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In 1903, Epstein published his proof of meromorphic continuation and a functional equation for Dirichlet series associated with quadratic forms, now called Epstein zeta-functions. However, already in 1889 (or even earlier) Hurwitz was aware of these results as his mathematical diaries and some unpublished notes (in an almost final form) found in his estate at the ETH Zurich show. In this article we present and analyze Hurwitz's notes and compare his reasoning with Epstein's paper in detail.

We correct an error in the paper of Z. Shen, S. Li, and J. Zhang published in [4]. In addition, we give an answer to a question posed by the authors.

We characterize functions of finite energy in the plane in terms of their traces on the lines that make up "graph paper" with squares of side length $mn$ for all $n$, and certain $\12-$order Sobolev norms on the graph paper lines. We also obtain analogous results for functions of finite energy on two classical fractals: the Sierpinski gasket and the Sierpinski carpet.

Let [Formula: see text] be a projective variety of dimension [Formula: see text] over an algebraically closed field of arbitrary characteristic. We prove a Fujiki–Lieberman type theorem on the structure of the automorphism group of [Formula: see text]. Let [Formula: see text] be a group of zero entropy automorphisms of [Formula: see text] and [Formula: see text] the set of elements in [Formula: see text] which are isotopic to the identity. We show that after replacing [Formula: see text] by a suitable finite-index subgroup, [Formula: see text] is a unipotent group of the derived length at most [Formula: see text]. This result was first proved by Dinh et al. for compact Kähler manifolds.

In a recent paper Robinson has obtained an explicit formula for the expression of an invariant matrix of an invariant matrix as a direct sum of invariant matrices. The object of the present note is to show that this formula may be deduced from known properties of Schur functions, with the aid of a result which the author has proved elsewhere.

The main goal of this short paper is to prove the existence of rank 2 simple and special Ulrich bundles on a wide class of elliptic surfaces: namely, on regular Weierstrass fibrations \(\pi : S\rightarrow \mathbb {P}^1\). Alongside we also show the existence of rank 2 weakly Ulrich sheaves on arbitrary Weierstrass fibrations \(S\rightarrow C_0\) and we deal with the (non-)existence of rank one Ulrich bundles on them.

0.1. How it all began. When the second author translated a result on algebraic sur faces into the language of lattice polygons using toric geometry, he got a simple inequality for lattice polygons. This inequality had originally been discovered by Scott [12]. The first author then found a third proof. Subsequently, both authors went through a phase of polygon addiction. Once you get started drawing lattice polygons on graph paper and discovering relations between their numerical invariants, it is not so easy to stop! (The gentle reader has been warned.) Thus, it was just unavoidable that the authors came up with new inequalities: Scott's inequality can be sharpened if one takes into account another invariant, which is de fined by peeling off the skins of the polygons like an onion (see Section 3).

In this short note we show that non-negative Ricci curvature is not preserved under Ricci flow for closed manifolds of dimensions four and above, strengthening a previous result of Knopf in \cite{K} for complete non-compact manifolds of bounded curvature. This brings down to four dimensions a similar result B\"ohm and Wilking have for dimensions twelve and above, \cite{BW}. Moreover, the manifolds constructed here are \Kahler manifolds and relate to a question raised by Xiuxiong Chen in \cite{XC}, \cite{XCL}., Comment: New version with added references and corrected typos

A large part of mathematical demography is built upon one fundamental theorem, the "strong ergodic theorem" of demography. If the fertility and mortality age-schedules of a population remain unchanged over time, its age distribution, no matter what its initial shape, will converge in time to a fixed and stable form. In brief, when demographic behavior remains unchanged, the population, it is said, converges to stability. This short paper presents a new argument for the convergence of the age structure, one that is self-contained, and that brings the mechanism behind convergence into full view. The idea is simple. Looked at directly, the dynamics of the age-distribution say little to our normal intuition. Looked at from a slightly different angle though, population dynamics define a smoothing or averaging process over the generations -- a process comfortable to our intuition. This smoothing and resmoothing turns out to be the mechanism that forces the age structure toward a fixed and final form.

The aim of this paper is to generalize results known for the symplectic involutions on K3 surfaces to the order 3 symplectic automorphisms on K3 surfaces. In particular, we will explicitly describe the action induced on the lattice $\Lambda_{K3}$, isometric to the second cohomology group of a K3 surface, by a symplectic automorphism of order 3; we exhibit the maps $\pi_*$ and $\pi^*$ induced in cohomology by the rational quotient map $\pi:X\dashrightarrow Y$, where $X$ is a K3 surface admitting an order 3 symplectic automorphism $\sigma$ and $Y$ is the minimal resolution of the quotient $X/\sigma$; we deduce the relation between the N\'eron--Severi group of $X$ and the one of $Y$. Applying these results we describe explicit geometric examples and generalize the Shioda--Inose structures, relating Abelian surfaces admitting order 3 endomorphisms with certain specific K3 surfaces admitting particular order 3 symplectic automorphisms., Comment: 28 pages. Version 2: this is the published version of the paper. The last section of the previous version (v1) was erased (the results are only stated) and it is now contained in arXiv:2209.10141

