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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 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]

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]

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]

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]

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]

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]

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]

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]

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]

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]