(X,d)

is quasi-homogeneous under the action of

G

. The essential assumption on

d

is that

d

is a minimal degree, i.e. that

d

is a degree which is minimal with the property that

q^d

occurs with non-zero coefficient in the quantum product

\sigma_u\star\sigma_v

of two Schubert classes

\sigma_u

and

\sigma_v

, where

\star

denotes the product in the (small) quantum cohomology ring

QH^\ast(X)

attached to

X

. We prove our main result on quasi-homogeneity by constructing an explicit morphism which has a dense open

G

-orbit in

\overline{M}_{0,3}(X,d)

. To carry out the construction of this morphism, we develop a combinatorial theory of generalized cascades of orthogonal roots which is interesting in its own right.
On Exact Pleijel's Constant for Some Domains
Authors: Bobkov Vladmir
Published: January 2023

We provide an explicit expression for the Pleijel constant for the planar disk and some of its sectors, as well as for

N

-dimensional rectangles. In particular, the Pleijel constant for the disk is equal to 0.4613019... Also, we characterize the Pleijel constant for some rings and annular sectors in terms of asymptotic behavior of zeros of certain cross-products of Bessel functions.
Abstract $\ell$-Adic 1-Motives and Tate's Canonical Class for Number Fields
Authors: Greither Cornelius , Popescu Cristian D.
Published: January 2023

In an earlier paper we constructed a new class of Iwasawa modules as

\ell

-adic realizations of waht we called abstract

\ell

-adic 1-motives in the number field setting. We proved in loc. cit. that the new Iwasawa modules satisfy an equivariant main conjecture. In this paper we link the new modules to the

\ell

-adified Tate canonical class, defined by Tate in 1960 [Ta1] and give an explicit construction of (the minus part og)

\ell

-adic Tate sequences for any Galois CM extension

K/k

of an arbitrary totally real number field

k

. These explicit constructions are significant and useful in their own right but also due to their applications (via results in [GP2]) to a proof of the minus part of the far reaching Equivariant Tamagawa Number Conjecture for the Artin motive associated to the Galois extension

K/k

.
Relative Homological Algebra via Truncations
Authors: Chachólski Wojciech , Neeman Amnon , Pitsch Wolfgang , Scherer Jérôme
Published: January 2023

To do homological algebra with unbounded chain complexes one needs to first find a way of constructing resolutions. Spaltenstein solved this problem for chain complexes of

R

-modules by truncating further and further to the left, resolving the pieces, and gluing back the partial resolutions. Our aim is to give a homotopy theoretical interpretation of this procedure, which may be extended to a relative setting. We work in an arbitrary abelian category

\mathcal{A}

and fix a class of “injective objects”

\mathcal{I}

. We show that Spaltenstein's construction can be captured by a pair of adjoint functors between unbounded chain complexes and towers of non-positively graded ones. This pair of adjoint functors forms what we call a Quillen pair and the above process of truncations, partial resolutions, and gluing, gives a meaningful way to resolve complexes in a relative setting up to a split error term. In order to do homotopy theory, and in particular to construct a well behaved relative derived category

D(\mathcal{A};\mathcal{I})

, we need more: the split error term must vanish. This is the case when

\mathcal I

is the class of all injective

R

-modules but not in general, not even for certain classes of injectives modules over a Noetherian ring. The key property is a relative analogue of Roos's AB4

^\ast

-

n

axiom for abelian categories. Various concrete examples such as Gorenstein homological algebra and purity are also discussed.
On Vector-Valued Siegel Modular Forms of Degree 2 and Weight $(j,2)$
Authors: Cléry Fabien , van der Geer Gerard
Published: January 2023

We formulate a conjecture that describes the vector-valued Siegel modular forms of degree 2 and level 2 of weight

\mathrm{Sym}^j\otimes\mathrm{det}^2

and provide some evidence for it. We construct such modular forms of weight

(j,2)

via covariants of binary sextics and calculate their Fourier expansions illustrating the effectivity of the approach via covariants. Two appendices contain related results of Chenevier; in particular a proof of the fact that every modular form of degree 2 and level 2 and weight

(j,1)

vanishes.
Walls in Milnor Fiber Complexes
Authors: Miller Alexander R.
Published: January 2023

For a real reflection group the reflecting hyperplanes cut out on the unit sphere a simplicial complex called the Coxeter complex. Abramenko showed that each reflecting hyperplane meets the Coxeter complex in another Coxeter complex if and only if the Coxeter diagram contains no subdiagram of type

