### Iwasawa Theory and F-Analytic Lubin-Tate (\varphi,\Gamma)-Modules

Berger & Fourquaux
Let $K$ be a finite extension of $\Qp$. We use the theory of $(\varphi,\Gamma)$-modules in the Lubin-Tate setting to construct some corestriction-compatible families of classes in the cohomology of $V$, for certain representations $V$ of $\Gal(\Qpbar/K)$. If in addition $V$ is crystalline, we describe these classes explicitly using Bloch-Kato's exponential maps. This allows us to generalize Perrin-Riou's period map to the Lubin-Tate setting.

### Positivity of Line Bundles and Newton-Okounkov Bodies

Küronya & Lozovanu
The purpose of this paper is to describe asymptotic base loci of line bundles on projective varieties in terms of Newton--Okounkov bodies. As a result, we obtain equivalent characterizations of ampleness and nefness via convex geometry.

### Characterizations of Morse Quasi-Geodesics via Superlinear Divergence and Sublinear Contraction

Arzhantseva, Cashen, Gruber & Hume
We introduce and begin a systematic study of sublinearly contracting projections. We give two characterizations of Morse quasi-geodesics in an arbitrary geodesic metric space. One is that they are sublinearly contracting; the other is that they have completely superlinear divergence. We give a further characterization of sublinearly contracting projections in terms of projections of geodesic segments.

### A Cycle Class Map from Chow Groups with Modulus to Relative $K$-Theory

Binda
Let $\ol{X}$ be a smooth quasi-projective $d$-dimensional variety over a field $k$ and let $D$ be an effective, non-reduced, Cartier divisor on it such that its support is strict normal crossing. In this note, we construct cycle class maps from (a variant of) the higher Chow group with modulus of the pair $(\ol{X};D)$ in the range $(d+n, n)$ to the relative $K$-groups $K_n(\ol{X}; D)$ for every $n\geq 0$.

### On the Milnor Monodromy of the Exceptional Reflection Arrangement of Type $G_{31}$

Dimca & Sticlaru
Combining recent results by A. M\u acinic, S. Papadima and R. Popescu with a spectral sequence and computer aided computations, we determine the monodromy action on $H^1(F,\C)$, where $F$ denotes the Milnor fiber of the hyperplane arrangement associated to the exceptional irreducible complex reflection group $G_{31}$. This completes the description given by the first author of such monodromy operators for all the other irreducible complex reflection groups.

### The $L^2$-Torsion Polytope of Amenable Groups

Florian Funke
We introduce the notion of groups of polytope class and show that torsion-free amenable groups satisfying the Atiyah Conjecture possess this property. A direct consequence is the homotopy invariance of the $L^2$-torsion polytope among $G$-CW-complexes for these groups. As another application we prove that the $L^2$-torsion polytope of an amenable group vanishes provided that it contains a non-abelian elementary amenable normal subgroup.

### Lax Colimits and Free Fibrations in infty-Categories

Gepner, Haugseng & Nikolaus
We define and discuss lax and weighted colimits of diagrams in $\infty$-categories and show that the coCartesian fibration corresponding to a functor is given by its lax colimit. A key ingredient, of independent interest, is a simple characterization of the free Cartesian fibration on a functor of $\infty$-categories. As an application of these results, we prove that 2-representable functors are preserved under exponentiation, and also that the total space of a presentable Cartesian fibration between...

### Equivariant $A$-Theory

Cary Malkiewich & Mona Merling
We give a new construction of the equivariant $K$-theory of group actions (cf. Barwick et al.), producing an infinite loop $G$-space for each Waldhausen category with $G$-action, for a finite group $G$. On the category $R(X)$ of retractive spaces over a $G$-space $X$, this produces an equivariant lift of Waldhausen's functor $A(X)$, and we show that the $H$-fixed points are the bivariant $A$-theory of the fibration $X_{hH}\to BH$. We then use the framework of spectral...

