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We define a “period-ring-valued beta function” and give a reciprocity law on its special values. The proof is based on some results of Rohrlich and Coleman concerning Fermat curves. We also have the following application. Stark’s conjecture implies that the exponentials of the derivatives at s = 0 s=0 of partial zeta functions are algebraic numbers which satisfy a reciprocity law under certain conditions. It follows from Euler’s formulas and properties of cyclotomic units when the base field is the rational number field. In this paper, we provide an alternative proof of a weaker result by using the reciprocity law on the period-ring-valued beta function. In other words, the reciprocity law given in this paper is a refinement of the reciprocity law on cyclotomic units. [ABSTRACT FROM AUTHOR]

Gromov’s compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this paper, we prove the analog of Gromov’s compactness theorem in non-archimedean analytic geometry. We work in the framework of Berkovich spaces. First, we introduce a notion of Kähler structure in non-archimedean analytic geometry using metrizations of virtual line bundles. Second, we introduce formal stacks and non-archimedean analytic stacks. Then we construct the moduli stack of non-archimedean analytic stable maps using formal models, Artin’s representability criterion and the geometry of stable curves. Finally, we reduce the non-archimedean problem to the known compactness results in algebraic geometry. The motivation of this paper is to provide the foundations for non-archimedean enumerative geometry. [ABSTRACT FROM AUTHOR]

Let j be the modular invariant. For the primes p ≦ 23 the q-expansion coefficients of Um( j – 744) are multiplicative as it was a Hecke eigenform modulo a power of p which increases with m. This was conjectured by Atkin on the basis of extensive numerical experiments, and is proved in this paper. The cases p = 5, 7 and 11 are under special consideration in this paper. [ABSTRACT FROM AUTHOR]

Let 𝐅 q \mathbf{F}_{q} be a finite field of q elements. For multiplicative characters χ 1 , … , χ m \chi_{1},\ldots,\chi_{m} of 𝐅 q × \mathbf{F}_{q}^{\times} , we let J ⁢ ( χ 1 , … , χ m ) J(\chi_{1},\ldots,\chi_{m}) denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for m = 2 m=2 , the normalized Jacobi sum q - 1 / 2 ⁢ J ⁢ ( χ 1 , χ 2 ) q^{-1/2}J(\chi_{1},\chi_{2}) ( χ 1 ⁢ χ 2 \chi_{1}\chi_{2} nontrivial) is asymptotically equidistributed on the unit circle as q → ∞ q\to\infty , when χ 1 \chi_{1} and χ 2 \chi_{2} run through all nontrivial multiplicative characters of 𝐅 q × \mathbf{F}_{q}^{\times}. In this paper, we show a similar property for m ≥ 2 m\geq 2. More generally, we show that the normalized Jacobi sum q - ( m - 1 ) / 2 ⁢ J ⁢ ( χ 1 , … , χ m ) q^{-(m-1)/2}J(\chi_{1},\ldots,\chi_{m}) ( χ 1 ⁢ ⋯ ⁢ χ m \chi_{1}\cdots\chi_{m} nontrivial) is asymptotically equidistributed on the unit circle, when χ 1 , … , χ m \chi_{1},\ldots,\chi_{m} run through arbitrary sets of nontrivial multiplicative characters of 𝐅 q × \mathbf{F}_{q}^{\times} with two of the sets being sufficiently large. The case m = 2 m=2 answers a question of Shparlinski. [ABSTRACT FROM AUTHOR]

Let A be an abelian variety. The abelian Ax–Lindemann theorem shows that the Zariski closure of an algebraic flow in A is a translate of an abelian subvariety of A. The paper discusses some conjectures on the usual topological closure of an algebraic flow in A. The main result is a proof of these conjectures when the algebraic flow is given by an algebraic curve. [ABSTRACT FROM AUTHOR]

The gauged Witten equation was essentially introduced by Witten in his formulation of the gauged linear σ-model (GLSM), which explains the so-called Landau–Ginzburg/Calabi–Yau correspondence. This is the first paper in a series towards a mathematical construction of GLSM, in which we set up a proper framework for studying the gauged Witten equation and its perturbations. We also prove several analytical properties of solutions and moduli spaces of the perturbed gauged Witten equation. We prove that solutions have nice asymptotic behavior on cylindrical ends of the domain. Under a good perturbation scheme, the energies of solutions are shown to be uniformly bounded by a constant depending only on the topological type. We prove that the linearization of the perturbed gauged Witten equation is Fredholm, and we calculate its Fredholm index. Finally, we define a notion of stable solutions and prove a compactness theorem for the moduli space of solutions over a fixed domain curve. [ABSTRACT FROM AUTHOR]

