Related papers: Notes on abelian class field theory

We introduce and study a natural class of fields in which certain first-order definable sets are existentially definable, and characterise this class by a number of equivalent conditions. We show that global fields belong to this class, and…

Expanded lecture notes. Preliminary version, comments are welcome.

This exposition begins with a systematic account of the theory of group schemes, ultimately specializing to algebraic tori.

In these lectures we present a few topics in Quantum Field Theory in detail. Some of them are conceptual and some more practical. They have been selected because they appear frequently in current applications to Particle Physics and String…

The Jacobian algebras are introduced and their various properties are studied.

In this paper we provide an algorithm to classify groups of points on abelian threefolds over finite fields. The classification is given in terms of the Weil polynomial of abelian varieties in a given $\mathbb{F}_q$-isogeny class. This work…

This is the text from a talk at the Arbeitstagung 2011, which can serve as an introduction to arxiv:1009.0736 and arXiv:1007.0907. I first discuss how a global field is determined by a certain dynamical system, and how this relates to…

Among abelian extensions of a congruence function field, an asymptotic relation of class number and genus is established. The proof is classical, employing well-known results from congruence function field theory.

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…

In this exposition we discuss the theory of algebraic extensions of valued fields. Our approach is mostly through Galois theory. Most of the results are well-known, but some are new. No previous knowledge on the theory of valuations is…

A comprehensive introduction to two-dimensional conformal field theory is given.

These are notes on some entanglement properties of quantum field theory, aiming to make accessible a variety of ideas that are known in the literature. The main goal is to explain how to deal with entanglement when -- as in quantum field…

We prove various characterizations of the period torsor of abelian varieties. This is the submitted version.

We study the arithmetic aspects of the finite group of extensions of abelian varieties defined over a number field. In particular, we establish relations with special values of L-functions and congruences between modular forms.

In this note we give a detailed proof of a theorem of Aubin.

We study base field extensions of ordinary abelian varieties defined over finite fields using the module theoretic description introduced by Deligne. As applications we give algorithms to determine the minimal field of definition of such a…

We give upper and lower bounds on the number of points on abelian varieties over finite fields, and lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian…

We give a rough description of the 'categories' formed by quantum field theories. A few recent mathematical conjectures derived from quantum field theories, some of which are now proven theorems, will be presented in this language.

Tate's theorem (Invent. Math. 1966)implies that the Tate conjecture holds for any abelian variety over a finite field whose Q_l-algebra of Tate classes is generated by those of degree 1. We construct families of abelian varieties over…

In this paper, we provide a classification of certain points on Hilbert modular varieties over finite fields under a mild assumption on Newton polygon. Furthermore, we use this characterization to prove estimates for the size of isogeny…