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Rational separability over a global field

Alexandra Shlapentokh
1996 Annals of Pure and Applied Logic  
function field over a perfect field of constants, the existence of the above-described collection of rational functions is equivalent to the requirement that the non-archimedean primes which do not appear  ...  Let F be a finitely generated field and let j:F -+ N be a weak presentation of F, i.e. an isomorphism from F onto a field whose universe is a subset of N and such that all the field operations are extendible  ...  We will next investigate the rational algebraic separability between holomorphy subrings of a global field.  ... 
doi:10.1016/0168-0072(95)00023-2 fatcat:bybgngvf7vczpgumxmi72rn5iq

Local-Global Principles for Zero-Cycles on Homogeneous Spaces over Arithmetic Function Fields [article]

Jean-Louis Colliot-Thélène, David Harbater, Julia Hartmann, Daniel Krashen, R. Parimala, V. Suresh
2018 arXiv   pre-print
In particular, we show that local-global principles hold for such zero-cycles provided that local-global principles hold for the existence of rational points over extensions of the function field.  ...  We study the existence of zero-cycles of degree one on varieties that are defined over a function field of a curve over a complete discretely valued field.  ...  Given a variety V over a field k, the index (resp. separable index ) of V is the greatest common divisor of the degrees of the finite (resp. finite separable) field extensions of k over which V has a rational  ... 
arXiv:1710.03173v2 fatcat:f2ebz7cd6vcericckaxjy2cjiu

On Rational Connectedness of Globally F-Regular Threefolds [article]

Yoshinori Gongyo, Zhiyuan Li, Zsolt Patakfalvi, Karl Schwede, Hiromu Tanaka, Hong R. Zong
2014 arXiv   pre-print
In this paper, we show that projective globally F-regular threefolds, defined over an algebraically closed field of characteristic p≥ 11, are rationally chain connected.  ...  Suppose that (X, ∆) is a proper, globally F -regular variety over an F -finite field k such that H 0 (X, O X ) ⊇ k is a separable field extension. Then (X, ∆) is geometrically globally F -regular.  ...  On separable rational connectedness In this section, we work over an algebraically closed field k of characteristic p > 0.  ... 
arXiv:1307.8188v2 fatcat:sh4dpxtfrzdjbnxhtdqfd5fkve

Curves over global fields violating the Hasse Principle [article]

Pete L. Clark
2009 arXiv   pre-print
In response to a question of B. Poonen, we exhibit for each global field k an algebraic curve over k which violates the Hasse Principle.  ...  We also use TAHP to construct further Hasse Principle violations, for instance among curves over any number field of any given genus g which is at least 2.  ...  Let k be a global field, l/k a finite separable field extension, and V /l a nice variety.  ... 
arXiv:0905.3459v1 fatcat:teyporm4yvfqzlzsmazu6cpmui

The local-global principle for symmetric determinantal representations of smooth plane curves in characteristic two [article]

Yasuhiro Ishitsuka, Tetsushi Ito
2016 arXiv   pre-print
We prove that a smooth plane curve over a global field of characteristic two is defined by the determinant of a symmetric matrix with entries in linear forms in three variables if and only if such a symmetric  ...  It is a special feature in characteristic two because analogous results are not true in other characteristics.  ...  The conic C B admits a symmetric determinantal representation over a separable quadratic extension of K because C B has a rational point over a separable quadratic extension by Bertini's theorem; see  ... 
arXiv:1412.8343v6 fatcat:k3snudkfk5grpd76oybec2dgqe

Page 5976 of Mathematical Reviews Vol. , Issue 99i [page]

1999 Mathematical Reviews  
Let k be a field of characteristic different from 2. A hyperelliptic function field over k is a quadratic extension of a rational function field over k of one variable.  ...  Let K be a global function field with full constant field F,, the finite field of g elements. Let N(K) be the number of rational places of K.  ... 

