### Abstract $\ell$--Adic $1$-Motives and Tate's Canonical Class for Number Fields

Greither & Popescu
In an earlier paper we constructed a new class of Iwasawa modules as l-adic realizations of what we called abstract l-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 l-adified Tate canonical class, defined by Tate in 1960 and give an explicit construction of (the minus part of) l-adic Tate sequences for...

### 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...

### 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.

### Hochschild Cohomology of Polynomial Representations of $\GL_2$

Miemietz & Turner
We compute the Hochschild cohomology algebras of Ringel-self-dual blocks of polynomial representations of $\GL_2$ over an algebraically closed field of characteristic $p>2$, that is, of any block whose number of simple modules is a power of $p$. These algebras are finite-dimensional and we provide an explicit description of their bases and multiplications.

### 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...

### 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...

### 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...

### 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.

### Numerical Dimension and Locally Ample Curves

Chung-Ching Lau
In the paper [the author, “Fujita vanishing theorems for q-ample divisors and applications on subvarieties with nef normal bundle”, Preprint, arXiv:1609.07797], it was shown that the restriction of a pseudoeffective divisor $D$ to a so-called nef subvariety $Y$ (e.g. $Y$ is lci in $X$ and has nef normal bundle) is pseudoeffective. Assuming the normal bundle is ample and that $D|_Y$ is not big, we prove that the numerical dimension of $D$ is bounded above by...

### Kähler Geometry on Hurwitz Spaces

Philipp Naumann
The classical Hurwitz space $\mathcal{H}^{n,b}$ is a fine moduli space for simple branched coverings of the Riemann sphere $\mathbb{P}^1$ by compact hyperbolic Riemann surfaces. In the article we study a generalized Weil-Petersson metric on the Hurwitz space, which was introduced in [R.~Axelsson, I.~Biswas, G.~Schumacher, K\"ahler structure on Hurwitz spaces, Manuscripta Math. 147, 63-79 (2015)]. For this purpose, Horikawa's deformation theory of holomorphic maps is refined in the presence of hermitian metrics in order to single...

### 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...

### 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...

### Arithmetic Families of $(\phi,\Gamma)$-Modules and Locally Analytic Representations of $GL_2(Q_p)$

Gaisin & Rodrigues Jacinto
Let $A$ be a $Q_p$-affinoid algebra, in the sense of Tate. We develop a theory of locally convex $A$-modules parallel to the treatment in the case of a field by Schneider and Teitelbaum. We prove that there is an integration map linking a category of locally analytic representations in $A$-modules and separately continuous relative distribution modules. There is a suitable theory of locally analytic cohomology for these objects and a version of Shapiro's Lemma, generalizing...

### 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...

### Big Cohen-Macaulay Modules, Morphisms of Perfect Complexes, and Intersection Theorems in Local Algebra

Avramov, Iyengar & Neeman
There is a well known link from the first topic in the title to the third one. In this paper we thread that link through the second topic. The central result is a criterion for the tensor nilpotence of morphisms of perfect complexes over commutative noetherian rings, in terms of a numerical invariant of the complexes known as their level. Applications to local rings include a strengthening of the Improved New Intersection Theorem, short direct...

### Differential Operators and Families of Automorphic Forms on Unitary Groups of Arbitrary Signature

Eischen, Fintzen, Mantovan & Varma
In the 1970's, Serre exploited congruences between $q$-expansion coefficients of Eisenstein series to produce $p$-adic families of Eisenstein series and, in turn, $p$-adic zeta functions. Partly through integration with more recent machinery, including Katz's approach to $p$-adic differential operators, his strategy has influenced four decades of developments. Prior papers employing Katz's and Serre's ideas exploiting differential operators and congruences to produce families of automorphic forms rely crucially on $q$-expansions of automorphic forms. The overarching goal...

### 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.

### Restrictions of Eisenstein Series and Rankin-Selberg Convolution

Rodney Keaton & Ameya Pitale
In a 2005 paper, Yang constructed families of Hilbert Eisenstein series, which when restricted to the diagonal are conjectured to span the underlying space of elliptic modular forms. One approach to these conjectures is to show the non-vanishing of an inner product of elliptic eigenforms with the restrictions of Eisenstein series. In this paper, we compute this inner product locally by using explicit values of new vectors in the Waldspurger model.

### Hermitian Forms and Systems of Quadratic Forms

Nokhodkar
We associate to every symmetric (antisymmetric) hermitian form a system of quadratic forms over the base field which determines its isotropy and metabolicity behaviour. It is shown that two even hermitian forms are isometric if and only if their associated systems are equivalent. As an application, it is also shown that an anisotropic symmetric hermitian form over a quaternion division algebra in characteristic two remains anisotropic over all odd degree extensions of the ground field.

### Green Functions and Higher Deligne--Lusztig Characters

Zhe Chen
We give a generalisation of the character formula of Deligne--Lusztig representations from the finite field case to the truncated formal power series case. Motivated by this generalisation, we give a definition of Green functions for these local rings, and prove some basic properties along the lines of the finite field case, like a summation formula. As an application we show that the higher Deligne--Lusztig characters and G\'erardin's characters agree at regular semisimple elements.

### On Endoscopic $p$-Adic Automorphic Forms for $\SL_2$

Ludwig
We show the existence of some non-classical cohomological $p$-adic automorphic eigenforms for $\SL_2$ using endoscopy and the geometry of eigenvarieties. These forms seem to account for some non-automorphic members of classical global $L$-packets.

### 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 K-Theory of Versal Flags and Cohomological Invariants of Degree 3

Baek , Devyatov & Zainoulline
Let $G$ be a split semisimple linear algebraic group over a field and let $X$ be a generic twisted flag variety of $G$. Extending the Hilbert basis techniques to Laurent polynomials over integers we give an explicit presentation of the Grothendieck ring $K0(X)$ in terms of generators and relations in the case $G=Gsc/\mu2$ is of Dynkin type ${A}$ or ${C}$ (here $Gsc$ is the simply-connected cover of $G$); we compute various groups of (indecomposable, semi-decomposable)...

### Homological Stability of Automorphism Groups of Quadratic Modules and Manifolds

Friedrich
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...

### Limit Linear Series on Chains of Elliptic Curves and Tropical Divisors on Chains of Loops

López Martín & Teixidor I Bigas
We describe the space of Eisenbud-Harris limit linear series on a chain of elliptic curves and compare it with the theory of divisors on tropical chains. Either model allows to compute some invariants of Brill-Noether theory using combinatorial methods. We introduce effective limit linear series.

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