Search

Let k be a field with algebraic closure k ¯ and D ⊂ P k ¯ 1 a reduced, effective divisor of degree n ≥ 3 , write k D for the field of moduli of D. A. Marinatto proved that when n is odd, or n = 4 , D descends to a divisor on P k D 1. We analyze completely the problem of when D descends to a divisor on a smooth, projective curve of genus 0 on k D , possibly with no rational points. In particular, we study the remaining cases n ≥ 6 even, and we obtain conceptual proofs of Marinatto's results and of a theorem by B. Huggins about the field of moduli of hyperelliptic curves. [ABSTRACT FROM AUTHOR]

Given a perfect field $k$ with algebraic closure $\overline {k}$ and a variety $X$ over $\overline {k}$ , the field of moduli of $X$ is the subfield of $\overline {k}$ of elements fixed by field automorphisms $\gamma \in \operatorname {Gal}(\overline {k}/k)$ such that the Galois conjugate $X_{\gamma }$ is isomorphic to $X$. The field of moduli is contained in all subextensions $k\subset k'\subset \overline {k}$ such that $X$ descends to $k'$. In this paper, we extend the formalism and define the field of moduli when $k$ is not perfect. Furthermore, Dèbes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and prove that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher-dimensional varieties, and to varieties with additional structures. In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As a consequence, we prove that a variety $X$ of dimension $d$ with a smooth marked point $p$ such that $\operatorname {Aut}(X,p)$ is finite, étale and of degree prime to $d!$ is defined over its field of moduli. [ABSTRACT FROM AUTHOR]

This is the second part of a two-part article formalizing existence and uniqueness of algebraic closures, using the Mizar [2], [1] formalism. Our proof follows Artin's classical one as presented by Lang in [3]. In the first part we proved that for a given field F there exists a field extension E such that every non-constant polynomial p ∈ F [X] has a root in E. Artin's proof applies Kronecker's construction to each polynomial p ∈ F [X]\F simultaneously. To do so we needed the polynomial ring F [X1, X2,...] with infinitely many variables, one for each polynomal p ∈ F [X]\F. The desired field extension E then is F [X1, X2, ...]\I, where I is a maximal ideal generated by all non-constant polynomials p ∈ F [X]. Note, that to show that I is maximal Zorn's lemma has to be applied. In this second part this construction is iterated giving an infinite sequence of fields, whose union establishes a field extension A of F, in which every non-constant polynomial p ∈ A[X] has a root. The field of algebraic elements of A then is an algebraic closure of F. To prove uniqueness of algebraic closures, e.g. that two algebraic closures of F are isomorphic over F, the technique of extending monomorphisms is applied: a monomorphism F → A, where A is an algebraic closure of F can be extended to a monomorphism E → A, where E is any algebraic extension of F. In case that E is algebraically closed this monomorphism is an isomorphism. Note that the existence of the extended monomorphism again relies on Zorn's lemma. [ABSTRACT FROM AUTHOR]

Given a prime number p let Cp be the topological completion of the algebraic closure of the field of p-adic numbers. Let O(T) be the Galois orbit of a transcendental element T of Cp with respect to the absolute Galois group. Our aim is to study the class of Galois equivariant functions defined on O(T) with values in Cp. We show that each function from this class is continuous and we characterize the class of Lipschitz functions, respectively the class of differentiable functions, with respect to a new orthonormal basis. Then we discuss some aspects related to analytic continuation for the functions of this class. [ABSTRACT FROM AUTHOR]

This is the first part of a two-part article formalizing existence and uniqueness of algebraic closures using the Mizar system [1], [2]. Our proof follows Artin's classical one as presented by Lang in [3]. In this first part we prove that for a given field F there exists a field extension E such that every non-constant polynomial p ∈ F [X] has a root in E. Artin's proof applies Kronecker's construction to each polynomial p ∈ F [X]\F simultaneously. To do so we need the polynomial ring F [X1, X2,...] with infinitely many variables, one for each polynomal p ∈ F [X]\F. The desired field extension E then is F [X1, X2,...]\I, where I is a maximal ideal generated by all non-constant polynomials p ∈ F [X]. Note, that to show that I is maximal Zorn's lemma has to be applied. In the second part this construction is iterated giving an infinite sequence of fields, whose union establishes a field extension A of F, in which every non-constant polynomial p ∈ A[X] has a root. The field of algebraic elements of A then is an algebraic closure of F. To prove uniqueness of algebraic closures, e.g. that two algebraic closures of F are isomorphic over F, the technique of extending monomorphisms is applied: a monomorphism F → A, where A is an algebraic closure of F can be extended to a monomorphism E → A, where E is any algebraic extension of F. In case that E is algebraically closed this monomorphism is an isomorphism. Note that the existence of the extended monomorphism again relies on Zorn's lemma. [ABSTRACT FROM AUTHOR]

