Related papers: A General Reciprocity Law on arbitrary Vector Spac…
Given a specific collection of curves on an oriented surface with punctures, we associate a power series by counting its intersections with multicurves. This paper presents a reciprocity formula on the power series when multicurves with no…
This article is the first article of a serie of articles on the generalization of Kato's Euler system. The main subject of this article is to construct a family of Kato's Euler systems and a family of Kato's explicit reciprocity laws over…
A generalization of the law of total covariance is presented and proved.
We associate to a full flag $\mathcal{F}$ in an $n$-dimensional variety $X$ over a field $k$, a "symbol map" $\mu_{\mathcal{F}}:K(F_X) \to \Sigma^n K(k)$. Here, $F_X$ is the field of rational functions on $X$, and $K(\cdot)$ is the…
The paper is devoted to vector fields on the spaces R^2 and R^3, their flow and invariants. Attention is plaid on the tensor representations of the group GL(2,R) and on fundamental vector fields. The rotation group on R^3 is generalized to…
We prove non-commutative reciprocity laws on an algebraic surface defined over a perfect field. These reciprocity laws claim the splittings of some central extensions of globally constructed groups over some subgroups constructed by points…
An introductory overview of vector spaces, algebras, and linear geometries over an arbitrary commutative field is given. Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space.…
We briefly review Artin's reciprocity law in the classical ideal theoretic language, and then study connections between Artin's reciprocity law and the proofs of the quadratic reciprocity law using Gauss's Lemma.
We study the general theorem about gravitational lensing which states the relationship between the numbers of images with different parities. Our formulation allows an extension to the nontransparent and singular model.
We sketch the construction of a derived enhancement of the reciprocity isomorphism of class field theory. Details will appear in a forthcoming joint paper of the authors with A. Raksit.
Using the quadratic reciprocity law as the motivating example, we convey an understanding of classical reciprocity laws.
The paper presents a general duality theory for vector measure spaces taking its origin in the author's papers written in the 1960s. The main result establishes a direct correspondence between the geometry of a measure in a vector space and…
We survey some recent progress in the theory of vector bundles on algebraic varieties and related questions in algebraic K-theory.
We study reciprocity laws involving complex line bundles on fibrations in oriented circles. In particularly, we prove the following reciprocity law. Let $B$ be a complex manifold and $\pi_i : M_i \to B$ be a fibration in oriented circles,…
We prove a combinatorial reciprocity theorem for the enumeration of non-intersecting paths in a linearly growing sequence of acyclic planar networks. We explain two applications of this theorem: reciprocity for fans of bounded Dyck paths,…
The notion of geometric version of an infinitely divisible law is introduced. Concepts parallel to attraction and partial attraction are developed and studied in the setup of geometric summing of random variables.
Three types of reciprocity laws for arithmetic surfaces are established. For these around a point or along a vertical curve, we first construct $K_2$ type central extensions, then introduce reciprocity symbols, and finally prove the law as…
The theory of total positivity for reductive groups is here extended to the case of symmetric spaces.
In this article we present the history of auxiliary primes used in proofs of reciprocity laws from the quadratic to Artin's reciprocity law. We also show that the gap in Legendre's proof can be closed with a simple application of Gauss's…
The theory of Space rotations is introduced. The relativity principle is generalized to satisfy to reference frames rotating in 3D space. It is shown that the most postulates and limitations of quantum theories are consequences of this…