Related papers: Le symbole de Hasse logarithmique

We give necessary and sufficient conditions for an integral polynomial without linear factors to be the characteristic polynomial of an isometry of some even, unimodular lattice of given signature. This gives rise to Hasse principle…

We consider a generalization of Hausdorff operator and introduce the notion of the symbol of such an operator. Using this notion we describe the structure and investigate important properties (such as invertibility, spectrum, norm, and…

By means of the Hardy-Littlewood method, we apply a new mean value theorem for exponential sums to confirm the truth, over the rational numbers, of the Hasse principle for pairs of diagonal cubic forms in thirteen or more variables.

We give an explicit formula for the symbol of a function of an operator. Given an operator {\hat A} on L^2(R^N) with symbol A, and a function f, we obtain the symbol of f(\hat A) in terms of A. As an application, Bohr-Sommerfeld…

In a beautiful paper Deligne and Illusie proved the degeneration of the Hodge-to-de Rham spectral sequence using positive characteristic methods. In a recent paper Arinkin, C\u{a}ld\u{a}raru and the author of this paper gave a geometric…

Based on the logarithmic algebraic geometry and the theory of Deligne systems, we define an abelian category of $\ell$-adic sheaves with weight filtrations on a logarithmic scheme over a finite field, which is similar to the category of…

Working on Berkovich analytic curves, we propose a geometric approach to the study of the Hasse principle over function fields of curves defined over a complete discretely valued field. Using it, we show the Hasse principle to be verified…

As a corollary of nonabelian Hodge theory, Simpson proved a strong Lefschetz theorem for complex polarized variations of Hodge structure. We show an arithmetic analog. Our primary technique is $p$-adic nonabelian Hodge theory. Conditional…

The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalised from modular curves to Hecke triangle surfaces. A sketch is included of how the modular symbols formalism gives rise to the…

We introduce a new kind of symbol in the framework of It\^o processes which are bounded on one side. The connection between this symbol and the infinitesimal generator is analyzed. Based on this concept, an integral criterion for invariant…

Recently, the authors of this paper introduced logarithmic Hochschild (co)homology of logarithmic spaces in a geometric way using formality of derived intersections. In this paper, the authors extend the decomposition theorem for the…

A central question in Arithmetic geometry is to determine for which polynomials $f \in \mathbb{Z}[t]$ and which number fields $K$ the Hasse principle holds for the affine equation $f(t) = N_{K/\mathbb{Q}}(\boldsymbol{x}) \neq 0$. Whilst…

In this paper, we investigate the solubility of homogeneous polynomial equations. The work of Browning, Le boudec, Sawin [3] shows that almost all homogeneous equations of degree $d\geq 4$ in $d+1$ or more variables satisfy the Hasse…

In this paper, we show the Hasse principle for the character group of a finitely generated field over the rational number field. By applying this result, we obtain an algebraic proof of unramified class field theory of arithmetical schemes.

We treat the notion of principal symbol mapping on a compact smooth manifold as a $\ast$-homomorphism of $C^{\ast}$-algebras. Principal symbol mapping is built from the ground, without referring to the pseudodifferential calculus on the…

Let $d$ and $n$ be natural numbers. Let $\nu_{d,n}: \mathbb{R}^n\rightarrow \mathbb{R}^{N}$ denote the Veronese embedding with $N=N_{n,d}:=\binom{n+d-1}{d}$, defined by listing all the monomials of degree $d$ in $n$ variables using the…

In this paper we shall consider some famous means such as arithmetic, harmonic, geometric, logarithmic means, etc. Inequalities involving logarithmic mean with differences among other means are presented

We survey old and new approaches to the study of symbolic powers of ideals. Our focus is on the symbolic Rees algebra of an ideal, viewed both as a tool to investigate its symbolic powers and as a source of challenging problems in its own…

Let K be a complete discretely valued field of mixed characteristic (0, p) with possibly imperfect residue field. We prove a Hasse-Arf theorem for the arithmetic ramification filtrations on G_K, except possibly in the absolutely unramified…

We establish the Hasse Principle for systems of r simultaneous diagonal cubic equations whenever the number of variables exceeds 6r and the associated coefficient matrix contains no singular r x r submatrix, thereby achieving the…