Recently, the theory of currents and the existence theory for Plateau's problem have been extended to the case of finite-dimensional currents in infinite-dimensional manifolds or even metric spaces; see [5] (and also [7, 39] for the most recent developments). In this paper, in the case when the ambient space is Hilbert, we provide the first partial regularity result, in a dense open set of the support, for n-dimensional integral currents which locally minimize the mass. Our proof follows with minor variants [34], implementing Lipschitz approximation and harmonic approximation without indirect arguments and with estimates which depend only on the dimension n and not on codimension or dimension of the target space. [ABSTRACT FROM AUTHOR]

In this paper, we prove that, if Deligne's “petits camarades conjecture” holds, then a Langlands type correspondence holds also for p-adic coefficients on a smooth curve over a finite field. As an application, we prove that any overconvergent F -isocrystal of rank less than or equal to 2 on a smooth curve is ı-mixed. [ABSTRACT FROM AUTHOR]

For a large class of geometric objects, the passage to categories of quasicoherent sheaves provides an embedding in the 2-category of abelian tensor categories. The notion of weakly Tannakian categories introduced by the author gives a characterization of tensor categories in the image of this embedding. However, this notion requires additional structure to be present, namely a fiber functor. For the case of classical Tannakian categories in characteristic zero, Deligne has found intrinsic properties-expressible entirely within the language of tensor categories-which are necessary and sufficient for the existence of a fiber functor. In this paper we generalize Deligne's result to weakly Tannakian categories in characteristic zero. The class of geometric objects whose tensor categories of quasi-coherent sheaves can be recognized in this manner includes both the gerbes arising in classical Tannaka duality and more classical geometric objects such as projective varieties over a field of characteristic zero. Our proof uses a different perspective on fiber functors, which we formalize through the notion of geometric tensor categories. A second application of this perspective allows us to describe categories of quasi-coherent sheaves on fiber products. [ABSTRACT FROM AUTHOR]

The goal of this paper is to show that the equivariant Tamagawa number conjecture implies the extended abelian Stark conjecture contained in [12] and [11]. In particular, this gives the first proof of the extended abelian Stark conjecture for the base field Q, since the equivariant Tamagawa number conjecture away from 2 was proved in this context by Burns and Greither in [8] and Flach completed their results at 2 in [13] and [14]. [ABSTRACT FROM AUTHOR]

We give a short and direct proof of Getzler and Pandharipande's λg-conjecture. The approach is through the Ekedahl-Lando-Shapiro-Vainshtein theorem, which establishes the “polynomiality” of Hurwitz numbers, from which we pick off the lowest degree terms. The proof is independent of Gromov-Witten theory. We briefly describe the philosophy behind our general approach to intersection numbers and how it may be extended to other intersection number conjectures. Ideas from this paper feature in two independent recent enlightening proofs of Witten's conjecture by Kazarian [Adv. Math.] and Chen, Li, and Liu [Asian J. Math. 12: 511–518, 2009]. [ABSTRACT FROM AUTHOR]

This paper studies linear classes of planar quasiregular mappings. We give a positive answer to a conjecture of K. Astala, T. Iwaniec, and G. Martin (2009) on reduced Beltrami equations. Moreover, we use it to prove a Wronsky-type theorem for general linear Beltrami systems. This is a key to show that the associated Beltrami equation of a linear quasiregular family is unique. [ABSTRACT FROM AUTHOR]

In this paper we give a splitting criterion for uniform vector bundles on Fano manifolds covered by lines. As a consequence, we classify low rank uniform vector bundles on Hermitian symmetric spaces and Fano bundles of rank two on Grassmannians. [ABSTRACT FROM AUTHOR]

In this article we prove that a connected and properly embedded translating soliton in ℝ 3 {{\mathbb{R}^{3}}} with uniformly bounded genus on compact sets which is C 1 {C^{1}} -asymptotic to two planes outside a cylinder, either is flat or coincide with the grim reaper cylinder. [ABSTRACT FROM AUTHOR]