D_4

,

F_4

or

H_4

. The present paper extends Abramenko's result to a wider class of complex reflection groups. These groups have a Coxeter-like presentation and a Coxeter-like complex called the Milnor fiber complex. Our first main theorem classifies the groups whose reflecting hyperplanes meet the Milnor fiber complex in another Milnor fiber complex. To understand better the walls that fail to be Milnor fiber complexes we introduce Milnor walls. Our second main theorem generalizes Abramenko's result in a second way. It says that each wall of a Milnor fiber complex is a Milnor wall if and only if the diagram contains no subdiagram of type

D_4

,

F_4

, or

H_4

.
A subshift of finite type in the Takens-Bogdanov bifurcation with $D_3$ symmetry
Authors: Matthies Karsten
Published: January 2023

We study the versal unfolding of a vector field of codimension two, that has an algebraically double eigenvalue 0 in the linearisation of the origin and is equivariant under a representation of the symmetry group

D_3

. A subshift of finite type is encountered near a clover of homoclinic orbits. The subshift encodes the itinerary along the three different homoclinic orbits. In this subshift all those symbol sequences are realized for which consecutive symbols are different. In the parameter space we also locate a transcritical, three different Hopf and two global (homoclinic) bifurcations.
The zeta Functions of Moduli Stacks of $G$-Zips and Moduli Stacks of Truncated Barsotti-Tate Groups
Authors: Lopuhaä-Zwakenberg Milan
Published: January 2023

We study stacks of truncated Barsotti-Tate groups and the

G

-zips defined by Pink, Wedhorn & Ziegler. The latter occur naturally when studying truncated Barsotti-Tate groups of height

1

with additional structure. By studying objects over finite fields and their automorphisms we determine the zeta functions of these stacks. These zeta functions can be expressed in terms of the Weyl group of the reductive group

G

and its action on the root system. The main ingredients are the classification of

G

-zips over algebraically closed fields and their automorphism groups by Pink, Wedhorn & Ziegler, and the study of truncated Barsotti-Tate groups and their automorphism groups by Gabber & Vasiu.
Subgroups of $\mathrm{Spin}(7)$ or $\mathrm{SO}(7)$ with Each Element Conjugate to Some Element of $\mathrm{G}_2$ and Applications to Automorphic Forms
Authors: Chenevier Gaëtan
Published: January 2023

As is well-known, the compact groups

\mathrm{Spin}(7)

and

\mathrm{SO}(7)

both have a single conjugacy class of compact subgroups of exceptional type

\mathrm{G}_2

. We first show that if

\Gamma

is a subgroup of

\mathrm{Spin}(7)

, and if each element of

\Gamma

is conjugate to some element of

\mathrm{G}_2

, then

\Gamma

itself is conjugate to a subgroup of

G_2

. The analogous statement for

\mathrm{SO}(7)

turns out be false, and our main result is a classification of all the exceptions. They are the following groups, embedded in each case in

\mathrm{SO}(7)

in a very specific way:

\mathrm{GL}_2(\mathbb{Z}/3\mathbb{Z})

,

\mathrm{SL}_2(\mathbb{Z}/3\mathbb{Z})

,

\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}

, as well as the nonabelian subgroups of

\mathrm{GO}_2(\mathbb{C})

with compact closure, similitude factors group

\{\pm 1\}

, and which are not isomorphic to the dihedral group of order 8. More generally, we consider the analogous problems in which the Euclidean space is replaced by a quadratic space of dimension 7 over an arbitrary field. This type of questions naturally arises in some formulation of a converse statement of Langlands' global functoriality conjecture, to which the results above have thus some applications. Moreover, we give necessary and sufficient local conditions on a cuspidal algebraic regular automorphic representation of

\mathrm{GL}_7

over a totally real number field so that its associated

\ell

-adic Galois representations can be conjugate into

\mathrm{G}_2(\overline{\mathbb{Q}_\ell})

. We provide 11 examples over

\mathbb{Q}

which are unramified at all primes.
Galois-Module Theory for Wildly Ramified Covers of Curves over Finite Fields (with an Appendix by Bernhard Köck and Adriano Marmora)
Authors: Fischbacher-Weitz Helena , Köck Bernhard , Marmora Adriano
Published: January 2023

