On some anabelian properties of arithmetic curves

In this paper we generalize an argument of Neukirch from birational anabelian geometry to the case of arithmetic curves. In contrast to the function field case, it seems to be more complicated to describe the position of decomposition groups of points at the boundary of the scheme \({{\rm Spec}\, \mathcal{O}_{K, S}}\) , where K is a number field and S a set of primes of K, intrinsically in terms of the fundamental group. We prove that it is equivalent to give the following pieces of information additionally to the fundamental group \({\pi_1({\rm Spec}\, \mathcal{O}_{K, S})}\) : the location of decomposition groups of boundary points inside it, the p-part of the cyclotomic character, the number of points on the boundary of all finite étale covers, etc. Under a certain finiteness hypothesis on Tate–Shafarevich groups with divisible coefficients, one can reconstruct all these quantities simply from the fundamental group.     

The Riemannian geometry of physical systems of curves

Summary The properties of a physical system S_k where k ≠−1, of ∞²ⁿ⁻¹ trajectories C. in a Riemannian space V_n are developed. The intrinsic differential equations and the equations of Lagrange, of a physical system S_k, are derived. The Lagrangian function L and the Hamiltonian function H, are studied in the conservative case. Also included are systems of the type (G), curvature trajectories, and natural families. The Appell transformation T of a dynamical system S ₀ in a Riemannian space V_n, is obtained. Finally, contact transformations and the transformation theory of a physical system S_k where k ≠−1, are considered in detail. To Enrico Bompiani on his scientific Jubilee Kasner,Differential geometric aspecte of dynamics, The Princeton Colloquium Lectures, 1909. Published by the ? American Mathematical Society, Providence, Rhode Island, 1913, and reprinted 1934.     

The local Gromov-Witten theory of curves

The local Gromov-Witten theory of curves is solved by localization and degeneration methods. Localization is used for the exact evaluation of basic integrals in the local Gromov-Witten theory of . A TQFT formalism is defined via degeneration to capture higher genus curves. Together, the results provide a complete and effective solution.

The local Gromov-Witten theory of curves is equivalent to the local Donaldson-Thomas theory of curves, the quantum cohomology of the Hilbert scheme points of , and the orbifold quantum cohomology of the symmetric product of . The results of the paper provide the local Gromov-Witten calculations required for the proofs of these equivalences.

     

A birational anabelian reconstruction theorem for curves over algebraically closed fields in Arbitrary Characteristic

The aim of Bogomolov’s programme is to prove birational anabelian conjectures for function fields K|k of varieties of dimension ≥ 2 over algebraically closed fields. The present article is concerned with the 1-dimensional case. While it is impossible to recover K|k from its absolute Galois group alone, we prove that it can be recovered from the pair (Aut(\(\bar K\)|k), Aut(\(\bar K\)|K)), consisting of the absolute Galois group of K and the larger group of field automorphisms fixing only the base field.     

On the geometry of non-classical curves

In this paper we relate the numerical invariants attached to a projective curve, called the order sequence of the curve, to the geometry of the varieties of tangent linear spaces to the curve and to the Gauss maps of the curve. partially supported by CNPq-Brazil, Proc. 301596-85-9     

The local index formula in noncommutative geometry

In noncommutative geometry a geometric space is described from a spectral vantage point, as a tripleA, H, D consisting of a *-algebraA represented in a Hilbert spaceH together with an unbounded selfadjoint operatorD, with compact resolvent, which interacts with the algebra in a bounded fashion. This paper contributes to the advancement of this point of view in two significant ways: (1) by showing that any pseudogroup of transformations of a manifold gives rise to such a spectral triple of finite summability degree, and (2) by proving a general, in some sense universal, local index formula for arbitrary spectral triples of finite summability degree, in terms of the Dixmier trace and its residue-type extension.We dedicate this paper to Misha Gromov     

Generalized planar curves and quaternionic geometry

Motivated by the analogies between the projective and the almost quaternionic geometries, we first study the generalized planar curves and mappings. We follow, recover, and extend the classical approach, see e.g., (Sov. Math. 27(1) 63–70 (1983), Rediconti del circolo matematico di Palermo, Serie II, Suppl. 54 75–81) (1998), Then we exploit the impact of the general results in the almost quaternionic geometry. In particular we show, that the natural class of ℍ-planar curves coincides with the class of all geodesics of the so called Weyl connections and preserving this class turns out to be the necessary and sufficient condition on diffeomorphisms to become morphisms of almost quaternionic geometries.     

The Igusa local zeta functions of elliptic curves

We determine the explicit form of the Igusa local zeta function associated to an elliptic curve. The denominator is known to be trivial. Here we determine the possible numerators and classify them according to the Kodaira-Néron classification of the special fibers of elliptic curves as determined by Tate's algorithm.

     

Generic equi-centro-affine differential geometry of plane curves

We study the equi-centro-affine invariants of plane curves from the view point of the singularity theory of smooth functions. We define the notion of the equi-centro-affine pre-evolute and pre-curve and establish the relationship between singularities of these objects and geometric invariants of plane curves.     

The local geometry of Carnot manifolds at singular points

In this paper we study the local geometry of Carnot manifolds in a neighborhood of a singular point in the case when horizontal vector fields are 2M-smooth. Here M is the depth of a Carnot manifold.     

Enumerative geometry of elliptic curves on toric surfaces

We establish the equality of classical and tropical curve counts for elliptic curves on toric surfaces with fixed j-invariant, refining results of Mikhalkin and Nishinou–Siebert. As an application, we determine a formula for such counts on P² and all Hirzebruch surfaces. This formula relates the count of elliptic curves with the number of rational curves on the surface satisfying a small number of tangency conditions with the toric boundary. Furthermore, the combinatorial tropical multiplicities of Kerber and Markwig for counts in P² are derived and explained algebro-geometrically, using Berkovich geometry and logarithmic Gromov–Witten theory. As a consequence, a new proof of Pandharipande’s formula for counts of elliptic curves in P² with fixed j-invariant is obtained.