### 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.

### Singularities of Moduli of Curves with a Universal Root

Galeotti
In a series of recent papers, Chiodo, Farkas and Ludwig carry out a deep analysis of the singular locus of the moduli space of stable (twisted) curves with an $\ell$-torsion line bundle. They show that for $\ell≤ 6$ and $\ell\neq 5$ pluricanonical forms extend over any desingularization. This opens the way to a computation of the Kodaira dimension without desingularizing, as done by Farkas and Ludwig for $\ell=2$, and by Chiodo, Eisenbud, Farkas and Schreyer...

### 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...

### $p$-Adic Fourier Theory of Differentiable Functions

Nagel
Let $K$ be a finite extension of $\Q_p$ of degree $d$ and $o_K$ its ring of integers; let $\C_p$ be the completed algebraic closure of $\Q_p$. The \emph{Fourier polynomials} $P_n \from o_K \to \C_p$ show that the topological algebra of all locally analytic distributions $\mu \from C^{la}(o_K,\C_p) \to \C_p$ is, by $\mu \mapsto \sum \mu(P_n) X^n$, isomorphic to that of all power series in $\C_p[[X]]$ that converge on the open unit disc of $\C_p$. Given...

### On the Invertibility of Motives of Affine Quadrics

Bachmann
We show that the reduced motive of a smooth affine quadric is invertible as an object of the triangulated category of motives $\DM(k, \ZZ[1/e])$ (where $k$ is a perfect field of exponential characteristic $e$). We also establish a motivic version of the conjectures of Po Hu on products of certain affine Pfister quadrics. Both of these results are obtained by studying a novel conservative functor on (a subcategory of) $\DM(k, \ZZ[1/e])$, the construction of which...

### On the Homotopy Exact Sequence for Log Algebraic Fundamental Groups

Di Proietto & Shiho
We construct a log algebraic version of the homotopy sequence for a normal crossing log variety over a log point of characteristic zero and prove some exactness properties of it. Our proofs are purely algebraic.

### Deformation Theory with Homotopy Algebra Structures on Tensor Products

Robert-Nicoud
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...

### Erratum for “Fano Threefolds with 2-Torus Action”

Hendrik Süß
The list of Fano threefolds given in Theorem 1.1 wrongly contained an element of the family 3.8.

### Normal Form for Infinite Type Hypersurfaces in C^2 with Nonvanishing Levi Form Derivative

Ebenfelt , Lamel & Zaitsev
In this paper, we study real hypersurfaces $M$ in ${\ C}2$ at points $p\in M$ of infinite type. The degeneracy of $M$ at $p$ is assumed to be the least possible, namely such that the Levi form vanishes to first order in the CR transversal direction. A new phenomenon, compared to known normal forms in other cases, is the presence of resonances as roots of a universal polynomial in the 7-jet of the defining function...

### On Free Resolutions of Iwasawa Modules

Alexandra Nichifor & Bharathwaj Palvannan
Let $\Lambda$ (isomorphic to $\mathbb{Z}_p[[T]]$) denote the usual Iwasawa algebra and $G$ denote the Galois group of a finite Galois extension $L/K$ of totally real fields. When the non-primitive Iwasawa module over the cyclotomic $\mathbb{Z}_p$-extension has a free resolution of length one over the group ring $\Lambda[G]$, we prove that the validity of the non-commutative Iwasawa main conjecture allows us to find a representative for the non-primitive $p$-adic $L$-function (which is an element of a...

### Prolongations of t-Motives and Algebraic Independence of Periods

Maurischat
In this article we show that the coordinates of a period lattice generator of the n-th tensor power of the Carlitz module are algebraically independent, if n is prime to the characteristic. The main part of the paper, however, is devoted to a general construction for t-motives which we call prolongation, and which gives the necessary background for our proof of the algebraic independence. Another incredient is a theorem which shows hypertranscendence for the Anderson-Thakur...

### Why do Solutions of the Maxwell--Boltzmann Equation Tend to be Gaussians? A Simple Answer

Hundertmark & Lee
The Maxwell-Boltzmann functional equation has recently attraction renewed interest since besides its importance in Boltzmann's kinetic theory of gases it also characterizes maximizers of certain bilinear estimates for solutions of the free Schrödinger equation. In this note we give a short and simple proof that, under some mild growth restrictions, any measurable complex-valued solution of the Maxwell-Boltzmann equation is a Gaussian. This covers most, if not all, of the applications.

### The Magnetic Laplacian Acting on Discrete Cusps

Gol\'Enia & Truc
We introduce the notion of discrete cusp for a weighted graph. In this context, we prove that the form-domain of the magnetic Laplacian and that of the non-magnetic Laplacian can be different. We establish the emptiness of the essential spectrum and compute the asymptotic of eigenvalues for the magnetic Laplacian.