In this paper we prove that a nodal hypersurface in P^4 with defect has at least (d-1)^2 nodes, and if it has at most 2(d-2)(d-1) nodes and d>6 then it contains either a plane or a quadric surface. Furthermore, we prove that a nodal double cover of P^3 ramified along a surface of degree 2d with defect has at least d(2d-1) nodes. We construct the largest dimensional family of nodal degree d hypersurfaces in P^(2n+2) with defect for d sufficiently large., v2: A proof for the Ciliberto-Di Gennaro conjecture is added (Section 5); Some minor corrections in the other sections. v3: some minor corrections in the abstract v4: The proof for the Ciliberto-Di Gennaro conjecture has been modified; The paper is split into two parts, the complete intersection case will be discussed in a different paper

We do three things in this paper: (1) study the analog of localization sequences (in the sense of algebraic $K$-theory of stable $\infty $-categories) for additive $\infty $-categories, (2) define the notion of nilpotent extensions for suitable $\infty $-categories and furnish interesting examples such as categorical square-zero extensions, and (3) use (1) and (2) to extend the Dundas–Goodwillie–McCarthy theorem for stable $\infty $-categories that are not monogenically generated (such as the stable $\infty $-category of Voevodsky’s motives or the stable $\infty $-category of perfect complexes on some algebraic stacks). The key input in our paper is Bondarko’s notion of weight structures, which provides a “ring-with-many-objects” analog of a connective $\mathbb{E}_1$-ring spectrum. As applications, we prove cdh descent results for truncating invariants of stacks extending the work by Hoyois–Krishna for homotopy $K$-theory and establish new cases of Blanc’s lattice conjecture.

Recently, two of the authors of this paper constructed cyclic cocycles on Harish-Chandra's Schwartz algebra of linear reductive Lie groups that detect all information in the $K$-theory of the corresponding group $C^*$-algebra. The main result in this paper is an index formula for the pairings of these cocycles with equivariant indices of elliptic operators for proper, cocompact actions. This index formula completely determines such equivariant indices via topological expressions., 40 pages; updates based on referee comments; expanded proof of Proposition 3.3

In our previous paper, viewing $D^b(K(l))$ as a noncommutative curve, where $K(l)$ is the Kronecker quiver with $l$-arrows, we introduced categorical invariants via counting of noncommutative curves. Roughly, these invariants are sets of subcategories in a given category and their quotients. The noncommutative curve-counting invariants are obtained by restricting the subcategories to be equivalent to $D^b(K(l))$. The general definition, however, defines a larger class of invariants and many of them behave properly with respect to fully faithful functors. Here, after recalling the definition, we focus on the examples and extend our studies beyond counting. We enrich our invariants with the following structures: the inclusion of subcategories makes them partially ordered sets and considering semi-orthogonal pairs of subcategories as edges amounts to directed graphs. It turns out that the problem for counting $D^b(A_k)$ in $D^b(A_n)$ has a geometric combinatorial parallel - counting of maps between polygons. Estimating the numbers counting noncommutative curves in $D^b({\mathbb P}^2)$ modulo the group of autoequivalences, we prove finiteness and that the exact determining of these numbers leads to a solution of Markov problem. Via homological mirror symmetry, this gives a new approach to this problem. Regarding the structure of a partially ordered set mentioned above, we initiate intersection theory of noncommutative curves focusing on the case of noncommutative genus zero. The above-mentioned structure of a directed graph (and related simplicial complex) is a categorical analogue of the classical curve complex, introduced by Harvey and Harrer. The paper contains pictures of the graphs in many examples and also presents an approach to Markov conjecture via counting of subgraphs in a graph associated with $D^b({{\mathbb{P}}}^2)$. Some of the results proved here were announced in a previous work.

We classify the Seifert fibrations of lens spaces where the base orbifold is non-orientable. This is an addendum to our earlier paper ‘Seifert fibrations of lens spaces’. We correct Lemma 4.1 of that paper and fill the gap in the classification that resulted from the erroneous lemma.

The discriminant group of a minimal equicontinuous action of a group $G$ on a Cantor set $X$ is the subgroup of the closure of the action in the group of homeomorphisms of $X$, consisting of homeomorphisms which fix a given point. The stabilizer and the centralizer groups associated to the action are obtained as direct limits of sequences of subgroups of the discriminant group with certain properties. Minimal equicontinuous group actions on Cantor sets admit a classification by the properties of the stabilizer and centralizer direct limit groups. In this paper, we construct new families of examples of minimal equicontinuous actions on Cantor sets, which illustrate certain aspects of this classification. These examples are constructed as actions on rooted trees. The acting groups are countable subgroups of the product or of the wreath product of groups. We discuss applications of our results to the study of attractors of dynamical systems and of minimal sets of foliations., 20 pages, 1 figure. The condition of finite generation in Thm 1.9 was replaced by countability. The proof of Thm 1.9 has been simplified. The notation used in 5 has been modified. Several minor corrections across the paper

