Volume 2018, Issue 741

Since the work of Mikhail Kapranov in [Compos. Math. 115 (1999), no. 1, 71–113], it is known that the shifted tangent complex 𝕋X\mathbb{T}_{X}[-1] of a smooth algebraic variety X is endowed with a weak Lie structure. Moreover, any complex of quasi-coherent sheaves E on X is endowed with a weak Lie action of this tangent Lie algebra. This Lie action is given by the Atiyah class of E . We will generalise this result to (finite enough) derived Artin stacks, without any smoothness assumption. This in particular applies to singular schemes. This work uses tools of both derived algebraic geometry and ∞{\infty}-category theory.

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.

Let 𝐅q\mathbf{F}_{q} be a finite field of q elements. For multiplicative characters χ1,…,χm\chi_{1},\ldots,\chi_{m} of 𝐅q×\mathbf{F}_{q}^{\times}, we let J⁢(χ1,…,χm)J(\chi_{1},\ldots,\chi_{m}) denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for m=2m=2, the normalized Jacobi sum q-1/2⁢J⁢(χ1,χ2)q^{-1/2}J(\chi_{1},\chi_{2}) (χ1⁢χ2\chi_{1}\chi_{2} nontrivial) is asymptotically equidistributed on the unit circle as q→∞q\to\infty, when χ1\chi_{1} and χ2\chi_{2} run through all nontrivial multiplicative characters of 𝐅q×\mathbf{F}_{q}^{\times}. In this paper, we show a similar property for m≥2m\geq 2. More generally, we show that the normalized Jacobi sum q-(m-1)/2⁢J⁢(χ1,…,χm)q^{-(m-1)/2}J(\chi_{1},\ldots,\chi_{m}) (χ1⁢⋯⁢χm\chi_{1}\cdots\chi_{m} nontrivial) is asymptotically equidistributed on the unit circle, when χ1,…,χm\chi_{1},\ldots,\chi_{m} run through arbitrary sets of nontrivial multiplicative characters of 𝐅q×\mathbf{F}_{q}^{\times} with two of the sets being sufficiently large. The case m=2m=2 answers a question of Shparlinski.

In this article, we generalize some works of Bertolini–Darmon and Vatsal on anticyclotomic L -functions attached to modular forms of weight two to higher weight case. We construct a class of anticyclotomic p -adic L -functions for ordinary modular forms and derive the functional equation and the interpolation formula at all critical specializations. Moreover, we prove results on the vanishing of μ-invariant of these p -adic L -functions and the non-vanishing of central L -values with anticyclotomic twists.

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.

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.

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.

We show that the cohomology of the perfect cone (also called first Voronoi) toroidal compactification 𝒜gPerf{{{\mathcal{A}}_{g}^{\operatorname{Perf}}}} of the moduli space of complex principally polarized abelian varieties stabilizes in close to the top degree. Moreover, we show that this stable cohomology is purely algebraic, and we compute it in degree up to 13. Our explicit computations and stabilization results apply in greater generality to various toroidal compactifications and partial compactifications, and in particular we show that the cohomology of the matroidal partial compactification 𝒜gMatr{{{\mathcal{A}}_{g}^{\operatorname{Matr}}}} stabilizes in fixed degree, and forms a polynomial algebra. For degree up to 8, we describe explicitly the generators of the cohomology, and discuss various approaches to computing all of the stable cohomology in general.

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=0s=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.

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 C2{C^{2}}-distance from a single sphere. The corresponding stability inequality is obtained with a sharp decay rate.