### Modular Equalities for Complex Reflection Arrangements

We compute the combinatorial Aomoto--Betti numbers $\betap(\A)$ of a complex reflection arrangement. When $\A$ has rank at least 3, we find that $\betap(\A)\le 2$, for all primes $p$. Moreover, $\betap(\A)=0$ if $p>3$, and $\beta2(\A)\ne 0$ if and only if $\A$ is the Hesse arrangement. We deduce that the multiplicity $ed(\A)$ of an order $d$ eigenvalue of the monodromy action on the first rational homology of the Milnor fiber is equal to the corresponding Aomoto--Betti number,...

### Commutative Algebraic Groups up to Isogeny

Brion
Consider the abelian category ${\mathcal C}k$ of commutative group schemes of finite type over a field $k$. By results of Serre and Oort, ${\mathcal C}k$ has homological dimension 1 (resp. 2) if $k$ is algebraically closed of characteristic 0 (resp. positive). In this article, we explore the abelian category of commutative algebraic groups up to isogeny, defined as the quotient of ${\mathcal C}k$ by the full subcategory ${\mathcal F}k$ of finite $k$-group schemes. We show...

Chintala
We introduce a notion of an \emph{embedding} of a quadratic space in an associative algebra. This simple notion allows us to capture the geometry of the quadratic space in terms of the algebra in which it is embedded. The general properties of such embeddings are analyzed by linking it to the Clifford algebra. Conversely, we describe the corresponding representation of the Spin group and the standard involution.

### The Motivic Hopf Map Solves the Homotopy Limit Problem for $K$-Theory

R{\"O}Ndigs, Spitzweck & {\O}Stv{\Ae}R
We solve affirmatively the homotopy limit problem for (effective) $K$-theory over fields of finite virtual cohomological dimension. Our solution uses the motivic slice filtration and the first motivic Hopf map.

### p-Adic Interpolation of Automorphic Periods for GL2

Order
We give a new and representation theoretic construction of $p$-adic interpolation series for central values of self-dual Rankin-Selberg $L$-functions for $GL2$ in dihedral towers of CM fields, using expressions of these central values as automorphic periods. The main novelty of this construction, apart from the level of generality in which it works, is that it is completely local. We give the construction here for a cuspidal automorphic representation of $GL2$ over a totally real field...

### Sur les Atomes Automorphes de Longueur 2 de GL2(Q_p)

Morra
Soit $p≥ 3$ un nombre premier. Le but de cet article est de donner une description des invariants, sous les sous-groupes de congruence principaux, des extensions entre séries principales apparaissant dans la correspondance de Langlands $p$-modulaire de $GL2(Qp)$. Comme application on décrit les espaces Hecke isotypiques de la cohomologie de la courbe modulaire sur $Q$ avec un niveau de ramification arbitraire en $p$. Let $p>3$ be a prime. The aim of this paper is to...

### On the Non Commutative Iwasawa Main Conjecture for Abelian Varieties over Function Fields

Fabien Trihan & David Vauclair
We establish the Iwasawa main conjecture for semistable abelian varieties over a function field of characteristic $p$ under certain restrictive assumptions. Namely we consider $p$-torsion free $p$-adic Lie extensions of the base field which contain the constant $\mathbb Z_p$-extension and are everywhere unramified. Under the usual $\mu=0$ hypothesis, we give a proof which mainly relies on the interpretation of the Selmer complex in terms of $p$-adic cohomology [F. Trihan, D. Vauclair, A comparison theorem for...

### Relative Homological Algebra via Truncations

Chach\'Olski, Neeman, Pitsch & Scherer
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 $A$ and fix a...

### Smooth Duals of Inner Forms of ${GL}_n$ and ${SL}_n$

Anne-Marie Aubert, Paul Baum, Roger Plymen & Maarten Solleveld
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 $\rm{GL}_n(F)$ 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 $\rm{SL}_n(F)$ we prove that each Bernstein component is canonically in bijection with the associated twisted extended quotient. In both cases, the bijections...

### Concordance Invariance of Levine-Tristram Signatures of Links

Nagel & Powell
We determine for which complex numbers on the unit circle the Levine-Tristram signature and the nullity give rise to link concordance invariants.

### Weil-\'Etale Cohomology and Zeta-Values of Proper Regular Arithmetic Schemes

Flach & Morin
We give a conjectural description of the vanishing order and leading Taylor coefficient of the Zeta function of a proper, regular arithmetic scheme $X$ at any integer $n$ in terms of Weil-\'etale cohomology complexes. This extends work of S. Lichtenbaum [Compos. Math. 141, No. 3, 689--702 (2005)] and T. Geisser [Math. Ann. 330, No. 4, 665--692 (2004)] for $X$ of characteristic $p$, of Lichtenbaum [Ann. Math. (2) 170, No. 2, 657--683 (2009)] for $X=Spec(O_F)$ and...

### Julia Sets for Polynomial Diffeomorphisms of $\mathbb{C}^2$ are not Semianalytic

Eric Bedford & Kyounghee Kim
For any polynomial diffeomorphism $f$ of $\mathbb{C}^2$ with positive entropy, neither the Julia set of $f$ nor of its inverse $f^{-1}$ is semianalytic.

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