In this paper, we prove that for the oper stratification of the de Rham moduli space $M_{\mathrm{dR}}(X,r)$, the closed oper stratum is the unique minimal stratum with dimension $r^2(g-1)+g+1$, and the open dense stratum consisting of irreducible flat bundles with stable underlying vector bundles is the unique maximal stratum., Comment: This paper comes from the last section of arXiv:1905.10765v1 as an independent paper. Comments are welcome! To appear in manuscripta mathematica

In this paper we study graded Bourbaki ideals. It is a well-known fact that for torsionfree modules over Noetherian normal domains, Bourbaki sequences exist. We give criteria in terms of certain attached matrices for a homomorphism of modules to induce a Bourbaki sequence. Special attention is given to graded Bourbaki sequences. In the second part of the paper, we apply these results to the Koszul cycles of the residue class field and determine particular Bourbaki ideals explicitly. We also obtain in a special case the relationship between the structure of the Rees algebra of a Koszul cycle and the Rees algebra of its Bourbaki ideal., Comment: 29 pages

The original definition of cluster algebras by Fomin and Zelevinsky has been categorified and generalised in several ways over the course of the past 20 years, giving rise to cluster theory. This study lead to Iyama and Yang's generalised cluster categories $\mathcal{T}/\mathcal{T}^{fd}$ coming from $n$-Calabi-Yau triples $(\mathcal{T}, \mathcal{T}^{fd}, \mathcal{M})$. In this paper, we use some classic tools of homological algebra to give a deeper understanding of such categories $\mathcal{T}/\mathcal{T}^{fd}$. Let $k$ be a field, $n\geq 3$ an integer and $\mathcal{T}$ a $k$-linear triangulated category with a triangulated subcategory $\mathcal{T}^{fd}$ and a subcategory $\mathcal{M}=\text{add}(M)$ such that $(\mathcal{T}, \mathcal{T}^{fd}, \mathcal{M})$ is an $n$-Calabi-Yau triple. In this paper, we prove some properties of the triangulated categories $\mathcal{T}$ and $\mathcal{T}/\mathcal{T}^{fd}$. Our first result gives a relation between the Hom-spaces in these categories, using limits and colimits. Our second result is a Gap Theorem in $\mathcal{T}$, showing when the truncation triangles split. Moreover, we apply our two theorems to present an alternative proof to a result by Guo, originally stated in a more specific setup of dg $k$-algebras $A$ and subcategories of the derived category of dg $A$-modules. This proves that $\mathcal{T}/\mathcal{T}^{fd}$ is Hom-finite and $(n-1)$-Calabi-Yau, its object $M$ is $(n-1)$-cluster tilting and the endomorphism algebras of $M$ over $\mathcal{T}$ and over $\mathcal{T}/\mathcal{T}^{fd}$ are isomorphic. Note that these properties make $\mathcal{T}/\mathcal{T}^{fd}$ a generalisation of the cluster category., Comment: 17 pages. Final accepted version to appear in the Pacific Journal of Mathematics

In this paper we consider the following problem: letXk, be a Banach space with a normalised basis (e(k, j))j, whose biorthogonals are denoted by${(e_{(k,j)}^*)_j}$, for$k\in\N$, let$Z=\ell^\infty(X_k:k\kin\N)$be theirl∞-sum, and let$T:Z\to Z$be a bounded linear operator with a large diagonal,i.e.,$$\begin{align*}\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.\end{align*}$$Under which condition does the identity onZfactor throughT? The purpose of this paper is to formulate general conditions for which the answer is positive.

In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data $$(\varSigma ,\gamma ,H)$$ . We prove that given a metric $$\gamma $$ on $${{\mathbf {S}}}^{n-1}$$ ( $$3\le n\le 7$$ ), $$({{\mathbf {S}}}^{n-1},\gamma ,H)$$ admits no fill-in of NNSC metrics provided the prescribed mean curvature H is large enough (Theorem 4). Moreover, we prove that if $$\gamma $$ is a positive scalar curvature (PSC) metric isotopic to the standard metric on $${{\mathbf {S}}}^{n-1}$$ , then the much weaker condition that the total mean curvature $$\int _{{{\mathbf {S}}}^{n-1}}H\,{{\mathrm {d}}}\mu _\gamma $$ is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (Four lectures on scalar curvature, 2019, see P. 23). In the second part of this paper, we investigate the $$\theta $$ -invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.

Special units are a sort of predecessor of Euler systems, and they are mainly used to obtain annihilators for class groups. So one is interested in finding as many special units as possible (actually we use a technical generalization called “semispecial”). In this paper we show that in any abelian field having a real genus field in the narrow sense all Washington units are semispecial, and that a slightly weaker statement holds true for all abelian fields. The group of Washington units is very often larger than Sinnott’s group of cyclotomic units. In a companion paper we will show that in concrete families of abelian fields the group of Washington units is much larger than that of Sinnott units, by giving lower bounds on the index. Combining this with the present paper gives strong annihilation results.

