Related papers: Notes on Nonlinear Number Fields
A version of the nonlinear Hodge equations is introduced in which the irrotationality condition is weakened. An elliptic estimate for solutions is derived.
We describe a new construction of families of Galois coverings of the line using basic properties of configuration spaces, covering theory, and the Grauert-Remmert Extension Theorem. Our construction provides an alternative to a previous…
This paper presents relevant modern mathematical formulations for (classical) gauge field theories, namely, ordinary differential geometry, noncommutative geometry, and transitive Lie algebroids. They provide rigorous frameworks to describe…
These lecture notes provide an introduction to logarithmic geometry with a view towards recent applications in the desingularization theory.
This is a largely expository paper about how groups arise or are of interest in model theory. Included are the following topics: classifying groups definable in specific structures or theories and the relation to algebraic groups, groups…
We prove an upper bound for the asymptotics of counting functions of number fields with nilpotent Galois groups.
We present some fundamental results on (possibly nonlinear) algebraic semigroups and monoids. These include a version of Chevalley's structure theorem for irreducible algebraic monoids, and the description of all algebraic semigroup…
Certain operator-valued functions and new generating structures (instead of generating functionals) are proposed for the analysis of equations for n-point information (n-pi). Some remarks are made concerning the intertwining of linearity…
We introduce Galois families of modular forms. They are a new kind of family coming from Galois representations of the absolute Galois groups of rational function fields over the rational field. We exhibit some examples and provide an…
A main problem in Galois theory is to characterize the fields with a given absolute Galois group. We apply a K-theoretic method for constructing valuations to study this problem in various situations. As a first application we obtain an…
A point on a plane curve is said to be Galois (for the curve) if the projection from the point as a map from the curve to a line induces a Galois extension of function fields. It is known that the number of Galois points is finite except…
We define vector fields, leaves and trajectories for schemes. With these tools, we are able to give a geometrical interpretation and to generalize several results of differential Galois theory and constructions on differential schemes. We…
In this paper we produce unconditionally new instances of Galois number field extensions exhibiting strong discrepancies in the distribution of Frobenius elements among conjugacy classes of the Galois group. We first prove an inverse Galois…
We show how to construct unramified qoaternion extensions of quadratic number fields.
In this paper, we give a geometrization and a generalization of a lemma of differential Galois theory. This geometrization, in addition of giving a nice insight on this result, offers us the occasion to investigate several points of…
A number field is said to be a CM-number field if it is a totally imaginary quadratic extension of a totally real number field. We define a totally imaginary number field to be of CM-type if it contains a CM-subfield, and of TR-type if it…
Apart from a few remarks on lattice systems with global or gauge symmetries, most of this talk is devoted to some interesting ancient examples of symmetries and their breakdowns in elasticity theory and hydrodynamics. Since Galois Theory is…
We estimate, in a number field, the number of elements and the maximal number of linearly independent elements, with prescribed bounds on their valuations. As a by-product, we obtain new bounds for the successive minima of ideal lattices.…
Over the last decade, non-relativistic theories have attracted considerable attention. In general, such theories can be obtained by contracting relativistic parent theories. In this work, we discuss features of non-relativistic…
We show that the image of the adelic Galois representation attached to a non-CM modular form is open in the adelic points of a suitable algebraic group. We also show a similar result for the adelic Galois representation attached to a finite…