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Rational separability over a global field
1996
Annals of Pure and Applied Logic
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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]
2018
arXiv
pre-print
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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]
2014
arXiv
pre-print
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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]
2009
arXiv
pre-print
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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]
2016
arXiv
pre-print
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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
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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]
2020
arXiv
pre-print
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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
ALGORITHMS FOR GALOIS EXTENSIONS OF GLOBAL FUNCTION FIELDS
2016
Bulletin of the Australian Mathematical Society
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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]
2015
arXiv
pre-print
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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
2001
Manuscripta mathematica
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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
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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]
2016
arXiv
pre-print
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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
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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. ...
A HASSE PRINCIPLE FOR PERIODIC POINTS
2013
International Journal of Number Theory
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2013 pre-print arXiv:1209.2399v3 fatcat:hrb2iv7qqfdrtcazhpmzw2bus4Other Versions
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
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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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