Related papers: Hasse principles for quadratic forms over function…

Given a number field $K$, a finite abelian group $G$ and finitely many elements $\alpha_1,\ldots,\alpha_t\in K$, we construct abelian extensions $L/K$ with Galois group $G$ that realise all of the elements $\alpha_1,\ldots,\alpha_t$ as…

We develop a theory of extensions of hyperfields that generalizes the notion of field extensions. Since hyperfields have a multivalued addition, we must consider two kinds of extensions that we call weak hyperfield extensions and strong…

We show how to express any Hasse-Schmidt derivation of an algebra in terms of a finite number of them under natural hypothesis. As an application, we obtain coefficient fields of the completion of a regular local ring of positive…

The Hasse principle in number theory states that information about integral solutions to Diophantine equations can be pieced together from real solutions and solutions modulo prime powers. We show that the Hasse principle holds for…

The ruled residue theorem characterises residue field extensions for valuations on a rational function field. Under the assumption that the characteristic of the residue field is different from $2$ this theorem is extended here to function…

The main purpose of the paper is to establish a closedness theorem over Henselian valued fields $K$ of equicharacteristic zero (not necessarily algebraically closed) with separated analytic structure. It says that every projection with a…

Let $R$ be a discrete valuation ring of field of fractions $K$ and of residue field $k$ of characteristic $p > 0$. In an earlier work, we studied the question of extending torsors on $K$-curves into torsors over $R$-regular models of the…

Let $p$ and $q$ be anisotropic quadratic forms of dimension $\geq 2$ over a field $F$. In a recent article, we formulated a conjecture describing the general constraints which the dimensions of $p$ and $q$ impose on the isotropy index of…

We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over Q and F_q(t), and conclude with a…

We give large families of Shimura curves defined by congruence conditions, all of whose twists lack $p$-adic points for some $p$. For each such curve we give analytically large families of counterexamples to the Hasse principle via the…

The paper presents a classification of quadratic extension algebras, also known as algebras of degree 2, as well as several characterizations of quaternion algebras over a field (of characteristic not 2). The presentation is not restricted…

Let G be an algebraic group defined over an algebraically closed field k of characteristic zero. We give a simple proof of the following result: if H^1(L, G) = {1} for some finitely generated field extension L/k of transcendence degree \ge…

This paper is concerned with algebraic geometry over complete discretely valued fields $K$ of equicharacteristic zero. Several results are given including: the canonical projection $K^{n} \times K\mathbb{P}^{m} \longrightarrow K^{n}$ and…

We present a general structure theorem for the Hardy field of an o-minimal expansion of the reals by restricted analytic functions and an unrestricted exponential. We proceed to analyze its residue fields with respect to arbitrary convex…

A theorem of Albert-Draxl states that if a tensor product of two quaternion division algebras $Q_1$, $Q_2$ over a field $F$ is not a division algebra, then there exists a separable quadratic extension of $F$ that embeds as a subfield in…

In this article we give an analogue of Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields. Let $K$ be a real quadratic field and $\Om_K$ its ring of integers. Let $\Gamma$ be a congruence subgroup of $\SL_2(\Om_K)$…

We prove a Hasse principle for binary direct summands of the Chow motive of a smooth projective quadric Q over a number field F. Besides, we show that such summands are twists of Rost motives. In the case when F has at most one real…

We define the universal exponential extension of an algebraically closed differential field and investigate its properties in the presence of a nice valuation and in connection with linear differential equations. Next we prove normalization…

This thesis is a contribution to the model theory of valued fields. We study forking in valued fields and some of their reducts. We focus particularly on pseudo-local fields, the ultraproducts of residue characteristic zero of the p-adic…

For an algebraic function field $F/K$ and a discrete valuation $v$ of $K$ with perfect residue field $k$, we bound the number of discrete valuations on $F$ extending $v$ whose residue fields are algebraic function fields of genus zero over…