Related papers: Class field theory for open curves over local fiel…
These notes are an introduction to higher dimensional local fields and higher dimensional adeles. As well as the foundational theory, we summarise the theory of topologies on higher dimensional local fields and higher dimensional local…
On one hand, we study the class of graphs on surfaces, satisfying tessellation properties, with positive Forman curvature on each edge. Via medial graphs, we provide a new proof for the finiteness of the class, and give a complete…
We study the distribution of algebraic points on curves in abelian varieties over finite fields.
We study geometric properties of linear strata of uni-singular curves. The singularities of closures of the strata are resolved and the resolutions are represent as projective bundles. This enables to study their geometry. In particular we…
This work derives closed-form expressions computing the expectation of co-presence and of number of co-occurrences of nodes on paths sampled from a network according to general path weights (a bag of paths). The underlying idea is that two…
Following the analogies between 3-dimensional topology and number theory, we study an id\`elic form of class field theory for 3-manifolds. For a certain set $\mathcal{K}$ of knots in a 3-manifold $M$, we first present a local theory for…
We study a dynamical system induced by the Artin reciprocity map for a global field. We translate the conjugacy of such dynamical systems into various arithmetical properties that are equivalent to field isomorphism, relating it to…
It is often the case that a covering map of the open annulus is semiconjugate to a map of the circle of the same degree. We investigate this possibility and its consequences on the dynamics. In particular, we address the problem of the…
A continuous cohomology theory for topological quandles is introduced, and compared to the algebraic theories. Extensions of topological quandles are studied with respect to continuous 2-cocycles, and used to show the differences in second…
This thesis develops some of the basic model theory of covers of algebraic curves. In particular, an equivalence between the good model-theoretic behaviour of the modular j-function, and the openness of certain Galois representations in the…
We determine the convergence regions of certain local integrals on the moduli spaces of curves in neighborhoods of fixed stable curves in terms of the combinatorics of the corresponding graphs.
We examine \'etale covers of genus two curves that occur in the linear system of a polarizing line bundle of type $(1,d)$ on a complex abelian surface. We give results counting fixed points of involutions on such curves as well as…
We study integrability for coactions of locally compact groups. For abelian groups, this corresponds to integrability of the associated action of the Pontrjagin dual group. The theory of integrable group actions has been previously studied…
The calculus of classes and closure operations has proved to be a useful tool in group theory and has led to a deep theory in the study of finite soluble groups. More recently, parallel theories have started to be developed in various…
We use knowledge of local fields to adapt Jonathan Lubin and Michael Rosen's proof of Mazur's Proposition 4.39. This changes the result about abelian varieties from only working over local fields with a finite residue field to working with…
Working over imperfect fields, we give a comprehensive classification of genus-one curves that are regular but not geometrically regular, extending the known case of geometrically reduced curves. The description is given intrinsically, in…
In recent papers, Fesenko has defined the non-abelian local reciprocity map for every totally-ramified arithmetically profinite ($APF$) Galois extension of a given local field $K$ by extending the works of Hazewinkel and Neukirch-Iwasawa.…
We prove, under some mild hypothesis, that an \'etale cover of curves defined over a number field has infinitely many specializations into an everywhere unramified extension of number fields. This constitutes an "absolute" version of the…
This note gives the complete projective classification of rational, cuspidal plane curves of degree at least 6, and having only weighted homogeneous singularities. It also sheds new light on some previous characterizations of free and…
The present work aims to exploit the interplay between the algebraic properties of rings and the graph-theoretic structures of their associated graphs. We introduce commutatively closed graphs and investigate properties of commutatively…