### Differential Embedding Problems over Complex Function Fields

Bachmayr, Harbater, Hartmann & Wibmer
We introduce the notion of differential torsors, which allows the adaptation of constructions from algebraic geometry to differential Galois theory. Using these differential torsors, we set up a general framework for applying patching techniques in differential Galois theory over fields of characteristic zero. We show that patching holds over function fields over the complex numbers. As the main application, we prove the solvability of all differential embedding problems over complex function fields, thereby providing new...

### Bivariant Theories in Motivic Stable Homotopy

D\'Eglise
The purpose of this work is to study the notion of bivariant theory introduced by Fulton and MacPherson in the context of motivic stable homotopy theory, and more generally in the broader framework of the Grothendieck six functors formalism. We introduce several kinds of bivariant theories associated with a suitable ring spectrum, and we construct a system of orientations (called fundamental classes) for global complete intersection morphisms between arbitrary schemes. These fundamental classes satisfy all...

### On an Analogue of the Conjecture of Birch and Swinnerton-Dyer for Abelian Schemes over Higher Dimensional Bases over Finite Fields

Timo Keller
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...

### A Local--Global Principle for Symplectic $\mathrm K_2$

Lavrenov
We prove that an element of the relative symplectic Steinberg group $g\in\mathrm{StSp}_{2n}(R[t],\,tR[t])$ is trivial if and only if its image under any maximal localisation homomorphism is trivial.

### On the Boundary and Intersection Motives of Genus 2 Hilbert-Siegel Varieties

Mattia Cavicchi
We study genus 2 Hilbert-Siegel varieties, i.e. Shimura varieties $S_K$ corresponding to the group $GSp_{4,F}$ over a totally real field $F$, along with the relative Chow motives $^\lambda \mathcal{V}$ of abelian type over $S_K$ obtained from irreducible representations $V_{\lambda}$ of$GSp_{4,F}$. We analyse the weight filtration on the degeneration of such motives at the boundary of the Baily-Borel compactification and we find a criterion on the highest weight $\lambda$, potentially generalisable to other families of Shimura...

### Potentially Crystalline Lifts of Certain Prescribed Types

Gee , Herzig, Liu & Savitt
We prove several results concerning the existence of potentially crystalline lifts of prescribed Hodge--Tate weights and inertial types of a given representation $\rbar:GK\to\GLn(\Fpbar)$, where $K/\Qp$ is a finite extension. Some of these results are proved by purely local methods, and are expected to be useful in the application of automorphy lifting theorems. The proofs of the other results are global, making use of automorphy lifting theorems.

### On the Center-Valued Atiyah Conjecture for L^2-Betti Numbers

Knebusch, Linnell & Schick
The so-called Atiyah conjecture states that the $\NG$-dimensions of the $L2$-homology modules of finite free $G$-CW-complexes belong to a certain set of rational numbers, depending on the finite subgroups of $G$. In this article we extend this conjecture to a statement for the center-valued dimensions. We show that the conjecture is equivalent to a precise description of the structure as a semisimple Artinian ring of the division closure $D(\{Q}[G])$ of $\Q[G]$ in the ring of...

### Morita Theory for Hopf Algebroids, Principal Bibundles, and Weak Equivalences

Kaoutit & Kowalzig
We show that two flat commutative Hopf algebroids are Morita equivalent if and only if they are weakly equivalent and if and only if there exists a principal bibundle connecting them. This gives a positive answer to a conjecture due to Hovey and Strickland. We also prove that principal (left) bundles lead to a bicategory together with a 2-functor from flat Hopf algebroids to trivial principal bundles. This turns out to be the universal solution...

### The Theta Function and the Weyl Law on Manifolds Without Conjugate Points

Bonthonneau
We prove that the usual $\Theta$ function on a Riemannian manifold without conjugate points is uniformly bounded from below. This extends a result of Green in two dimensions. We deduce that the Bérard remainder in the Weyl law is valid for a manifold without conjugate points, without any restriction on the dimension.