Compatibility of weak approximation for zero-cycles on products of varieties [article]

Yongqi Liang
2020 arXiv   pre-print
Zero-cycles are conjectured to satisfy weak approximation with Brauer-Manin obstruction for proper smooth varieties defined over number fields.  ...  Roughly speaking, we prove that the conjecture is compatible for products of rationally connected varieties, K3 surfaces, Kummer varieties, and one curve.  ...  Suppose that y ∞ is a global effective 0-cycle of C of degree ∆ > 2g and z v is a separable effective 0-cycle rationally equivalent to y ∞ on C kv for all v ∈ S.  ... 
arXiv:2004.09343v1 fatcat:q625j3kw4jfptjowyotpjt4lsy


2016 Bulletin of the Australian Mathematical Society  
Global function fields, which are finite separable extensions of a global rational function field, are interesting because they provide a basis for designing efficient algorithms for algebraic curves.  ...  We investigate the efficient computation of integral closures, or maximal orders, in cyclic extensions of global fields and the determination of Galois groups for polynomials over global function fields  ...  Function fields defined over a finite field k (with characteristic p > 0), along with number fields, are global fields.  ... 
doi:10.1017/s0004972715001793 fatcat:ex7ucffxorgjxkak3yo5ouirry

Weak Approximation for Cubic Hypersurfaces and Degree 4 del Pezzo Surfaces [article]

Letao Zhang, Zhiyu Tian
2015 arXiv   pre-print
over global function fields, if there is a rational point, then weak approximation holds at places of good reduction whose residual field has at least 11 elements. (2) For del Pezzo surfaces of degree  ...  4 defined over global function fields, if there is a rational point, then weak approximation holds at places of good reduction whose residual field has at least 13 elements. (3) Weak approximation holds  ...  By Lemma 3.4, the set of rational points of a del Pezzo surface of degree at least 4 (in fact any smooth projective separably rationally connected variety) defined over a global function field is either  ... 
arXiv:1511.08156v1 fatcat:55tmva5arbawllurytbv2iajda

The Mordel-Weil theorems for Drinfeld modules over finitely generated function fields

Julie Tzu-Yueh Wang
2001 Manuscripta mathematica  
We generalize Poonen's analogue of Mordell-Weil theorems for Drinfeld modules over global function fields to the case of Drinfeld modules over finitely generated function fields.  ...  In addition, the A-characteristic of the function fields under our consideration can be arbitrary.  ...  K = a global function field with field of constant F q . ∞ = a fixed place of K.  ... 
doi:10.1007/s00229-001-0207-2 fatcat:y4rx5z6ppzbdxaxxo6diaoytju

Page 52 of Mathematical Reviews Vol. 49, Issue 1 [page]

1975 Mathematical Reviews  
A global field is either a finite extension field of Q (the rational number field) or a field of algebraic functions in one variable over a finite field of constants.  ...  pairing for elliptic curves over local fields in such a way that one can define analogous things for elliptic curves over generalized local fields, quasi-local fields, global and quasi-global fields.  ... 

Morphisms to Brauer-Severi Varieties, with Applications to Del Pezzo Surfaces [article]

Christian Liedtke
2016 arXiv   pre-print
As an application, we study del Pezzo surfaces of large degree with a view towards Brauer-Severi varieties, and recover classical results on rational points, the Hasse principle, and weak approximation  ...  We classify morphisms from proper varieties to Brauer-Severi varieties, which generalizes the classical correspondence between morphisms to projective space and globally generated invertible sheaves.  ...  By our conventions above, a variety over a field k is a scheme X that is of finite type, separated, and geometrically integral over k.  ... 
arXiv:1602.07491v2 fatcat:ycotziufszbxhf7st4jlnoxhzm

Page 2939 of Mathematical Reviews Vol. , Issue 2001E [page]

2001 Mathematical Reviews  
With similar considerations they also prove the rationality of K(x, y) over K (any field) when a is defined on x and y by certain fractional linear transformations.  ...  Let k be a field of characteristic p > 0 and F = k(x) be the field of rational functions in one variable.  ... 


2013 International Journal of Number Theory  
Let F be a global field, let ∈ be a rational map of degree at least 2, and let ∈ F. We say that is periodic if () = for some n ≥ 1.  ...  A Hasse principle is the idea, or hope, that a phenomenon which happens everywhere locally should happen globally as well.  ...  Let F be a function field with a finite field of constants, letp be a prime of o F , and let ϕ : P 1 → P 1 be a rational function defined over F with separable degree at least 2 and with separable good  ... 
doi:10.1142/s1793042113500747 fatcat:rk2cyuiubzc63iftfh24ngiv5u

Page 101 of American Journal of Mathematics Vol. 81, Issue 1 [page]

1959 American Journal of Mathematics  
To prove Theorem 1 or 2, we may assume that all point of A,» are rational over K, because we may deal with the finite separable extension K(A,,) of K instead of K itself.  ...  By a global field we shall mean a field K which is either a function field over an algebraically closed constant field k (i.e. a finitely generated regular extension of k) or an algebraic number field  ... 
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