We study the relationship between derived categories of factorizations on gauged Landau–Ginzburg models related by variations of the linearization in Geometric Invariant Theory. Under assumptions on the variation, we show the derived categories are comparable by semi-orthogonal decompositions and we completely describe all components appearing in these semi-orthogonal decompositions. We show how this general framework encompasses many well-known semi-orthogonal decompositions. We then proceed to give applications of this complete description. In this setting, we verify a question posed by Kawamata: we show that D-equivalence and K-equivalence coincide for such variations. The results are applied to obtain a simple inductive description of derived categories of coherent sheaves on projective toric Deligne–Mumford stacks. This recovers Kawamata's theorem that all projective toric Deligne–Mumford stacks have full exceptional collections. Using similar methods, we prove that the Hassett moduli spaces of stable symmetrically-weighted rational curves also possess full exceptional collections. As a final application, we show how our results recover and extend Orlov's σ-model/Landau–Ginzburg model correspondence. [ABSTRACT FROM AUTHOR]

We prove that there are at most two possibilities for the base of a Lagrangian fibration from a complex projective irreducible symplectic fourfold. [ABSTRACT FROM AUTHOR]

Following Jacquet, Lapid and Rogawski, we define regularized periods of automorphic forms on U n + 1 \mathrm{U}_{n+1} × \times U n \mathrm{U}_{n} along the diagonal subgroup U n {\mathrm{U}_{n}} and compute the regularized periods of cuspidal Eisenstein series and their residues. The formula for the periods of residues has an application to the Gan–Gross–Prasad conjecture. [ABSTRACT FROM AUTHOR]

We prove that the boundary of a (not necessarily connected) bounded smooth set with constant nonlocal mean curvature is a sphere. More generally, and in contrast with what happens in the classical case, we show that the Lipschitz constant of the nonlocal mean curvature of such a boundary controls its C 2 {C^{2}} -distance from a single sphere. The corresponding stability inequality is obtained with a sharp decay rate. [ABSTRACT FROM AUTHOR]

We prove that the geometric Bogomolov conjecture for any abelian varieties is reduced to that for nowhere degenerate abelian varieties with trivial trace. In particular, the geometric Bogomolov conjecture holds for abelian varieties whose maximal nowhere degenerate abelian subvariety is isogenous to a constant abelian variety. To prove the results, we investigate closed subvarieties of abelian schemes over constant varieties, where constant varieties are varieties over a function field which can be defined over the constant field of the function field. [ABSTRACT FROM AUTHOR]

We characterize Siegel cusp forms in the space of Siegel modular forms of large weight k > 2 ⁢ n {k>2n} on any Siegel congruence subgroup Γ of any degree n and any level N, by a suitable growth of their Fourier coefficients (e.g., by the well-known Hecke bound) at any one of the cusps. For this, we use a ‘local’ approach as compared to our previous results on this topic. We also touch upon the question in the context of vector-valued modular forms. [ABSTRACT FROM AUTHOR]

We prove an integral R = 𝐓 {R=\mathbf{T}} theorem for odd two-dimensional p-adic representations of G 𝐐 {G_{\mathbf{Q}}} which are unramified at p, extending results of [5] to the non-minimal case. We prove, for any p, the existence of Katz modular forms modulo p of weight one which do not lift to characteristic zero. [ABSTRACT FROM AUTHOR]

Motivated by problems arising in conductivity imaging, we prove existence, uniqueness, and comparison theorems - under certain sharp conditions - for minimizers of the general least gradient problem infu∈BVf(Ω)∫Ωφ(x,Du), where f:∂Ω→R is continuous, and φ(x,ξ) is a function that, among other properties, is convex and homogeneous of degree 1 with respect to the ξ variable. In particular we prove that if a∈C1,1(Ω) is bounded away from zero, then minimizers of the weighted least gradient problem infu∈BVf∫Ωa∣Du∣ are unique in BVf(Ω). We construct counterexamples to show that the regularity assumption a∈C1,1 is sharp, in the sense that it can not be replaced by a∈C1,α(Ω) with any α<1.a∈C1,α(Ω) with any α<1. [ABSTRACT FROM AUTHOR]