Projective varieties with ample cotangent bundle satisfy many notions of hyperbolicity, and one goal of this paper is to discuss generalizations to quasi-projective varieties. A major hurdle is that the naive generalization fails, i.e. the log cotangent bundle is never ample. Instead, we define a notion called almost ample which roughly asks that the log cotangent is as positive as possible. We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type. In addition, if one assumes globally generated then we obtain that such varieties contain finitely many integral points. In another direction, we show that the Lang-Vojta conjecture implies the number of stably integral points on curves of log general type, and surfaces of log general type with almost ample log cotangent sheaf are uniformly bounded., v5: exposition greatly improved. Previous section on function fields removed, to be expanded upon in a future paper. To appear in IMRN

In this paper, we study noncommutative surface singularities arising from orders. The singularities we study are mild in the sense that they have finite representation type or, equivalently, are log terminal in the sense of the Mori minimal model program for orders (Chan and Ingalls in Invent Math 161(2):427–452, 2005). These were classified independently by Artin (in terms of ramification data) and Reiten–Van den Bergh (in terms of their AR-quivers). The first main goal of this paper is to connect these two classifications, by going through the finite subgroups $$G \subset {{{\,\mathrm{GL}\,}}_2}$$ , explicitly computing $$H^2(G,k^*)$$ , and then matching these up with Artin’s list of ramification data and Reiten–Van den Bergh’s AR-quivers. This provides a semi-independent proof of their classifications and extends the study of canonical orders in Chan et al. (Proc Lond Math Soc (3) 98(1):83–115, 2009) to the case of log terminal orders. A secondary goal of this paper is to study noncommutative analogues of plane curves which arise as follows. Let $$B = k_{\zeta } \llbracket x,y \rrbracket $$ be the skew power series ring where $$\zeta $$ is a root of unity, or more generally a terminal order over a complete local ring. We consider rings of the form $$A = B/(f)$$ where $$f \in Z(B)$$ which we interpret to be the ring of functions on a noncommutative plane curve. We classify those noncommutative plane curves which are of finite representation type and compute their AR-quivers.

In this paper, we consider the steady incompressible Navier–Stokes equations in a smooth bounded domain $$\Omega \subset \mathbb R^n$$ Ω ⊂ R n with the dimension $$n\ge 3$$ n ≥ 3 . We first establish asymptotic expansion formulae of Sobolev regular finite energy solutions in $$\Omega$$ Ω . In the second part of this paper, explicit representation formulae of Sobolev regular solutions are showed in the regular polyhedron $$\Omega :=[0,T]^n$$ Ω : = [ 0 , T ] n .

The classical Morse–Sard theorem claims that for a mapping v : R n → R m + 1 of class C k the measure of critical values v ( Z v , m ) is zero under condition k ≥ n − m . Here the critical set, or m-critical set is defined as Z v , m = { x ∈ R n : rank ∇ v ( x ) ≤ m } . Further Dubovitskiĭ in 1957 and independently Federer and Dubovitskiĭ in 1967 found some elegant extensions of this theorem to the case of other (e.g., lower) smoothness assumptions. They also established the sharpness of their results within the C k category. Here we formulate and prove a bridge theorem that includes all the above results as particular cases: namely, if a function v : R n → R d belongs to the Holder class C k , α , 0 ≤ α ≤ 1 , then for every q > m the identity H μ ( Z v , m ∩ v − 1 ( y ) ) = 0 holds for H q -almost all y ∈ R d , where μ = n − m − ( k + α ) ( q − m ) . Intuitively, the sense of this bridge theorem is very close to Heisenberg's uncertainty principle in theoretical physics: the more precise is the information we receive on measure of the image of the critical set, the less precisely the preimages are described, and vice versa. The result is new even for the classical C k -case (when α = 0 ); similar result is established for the Sobolev classes of mappings W p k ( R n , R d ) with minimal integrability assumptions p = max ⁡ ( 1 , n / k ) , i.e., it guarantees in general only the continuity (not everywhere differentiability) of a mapping. However, using some N-properties for Sobolev mappings, established in our previous paper, we obtained that the sets of nondifferentiability points of Sobolev mappings are fortunately negligible in the above bridge theorem. We cover also the case of fractional Sobolev spaces. The proofs of the most results are based on our previous joint papers with J. Bourgain and J. Kristensen (2013, 2015). We also crucially use very deep Y. Yomdin's entropy estimates of near critical values for polynomials (based on algebraic geometry tools).

Recently, the authors have established $$L^p$$ -boundedness of vector-valued q-variational inequalities for averaging operators which take values in the Banach space satisfying the martingale cotype q property in Hong and Ma (Math Z 286(1–2):89–120, 2017). In this paper, we prove that the martingale cotype q property is also necessary for the vector-valued q-variational inequalities, which was a question left open in the previous paper. Moreover, we also prove that the UMD property and the martingale cotype q property can be characterized in terms of vector valued q-variational inequalities for the Hilbert transform.