Geyer and Jarden proved several results for torsion points of elliptic curves defined over the fixed field by finitely many elements in the absolute Galois group of a finitely generated field over the prime field in its algebraic closure. As an analogue of these results, this paper studies torsion points of Drinfeld modules defined over the fixed field by finitely many elements in the absolute Galois group of a finitely generated function field in its algebraic closure. We prove some results which are similar to those of Geyer and Jarden. [ABSTRACT FROM AUTHOR]

Let p be a prime number. Let k = F ¯ p , the algebraic closure of field F p of p elements, and G be a connected reductive group defined over F p and B be a Borel subgroup of G. Let θ be a (one dimensional) character of B (may not be rational). We give the necessary and sufficient condition for M (θ) = k G ⊗ k B θ being finite length. [ABSTRACT FROM AUTHOR]

Let k be a perfect field with algebraic closure ... If H is a subgroup of plane automorphisms over k and k and ... is a point, we describe the subgroup consisting of plane automorphisms which stabilize the orbit of p under H, when this orbit has irreducible closure in ... As an application, we treat the case where H is cyclic and the closure of the orbit of p is an arbitrary (non-necessarily irreducible) curve. [ABSTRACT FROM AUTHOR]

Let be a field of characteristic with algebraic closure. Let be a connected reductive -group, and let be a spherical variety over (a spherical homogeneous space or a spherical embedding). Let be a -model (-form) of. We give necessary and sufficient conditions for the existence of a -equivariant -model of. [ABSTRACT FROM AUTHOR]

In anabelian geometry, various strong/desired forms of Grothendieck Conjecture-type results for hyperbolic curves over relatively small arithmetic fields — for instance, finite fields, number fields, or p -adic local fields — have been obtained by many researchers, especially by A. Tamagawa and S. Mochizuki. Let us recall that, in their proofs, the Weil Conjecture or p -adic Hodge theory plays an essential role. Therefore, to obtain such Grothendieck Conjecture-type results, it appears that the condition that the cyclotomic characters of the absolute Galois groups of the base fields are highly nontrivial is indispensable. On the other hand, in an author's recent joint work with Y. Hoshi and S. Mochizuki, we introduced the notion of TKND-AVKF-field [concerning the divisible subgroups of the groups of rational points of semi-abelian varieties] and obtained the semi-absolute version of the Grothendieck Conjecture for higher dimensional (≥2) configuration spaces associated to hyperbolic curves of genus 0 over TKND-AVKF-fields contained in the algebraic closure of the field of rational numbers. For instance, every [possibly, infinite] cyclotomic extension field of a number field is such a TKND-AVKF-field. In particular, this Grothendieck Conjecture-type result suggests that the condition that the cyclotomic character of the absolute Galois group of the base field under consideration is sufficiently nontrivial is, in fact, not indispensable for strong/desired form of anabelian phenomena. In the present paper, to pose another evidence for this observation, we prove the relative birational version of the Grothendieck Conjecture for smooth curves over TKND-AVKF-fields with a certain mild condition that every cyclotomic extension field of a number field satisfies. From the viewpoint of the condition on base fields, this result may be regarded as a partial generalization of F. Pop and S. Mochizuki's results on the birational version of the Grothendieck Conjecture for smooth curves. [ABSTRACT FROM AUTHOR]

Let K/Q be an algebraic extension of fields, and let α ≠ 0 be contained in an algebraic closure of K. If α can be approximated by roots of numbers in K× with respect to the Weil height, we prove that some nonzero integer power of α must belong to K×. More generally, let K1, K2, ⋅⋅⋅, KN, be algebraic extensions of Q such that each pair of extensions includes one which is a (possibly infinite) Galois extension of a common subfield. If α ≠ 0 can be approximated by a product of roots of numbers from each Kn with respect to the Weil height, we prove that some nonzero integer power of α must belong to the multiplicative group K1× K2× ⋅⋅⋅ KN×. Our proof of the more general result uses methods from functional analysis. [ABSTRACT FROM AUTHOR]

We prove the existence of global in time weak solutions to a compressible two-fluid Stokes system with a single velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. The system appears to be outside the class of problems that can be treated using the classical Lions-Feireisl approach. Adapting the novel compactness tool developed by the first author and P.-E. Jabin in the mono-fluid compressible Navier-Stokes setting, we first prove the weak sequential stability of solutions. Next, we construct weak solutions via an unconventional approximation using the Lagrangian formulation, truncations, and a stability result of trajectories for rough velocity fields. [ABSTRACT FROM AUTHOR]