We prove the compactness of solutions of general fourth order elliptic equations which are L1-perturbations of the Q-curvature equation on compact Riemannian 4-manifolds. Consequently, we prove the global existence and convergence of the Q-curvature flow on a generic class of Riemannian 4-manifolds. As a by-product, we give a positive answer to an open question by A. Malchiodi [12] on the existence of bounded Palais-Smale sequences for the Q-curvature problem when the Paneitz operator is positive with trivial kernel. [ABSTRACT FROM AUTHOR]

We prove the K-theoretic Farrell-Jones conjecture for groups with the Haagerup approximation property and coefficient rings and C*-algebras which are stable with respect to compact operators. We use this and Higson-Kasparov's result that the Baum-Connes conjecture holds for such a group G, to show that the algebraic and the C*-crossed product of G with a stable separable G-C*-algebra have the same K-theory. [ABSTRACT FROM AUTHOR]

Given a locally compact quantum group G, we define and study representations and C*-completions of the convolution algebra L1(G) associated with various linear subspaces of the multiplier algebra Cb(G). For discrete quantum groups G, we investigate the left regular representation, amenability and the Haagerup property in this framework. When G is unimodular and discrete, we study in detail the C*-completions of L1(G) associated with the non-commutative Lp-spaces Lp(G). As an application of this theory, we characterize (for each p ∊ [ 1;∊) the positive definite functions on unimodular orthogonal and unitary free quantum groups G that extend to states on the Lp-C*-algebra of G. Using this result, we construct uncountably many new examples of exotic quantum group norms for compact quantum groups. [ABSTRACT FROM AUTHOR]

As a first step in a general verification of the Alperin-McKay conjecture, we prove a reduction of the conjecture to statements about simple groups. Furthermore we show in numerous cases, that simple groups satisfy the required conditions. The methods are also applied to obtain similar results for refinements due to Isaacs and Navarro. [ABSTRACT FROM AUTHOR]

The essential dimension is a numerical invariant of an algebraic group G which may be thought of as a measure of complexity of G-torsors over fields. A recent theorem of N. Karpenko and A. Merkurjev gives a simple formula for the essential dimension of a finite p-group. We obtain similar formulas for the essential p-dimension of a broad class of groups, which includes all algebraic tori. [ABSTRACT FROM AUTHOR]

We consider continuum one-dimensional Schrödinger operators with potentials that are given by a sum of a suitable background potential and an Anderson-type potential whose single-site distribution has a continuous and compactly supported density. We prove exponential decay of the expectation of the finite volume correlators, uniform in any compact energy region, and deduce from this dynamical and spectral localization. The proofs implement a continuum analog of the method Kunz and Souillard developed in 1980 to study discrete one-dimensional Schrödinger operators with potentials of the form background plus random. [ABSTRACT FROM AUTHOR]

Let A be a quotient of J0( N) associated to a newform ƒ such that the special L-value of A (at s = 1) is non-zero. We give a formula for the ratio of the special L-value to the real period of A that expresses this ratio as a rational number. We extract an integer factor from the numerator of this formula; this factor is non-trivial in general and is related to certain congruences of ƒ with eigenforms of positive analytic rank. We use the techniques of visibility to show that, under certain hypotheses (which includes the first part of the Birch and Swinnerton-Dyer conjecture on rank), if an odd prime q divides this factor, then q divides either the order of the Shafarevich-Tate group or the order of a component group of A. Suppose p is an odd prime such that p2 does not divide N, p does not divide the order of the rational torsion subgroup of A, and ƒ is congruent modulo a prime ideal over p to an eigenform whose associated abelian variety has positive Mordell-Weil rank. Then we show that p divides the factor mentioned above; in particular, p divides the numerator of the ratio of the special L-value to the real period of A. Both of these results are as implied by the second part of the Birch and Swinnerton-Dyer conjecture, and thus provide theoretical evidence towards the conjecture. [ABSTRACT FROM AUTHOR]

By analogy with Thompson's classification of nonsolvable finite N-groups, we classify groups of finite Morley rank with solvable local subgroups of even and of mixed type. We also consider several aspects reminiscent of “small” groups of finite Morley rank of odd type. [ABSTRACT FROM AUTHOR]