Let $\unicode[STIX]{x1D703}$ be an irrational real number. The map $T_{\unicode[STIX]{x1D703}}:y\mapsto (y+\unicode[STIX]{x1D703})\!\hspace{0.6em}{\rm mod}\hspace{0.2em}1$ from the unit interval $\mathbf{I}= [\!0,1\![$ (endowed with the Lebesgue measure) to itself is ergodic. In a short paper [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys.16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift when $\unicode[STIX]{x1D703}$ is extremely well approximated by the rational numbers, namely, if $$\begin{eqnarray}\inf _{q\geq 1}q^{4}4^{q^{2}}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ A few years later, Hoffman and Rudolph [Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2)156 (2002), 79–101] showed that for every irrational number, the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ is isomorphic to the unilateral dyadic Bernoulli shift. Their proof is not constructive. In the present paper, we relax notably Parry’s condition on $\unicode[STIX]{x1D703}$: the explicit map provided by Parry’s method is an isomorphism between the map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift whenever $$\begin{eqnarray}\inf _{q\geq 1}q^{4}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ This condition can be relaxed again into $$\begin{eqnarray}\inf _{n\geq 1}q_{n}^{3}~(a_{1}+\cdots +a_{n})~|q_{n}\unicode[STIX]{x1D703}-p_{n}| where $[0;a_{1},a_{2},\ldots ]$ is the continued fraction expansion and $(p_{n}/q_{n})_{n\geq 0}$ the sequence of convergents of $\Vert \unicode[STIX]{x1D703}\Vert :=\text{dist}(\unicode[STIX]{x1D703},\mathbb{Z})$. Whether Parry’s map is an isomorphism for every $\unicode[STIX]{x1D703}$ or not is still an open question, although we expect a positive answer.

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers, the current authors and T.-C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number $\sqrt{q}$. This establishes a key step that one may extend our previous explicit calculations of superspecial abelian surfaces to those of supersingular abelian surfaces.The second part is to introduce the notion of genera and ideal complexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the results of Yu on abelian varieties with additional structures to similitude classes, which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigation., Comment: 23 pages. Section 5.4 corrected

In this paper, we introduce the associated geodesic-Einstein flow for a relative ample line bundle L over the total space $$\mathcal {X}$$ of a holomorphic fibration and obtain a few properties of that flow. In particular, we prove that the pair $$(\mathcal {X}, L)$$ is nonlinear semistable if the associated Donaldson type functional is bounded from below and the geodesic-Einstein flow has long-time existence property. We also define the associated S-classes and C-classes for $$(\mathcal {X}, L)$$ and obtain two inequalities between them when L admits a geodesic-Einstein metric. Finally, in the appendix of this paper, we prove that a relative ample line bundle is geodesic-Einstein if and only if an associated infinite rank bundle is Hermitian–Einstein.

The main motive of the paper is to look on rectifying and osculating curves on a smooth surface. In this paper we find the normal and geodesic curvature for a rectifying curve on a smooth surface and we also prove that geodesic curvature is invariant under the isometry of surfaces such that rectifying curves remain. We find a sufficient condition for which an osculating curve on a smooth surface remains invariant under isometry of surfaces and also we prove that the component of the position vector of an osculating curve α(s) on a smooth surface along any tangent vector to the surface at α(s) is invariant under such isometry.

This paper classifies Poisson structures with the reduced simple normal crossing divisor on a product of Fano varieties of Picard number 1. The characterization of even-dimensional projective spaces from the viewpoint of Poisson structures is given by Lima and Pereira. In this paper, we generalize the characterization of projective spaces to any dimension.

T. Ikeda and H. Katsurada have developed the theory of the Gross-Keating invariant of a quadratic form in their recent papers [IK1] and [IK2]. In particular, they prove that the local factor of the Fourier coefficients of the Siegel-Eisenstein series is completely determined by the Gross-Keating invariant with extra datum, called the extended GK datum, in [IK2]. On the other hand, such local factor is a special case of the local densities for a pair of two quadratic forms. Thus we propose a general question if the local density can be determined by certain series of the Gross-Keating invariants and the extended GK datums. In this paper, we prove that the answer to this question is affirmative, for the local density of a single quadratic form defined over an unramified finite extension of $\mathbb{Z}_2$. In the appendix, T. Ikeda and H. Katsurada compute the local density formula of a single binary quadratic form defined over any finite extension of $\mathbb{Z}_2$., 32 pages

A conjecture of Kotschick predicts that a compact Kähler manifold X fibres smoothly over the circle if and only if it admits a holomorphic one-form without zeros. In this paper we develop an approach to this conjecture and verify it in dimension two. In a joint paper with Hao [10], we use our approach to prove Kotschick’s conjecture for smooth projective three-folds.