Given a Galois cover of curves over

\mathbb{F}_p

, we relate the

p

-adic valuation of epsilon constants appearing in functional equations of Artin L-functions to an equivariant Euler characteristic. Our main theorem generalises a result of Chinburg from the tamely to the weakly ramified case. We furthermore apply Chinburg's result to obtain a 'weak' relation in the general case. In the Appendix, we study, in this arbitrarily wildly ramified case, the integrality of

p

-adic valuations of epsilon constants.
The Spectral Side of Stable Local Trace Formula
Authors: Mok Chung Pang , Peng Zhifeng
Published: January 2023

Let

G

be a connected quasi-split reductive group over

\mathbb{R}

, and more generally, a quasi-split

K

-group over

\mathbb{R}

. Arthur had obtained the formal formula for the spectral side of the stable local trace formula, by using formal substitute of Langlands parameters. In this paper, we construct the spectral side of the stable local trace formula and endoscopic local trace formula directly for quasi-split

K

-groups over

\mathbb{R}

, by incorporating the works of Shelstad. In particular we give the explicit expression for the spectral side of the stable local trace formula, in terms of Langlands parameters.
Smooth Duals of Inner Forms of $\mathrm{GL}_n$ and $\mathrm{SL}_n$
Authors: Aubert Anne-Marie , Baum Paul , Plymen Roger , Solleveld Maarten
Published: January 2023

Let

F

be a non-archimedean local field. We prove that every Bernstein component in the smooth dual of each inner form of the general linear group

\mathrm{GL}_n(F)

is canonically in bijection with the extended quotient for the action, given by Bernstein, of a finite group on a complex torus. For inner forms of

\mathrm{SL}_n(F)

we prove that each Bernstein component is canonically in bijection with the associated twisted extended quotient. In both cases, the bijections satisfy naturality properties with respect to the tempered dual, parabolic induction, central character, and the local Langlands correspondence.
Oriented Cohomology Sheaves on Double Moment Graphs
Authors: Devyatov Rostislav , Lanini Martina , Zainoulline Kirill
Published: January 2023

In the present paper we extend the theory of sheaves on moment graphs due to Braden–MacPherson and Fiebig to the context of an arbitrary oriented equivariant cohomology

\mathtt h

(e.g. to algebraic cobordism). We introduce and investigate structure

\mathtt h

-sheaves on double moment graphs to describe equivariant oriented cohomology of products of flag varieties. We show that in the case of a total flag variety

X

of Dynkin type

A

the space of global sections of the double structure

\mathtt h

-sheaf also describes the endomorphism ring of the equivariant

\mathtt h

-motive of

X

.
On an Analogue of the Conjecture of Birch and Swinnerton-Dyer for Abelian Schemes over Higher Dimensional Bases over Finite Fields
Authors: Keller Timo
Published: January 2023

We formulate an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields of characteristic

p

. We prove the prime-to-

p

part conditionally on the finiteness of the

p

-primary part of the Tate-Shafarevich group or the equality of the analytic and the algebraic rank. If the base is a product of curves, Abelian varieties and K3 surfaces, we prove the prime-to-

p

part of the conjecture for constant or isoconstant Abelian schemes, in particular the prime-to-

p

part for (1) relative elliptic curves with good reduction or (2) Abelian schemes with constant isomorphism type of

\mathscr{A}[p]

or (3) Abelian schemes with supersingular generic fibre, and the full conjecture for relative elliptic curves with good reduction over curves and for constant Abelian schemes over arbitrary bases. We also reduce the conjecture to the case of surfaces as the basis.
On the Average Number of 2-Selmer Elements of Elliptic Curves over $\mathbb F_q(X)$ with Two Marked Points
Authors: Thorne Jack A.
Published: January 2023

We consider elliptic curves over global fields of positive characteristic with two distinct marked non-trivial rational points. Restricting to a certain subfamily of the universal one, we show that the average size of the 2-Selmer groups of these curves exists, in a natural sense, and equals 12. Along the way, we consider a map from these 2-Selmer groups to the moduli space of

G

-torsors over an algebraic curve, where

G

is isogenous to

\mathrm{SL}_2^4

, and show that the images of 2-Selmer elements under this map become equidistributed in the limit.
Integral representation formulas for Cartan domains
Authors: Affane Atallah
Published: January 2023

For a bounded, symmetric and circled domain

D

in

\mathbf{C}^{n},\;