We give bounds for the number and the size of the primes p such that a reduction modulo p of a system of multivariate polynomials over the integers with a finite number T of complex zeros does not have exactly T zeros over the algebraic closure of the field with p elements. We apply these bounds to study the periodic points and the intersection of orbits of algebraic dynamical systems over finite fields. In particular, we establish some links between these problems and the uniform dynamical Mordell-Lang conjecture. [ABSTRACT FROM AUTHOR]

The canonical height associated to a polarized endomorphism of a projective variety, constructed by Call and Silverman and generalizing the Néron-Tate height on a polarized abelian variety, plays an important role in the arithmetic theory of dynamical systems. We generalize this construction to polarized correspondences, prove various fundamental properties, and show how the global canonical height decomposes as an integral of a local height over the space of absolute values on the algebraic closure of the field of definition. [ABSTRACT FROM AUTHOR]

We study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational numbers. As an application we confirm finiteness conjectures of Shafarevich and Coleman in the CM case. In addition we prove the uniform boundedness of the Galois invariant subgroup of the geometric Brauer group for forms of a smooth projective variety satisfying the integral Mumford–Tate conjecture. When applied to K3 surfaces, this affirms a conjecture of Várilly-Alvarado in the CM case. [ABSTRACT FROM AUTHOR]

Let K be a fixed algebraic closure of an infinite field K. We consider an absolutely integral curve Γ in PKn with n ≥ 2. The curve ΓK should have only finitely many inflection points, finitely many double tangents, and there exists no point in nK through which infinitely many tangents to ΓK go. In addition there exists a prime number q such that ΓK has a cusp of multiplicity q and themultiplicities of all other points of ΓK are at most q. Under these assumptions, we construct a non-empty Zariski-open subset O of nK such that if n ≥ 3, the projection from each point o ∈ O(K) birationally maps Γ onto an absolutely integ ral curve Γ' in PKn-1 with the same properties as Γ (keeping q unchanged). If n = 2, then the projection from each o ∈ O(K) maps Γ onto PK¹ and leads to a stabilizing element t of the function field F of Γ over K. The latter means that F/K(t) is a finite separable extension whose Galois closure F is regular over K. [ABSTRACT FROM AUTHOR]

Let v be a henselian valuation of arbitrary rank of a field K with value group G(K) and residue field R(K) and be the unique extension of v to a fixed algebraic closure of K with value group . It is known that a complete distinguished chain for an element θ belonging to with respect to (K, v) gives rise to several invariants associated to θ, including a chain of subgroups of , a tower of fields, together with a sequence of elements belonging to which are the same for all K-conjugates of θ. These invariants satisfy some fundamental relations. In this paper, we deal with the converse: Given a chain of subgroups of containing G(K), a tower of extension fields of R(K), and a finite sequence of elements of satisfying certain properties, it is shown that there exists a complete distinguished chain for an element associated to these invariants. We use the notion of lifting of polynomials to construct it. [ABSTRACT FROM AUTHOR]

In this paper, we shall study type-definable groups in a simple theory with respect to one or several stable reducts. While the original motivation came from the analysis of definable groups in structures obtained by Hrushovski's amalgamation method, the notions introduced are in fact more general, and in particular can be applied to certain expansions of algebraically closed fields by operators. We prove the following (Theorem 3.1): Theorem. Let T be simple and T0 be a stable reduct of T. If G is a type-definable group in T, then there are a *-interpretable group H in T0 and a definable homomorphism φ V G0 ! H such that for independent generic elements g, g0 of G we can name a set D independent of g, g0 with acl.g/; acl.g0/j0 acl.φ.gg0//\acl0.acl.g//;acl.g0//acl.gg0/; where j0 and acl0 denote independence and algebraic closure respectively in the reduct T0 over D. Thus φ captures all possible 0-dependence relations resulting from the group operation. In order to ensure that the morphism is non-trivial, we need to add further hypotheses on the geometric complexity of T over T0: relative one-basedness and relative CM-triviality. The kernel of φ is finite in the first case (Theorem 4.9) and virtually central in the latter (Theorem 5.7). Examples for relatively one-based theories are differentially closed fields in characteristic 0 and fields with a generic automorphism. We therefore recover, up to isogeny, the characterisation of definable groups in these structures from [20, 24]. Finally, we show that all coloured fields and fusions constructed by Hrushovski amalgamation are relatively CM-trivial (Theorem 6.4). [ABSTRACT FROM AUTHOR]