In this paper, we propose and study the polar Orlicz-Minkowski problems: under what conditions on a nonzero finite measure $\mu$ and a continuous function $\varphi:(0,\infty)\rightarrow(0,\infty)$, there exists a convex body $K\in\mathcal{K}_0$ such that $K$ is an optimizer of the following optimization problems: \begin{equation*} \inf/\sup \bigg\{\int_{S^{n-1}}\varphi\big( h_L \big) \,d \mu: L \in \mathcal{K}_{0} \ \text{and}\ |L^\circ|=\omega_{n}\bigg\}. \end{equation*} The solvability of the polar Orlicz-Minkowski problems is discussed under different conditions. In particular, under certain conditions on $\varphi,$ the existence of a solution is proved for a nonzero finite measure $\mu$ on $S^{n-1}$ which is not concentrated on any hemisphere of $S^{n-1}.$ Another part of this paper deals with the $p$-capacitary Orlicz-Petty bodies. In particular, the existence of the $p$-capacitary Orlicz-Petty bodies is established and the continuity of the $p$-capacitary Orlicz-Petty bodies is proved., Comment: This paper has been accepted by Indiana University Mathematics Journal

The standard $L$-functions of $\mathrm{GL}_{2n}$ expressed in terms of the Friedberg-Jacquet global zeta integrals have better structure for arithmetic applications, due to the relation of the linear periods with the modular symbols. The most technical obstacles towards such arithmetic applications are (1) non-vanishing of modular symbols at infinity and (2) the existance or construction of uniform cohomological test vectors. Problem (1) is also called the non-vanishing hypothesis at infinity, which was proved by Binyong Sun, by establishing the existence of certain cohomological test vectors. In this paper, we explicitly construct an archimedean local integral that produces a new type of a twisted linear functional $\Lambda_{s,\chi}$, which, when evaluated with our explicitly constructed cohomological vector, is equal to the local twisted standard $L$-function $L(s,\pi\otimes\chi)$ as a meromorphic function of $s\in \mathbb{C}$. With the relations between linear models and Shalika models, we establish (1) with an explicitly constructed cohomological vector, and hence recovers a non-vanishing result of Binyong Sun via a completely different method. Our main result indicates a complete solution to (2), which will be presented in a paper of Dihua Jiang, Binyong Sun and Fangyang Tian with full details and with applications to the global period relations for the twisted standard $L$-functions at critical places., Comment: 39 pages. The current version of this paper is significantly shorter than the previous one, as the first author pointed out a conceptual intepretation of construction of cohomological test vector in the old version of this paper. Section 4 is completely rewritten. Also fix some inaccuracies

In this paper we provide some quantitative mixed weak-type estimates assuming conditions that imply that $$uv\in A_{\infty }$$ for Calderon–Zygmund operators, rough singular integrals and commutators. The main novelty of this paper lies in the fact that we rely upon sparse domination results, pushing an approach to endpoint estimates that was introduced in Domingo-Salazar et al. (Bull Lond Math Soc 48(1):63–73, 2016) and extended in Lerner et al. (Adv Math 319:153–181, 2017) and Li et al. (J Geom Anal, 2018).

If $G$ is a group, a virtual retract of $G$ is a subgroup which is a retract of a finite index subgroup. Most of the paper focuses on two group properties: property (LR), that all finitely generated subgroups are virtual retracts, and property (VRC), that all cyclic subgroups are virtual retracts. We study the permanence of these properties under commensurability, amalgams over retracts, graph products and wreath products. In particular, we show that (VRC) is stable under passing to finite index overgroups, while (LR) is not. The question whether all finitely generated virtually free groups satisfy (LR) motivates the remaining part of the paper, studying virtual free factors of such groups. We give a simple criterion characterizing when a finitely generated subgroup of a virtually free group is a free factor of a finite index subgroup. We apply this criterion to settle a conjecture of Brunner and Burns., 30 pages, 1 figure. v3: added Lemma 5.8 and made minor corrections following referee's comments. This version of the paper has been accepted for publication

In our previous paper we have elaborated a certain signed count of real lines on real projective n-dimensional hypersurfaces of degree 2n-1. Contrary to the honest "cardinal" count, it is independent of the choice of a hypersurface, and by this reason provides a strong lower bound on the honest count. In this count the contribution of a line is its local input to the Euler number of a certain auxiliary vector bundle. The aim of this paper is to present other, in a sense more geometric, interpretations of this local input. One of them results from a generalization of Segre species of real lines on cubic surfaces and another from a generalization of Welschinger weights of real lines on quintic threefolds., Comment: 20 pages, typos are corrected (most essential, in Proposition 4.3.3)

Let $f\in\mathbb{Z}[T]$ be any polynomial of degree $d>1$ and $F\in\mathbb{Z}[X_{0},...,X_{n}]$ an irreducible homogeneous polynomial of degree $e>1$ such that the projective hypersurface $V(F)$ is smooth. In this paper we give a bound for \[ N(f,F,B):=|\{\textbf{x}\in\mathbb{Z}^{n+1}:\max_{0\leq i\leq n}|x_{i}|\leq B,\exists t\in\mathbb{Z}\text{ such that }f(t)=F(\textbf{x})\}|, \] To do this, we introduce a generalization of the Heath-Brown and Munshi's power sieve and we extend two results by Deligne and Katz on estimates for additive and multiplicative characters in many variables., Theorem 1 has been improved. The paper has been reorganized to improve the exposition