### Topological Conjugacy of Topological Markov Shifts and Cuntz-Krieger Algebras

Matsumoto
For an irreducible non-permutation matrix $A$, the triplet $(\OA,\DA,\rhoA)$ for the Cuntz-Krieger algebra $\OA$, its canonical maximal abelian $C^*$-subalgebra $\DA$, and its gauge action $\rhoA$ is called the Cuntz--Krieger triplet. We introduce a notion of strong Morita equivalence in the Cuntz--Krieger triplets, and prove that two Cuntz--Krieger triplets $(\OA,\DA,\rhoA)$ and $(\OB,\DB,\rhoB)$ are strong Morita equivalent if and only if $A$ and $B$ are strong shift equivalent. We also show that the generalized gauge actions on...

### Cohomological Support and the Geometric Join

Dao & Sanders
Let $M,N$ be finitely generated modules over a local complete intersection $R$. Assume that all the modules $\tor^R_i(M,N)$ are zero for $i>0$. We prove that the cohomological support of $M\otimes_R N$ (in the sense of Avramov-Buchweitz) is equal to the geometric join of the cohomological supports of $M,N$. This result gives a new connection between two active areas or research, and immediately produces several interesting corollaries. Naturally, it also raises many intriguing new questions about...

### On the Rank of Universal Quadratic Forms over Real Quadratic Fields

Blomer & Kala
We study the minimal number of variables required by a totally positive definite diagonal universal quadratic form over a real quadratic field $\Q(\sqrt D)$ and obtain lower and upper bounds for it in terms of certain sums of coefficients of the associated continued fraction. We also estimate such sums in terms of $D$ and establish a link between continued fraction expansions and special values of $L$-functions in the spirit of Kronecker's limit formula.

### Spectral Asymptotics for the Schrödinger Operator on the Line with Spreading and Oscillating Potentials

Duchêne & Raymond
This study is devoted to the asymptotic spectral analysis of multiscale Schrödinger operators with oscillating and decaying electric potentials. Different regimes, related to scaling considerations, are distinguished. By means of a normal form filtrating most of the oscillations, a reduction to a non-oscillating effective Hamiltonian is performed.

### Limiting Absorption Principle for Schrödinger Operators with Oscillating Potentials

Jecko & Mbarek
Making use of the localised Putnam theory developed in [gj], we show the limiting absorption principle for Schrödinger operators with perturbed oscillating potential on appropriate energy intervals. We focus on a certain class of oscillating potentials (larger than the one in [gj2]) that was already studied in [bd,dmr,dr1,dr2,mu,ret1,ret2]. Allowing long-range and short-range components and local singularities in the perturbation, we improve known results. A subclass of the considered potentials actually cannot be treated by the...

### Quasi-Homogeneity of the Moduli Space of Stable Maps to Homogeneous Spaces

Baerligea
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...

### Some Non-Special Cubic Fourfolds

In \cite{rv}, Ranestad and Voisin showed, quite surprisingly, that the divisor in the moduli space of cubic fourfolds consisting of cubics ''apolar to a Veronese surface'' is not a Noether--Lefschetz divisor. We give an independent proof of this by exhibiting an explicit cubic fourfold $X$ in the divisor and using point counting methods over finite fields to show $X$ is Noether--Lefschetz general. We also show that two other divisors considered in \cite{rv} are not Noether--Lefschetz...

### Higher Zigzag Algebras

Joseph Grant
Given a Koszul algebra of finite global dimension we define its higher zigzag algebra as a twisted trivial extension of the Koszul dual. If our original algebra is the path algebra of a tree-type quiver, this construction recovers the zigzag algebras of Huerfano-Khovanov. We study examples of higher zigzag algebras coming from Iyama's type A higher representation finite algebras, give their presentations by quivers and relations, and describe relations between spherical twists acting on their...

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