We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space $\mathcal{Q}_{g,n}$ of genus $g$ meromorphic quadratic differentials with $n$ simple poles as polynomials in the intersection numbers of $\psi$-classes with explicit rational coefficients. The formulae obtained in this article result from lattice point counts involving the Kontsevich volume polynomials that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli spaces of bordered hyperbolic surfaces with geodesic boundaries. A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by Mirzakhani via completely different approach. Furthermore, we prove that the density of the mapping class group orbit of any simple closed multicurve $\gamma$ inside the ambient set of integral measured laminations computed by Mirzakhani coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to $\gamma$ among all square-tiled surfaces in $\mathcal{Q}_{g,n}$. We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case $n=0$. In particular, we compute the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus $g$ for small $g$ and we show that for large genera the separating closed geodesics are $\sqrt{\frac{2}{3\pi g}}\cdot\frac{1}{4^g}$ times less frequent., Comment: The current paper (as well as the companion paper arXiv:2007.04740) has grown from arxiv:1908.08611. The conjectures stated in arXiv:1908.08611 are proved by A. Aggarwal in arXiv:2004.05042

We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal $L$-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal $L$-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of $L$-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for $\ell$-torsion in class groups, for all integers $\ell \geq 1$, applicable to infinite families of fields of arbitrarily large degree., Comment: 52 pages. This shorter version aligns with the published paper. Note that portions of Section 8 of the longer v1 have been developed as a separate paper with identifier arXiv:1902.02008

International audience; This article is the continuation of our first work on the determination of the cases where there is equality in Courant's Nodal Domain theorem in the case of a Robin boundary condition (with Robin parameter h). For the square, our first paper focused on the case where h is large and extended results that were obtained by Pleijel, Bérard-Helffer, for the problem with a Dirichlet boundary condition. There, we also obtained some general results about the behaviour of the nodal structure (for planar domains) under a small deformation of h, where h is positive and not close to 0. In this second paper, we extend results that were obtained by Helffer-Persson-Sundqvist for the Neumann problem to the case where h > 0 is small. MSC classification (2010): 35P99, 58J50, 58J37.

Our main purpose in this paper is to establish the existence and nonexistence of extremal functions (also known as maximizers) and symmetry of extremals for several Trudinger-Moser type inequalities on the entire space R n , including both the critical and subcritical Trudinger-Moser inequalities (see Theorems 1.1, 1.2, 1.3, 1.4, 1.5). Most of earlier works on existence of maximizers in the literature rely on the complicated blow-up analysis of PDEs for the associated Euler-Lagrange equations of the corresponding Moser functionals. The new approaches developed in this paper are using the identities and relationship between the supremums of the subcritical Trudinger-Moser inequalities and the critical ones established by the same authors in [25] , combining with the continuity of the supremum function that is observed for the first time in the literature. These allow us to establish the existence and nonexistence of the maximizers for the Trudinger-Moser inequalities in different ranges of the parameters (including those inequalities with the exact growth). This method is considerably simpler and also allows us to study the symmetry problem of the extremal functions and prove that the extremal functions for the subcritical singular Truddinger-Moser inequalities are symmetric. Moreover, we will be able to calculate the exact values of the supremums of the Trudinger-Moser type in certain cases. These appear to be the first results in this direction.

Liouville quantum gravity (LQG) and the Brownian map (TBM) are two distinct models of measure-endowed random surfaces. LQG is defined in terms of a real parameter $$\gamma $$, and it has long been believed that when $$\gamma = \sqrt{8/3}$$, the LQG sphere should be equivalent (in some sense) to TBM. However, the LQG sphere comes equipped with a conformal structure, and TBM comes equipped with a metric space structure, and endowing either one with the other’s structure has been an open problem for some time. This paper is the first in a three-part series that unifies LQG and TBM by endowing each object with the other’s structure and showing that the resulting laws agree. The present work considers a growth process called quantum Loewner evolution (QLE) on a $$\sqrt{8/3}$$-LQG surface $${\mathcal {S}}$$ and defines $$d_{{\mathcal {Q}}}(x,y)$$ to be the amount of time it takes QLE to grow from $$x \in {\mathcal {S}}$$ to $$y \in {\mathcal {S}}$$. We show that $$d_{{\mathcal {Q}}}(x,y)$$ is a.s. determined by the triple $$({\mathcal {S}},x,y)$$ (which is far from clear from the definition of QLE) and that $$d_{{\mathcal {Q}}}$$ a.s. satisfies symmetry (i.e., $$d_{{\mathcal {Q}}}(x,y) = d_{{\mathcal {Q}}}(y,x)$$) for a.a. (x, y) pairs and the triangle inequality for a.a. triples. This implies that $$d_{{\mathcal {Q}}}$$ is a.s. a metric on any countable sequence sampled i.i.d. from the area measure on $${\mathcal {S}}$$. We establish several facts about the law of this metric, which are in agreement with similar facts known for TBM. The subsequent papers will show that this metric a.s. extends uniquely and continuously to the entire $$\sqrt{8/3}$$-LQG surface and that the resulting measure-endowed metric space is TBM.

This paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal. Our investigation is focused on the behavior of the function $${{\,\mathrm{depth}\,}}R/I^{(t)} = \dim R -{{\,\mathrm{pd}\,}}I^{(t)} - 1$$ , where $$I^{(t)}$$ denotes the t-th symbolic power of a homogeneous ideal I in a noetherian polynomial ring R and $${{\,\mathrm{pd}\,}}$$ denotes the projective dimension. It has been an open question whether the function $${{\,\mathrm{depth}\,}}R/I^{(t)}$$ is non-increasing if I is a squarefree monomial ideal. We show that $${{\,\mathrm{depth}\,}}R/I^{(t)}$$ is almost non-increasing in the sense that $${{\,\mathrm{depth}\,}}R/I^{(s)} \ge {{\,\mathrm{depth}\,}}R/I^{(t)}$$ for all $$s \ge 1$$ and $$t \in E(s)$$ , where $$\begin{aligned} E(s) = \bigcup _{i \ge 1}\{t \in {\mathbb {N}}|\ i(s-1)+1 \le t \le is\} \end{aligned}$$ (which contains all integers $$t \ge (s-1)^2+1$$ ). The range E(s) is the best possible since we can find squarefree monomial ideals I such that $${{\,\mathrm{depth}\,}}R/I^{(s)} < {{\,\mathrm{depth}\,}}R/I^{(t)}$$ for $$t \not \in E(s)$$ , which gives a negative answer to the above question. Another open question asks whether the function $${{\,\mathrm{depth}\,}}R/I^{(t)}$$ is always constant for $$t \gg 0$$ . We are able to construct counter-examples to this question by monomial ideals. On the other hand, we show that if I is a monomial ideal such that $$I^{(t)}$$ is integrally closed for $$t \gg 0$$ (e.g. if I is a squarefree monomial ideal), then $${{\,\mathrm{depth}\,}}R/I^{(t)}$$ is constant for $$t \gg 0$$ with $$\begin{aligned} \lim _{t \rightarrow \infty }{{\,\mathrm{depth}\,}}R/I^{(t)} = \dim R - \dim \oplus _{t \ge 0}I^{(t)}/{\mathfrak {m}}I^{(t)}. \end{aligned}$$ Our last result (which is the main contribution of this paper) shows that for any positive numerical function $$\phi (t)$$ which is periodic for $$t \gg 0$$ , there exist a polynomial ring R and a homogeneous ideal I such that $${{\,\mathrm{depth}\,}}R/I^{(t)} = \phi (t)$$ for all $$t \ge 1$$ . As a consequence, for any non-negative numerical function $$\psi (t)$$ which is periodic for $$t \gg 0$$ , there is a homogeneous ideal I and a number c such that $${{\,\mathrm{pd}\,}}I^{(t)} = \psi (t) + c$$ for all $$t \ge 1$$ .

Following a result of Hatori et al. (J Math Anal Appl 326:281–296, 2007), we give here a spectral characterization of an isomorphism from a $$C^\star $$ -algebra onto a Banach algebra. We then use this result to show that a $$C^\star $$ -algebra A is isomorphic to a Banach algebra B if and only if there exists a surjective function $$\phi :A\rightarrow B$$ satisfying (i) $$\sigma \left( \phi (x)\phi (y)\phi (z)\right) =\sigma \left( xyz\right) $$ for all $$x,y,z\in A$$ (where $$\sigma $$ denotes the spectrum), and (ii) $$\phi $$ is continuous at $$\mathbf 1$$ . In particular, if (in addition to (i) and (ii)) $$\phi (\mathbf 1)=\mathbf 1$$ , then $$\phi $$ is an isomorphism. An example shows that (i) cannot be relaxed to products of two elements, as is the case with commutative Banach algebras. The results presented here also elaborate on a paper of Bresar and Spenko (J Math Anal Appl 393:144–150, 2012), and a paper of Bourhim et al. (Arch Math 107:609–621, 2016).

In the present paper, we study fundamental groups of curves in positive characteristic. Let $$X^{\bullet }$$ be a pointed stable curve of type $$(g_{X}, n_{X})$$ over an algebraically closed field of characteristic $$p>0$$, $$\Gamma _{X^{\bullet }}$$ the dual semi-graph of $$X^{\bullet }$$, and $$\Pi _{X^{\bullet }}$$ the admissible fundamental group of $$X^{\bullet }$$. In the present paper, we study a kind of group-theoretical invariant $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ associated to the isomorphism class of $$\Pi _{X^{\bullet }}$$ called the limit of p-averages of $$\Pi _{X^{\bullet }}$$, which plays a central role in the theory of anabelian geometry of curves over algebraically closed fields of positive characteristic. Without any assumptions concerning $$\Gamma _{X^{\bullet }}$$, we give a lower bound and a upper bound of $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$. In particular, we prove an explicit formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ under a certain assumption concerning $$\Gamma _{X^{\bullet }}$$ which generalizes a formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ obtained by Tamagawa. Moreover, if $$X^{\bullet }$$ is a component-generic pointed stable curve, we prove an explicit formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ without any assumptions concerning $$\Gamma _{X^{\bullet }}$$, which can be regarded as an averaged analogue of the results of Nakajima, Zhang, and Ozman–Pries concerning p-rank of abelian etale coverings of projective generic curves for admissible coverings of component-generic pointed stable curves.