scholarly journals Finiteness theorems for algebraic tori over function fields

2021 ◽  
Vol 359 (8) ◽  
pp. 939-944
Author(s):  
Andrei S. Rapinchuk ◽  
Igor A. Rapinchuk
Keyword(s):  
Function Fields ◽  
Algebraic Tori ◽  
Finiteness Theorems

scholarly journals Finiteness theorems for algebraic groups over function fields

2011 ◽  
Vol 148 (2) ◽  
pp. 555-639 ◽  
Author(s):  
Brian Conrad
Keyword(s):  
Function Fields ◽  
Simple Groups ◽  
Structure Theory ◽  
Reductive Groups ◽  
Compactness Criterion ◽  
Group Schemes ◽  
Global Function Fields ◽  
Finiteness Theorems ◽  
Tamagawa Numbers

AbstractWe prove the finiteness of class numbers and Tate–Shafarevich sets for all affine group schemes of finite type over global function fields, as well as the finiteness of Tamagawa numbers and Godement’s compactness criterion (and a local analogue) for all such groups that are smooth and connected. This builds on the known cases of solvable and semi-simple groups via systematic use of the recently developed structure theory and classification of pseudo-reductive groups.

scholarly journals On a classification of the function fields of algebraic tori

1975 ◽  
Vol 56 ◽  
pp. 85-104 ◽  
Author(s):  
Shizuo Endo ◽  
Takehiko Miyata
Keyword(s):  
Finite Group ◽  
Equivalence Relation ◽  
Galois Extension ◽  
Function Fields ◽  
Equivalence Classes ◽  
Finitely Generated ◽  
Abelian Semigroup ◽  
Algebraic Tori ◽  
A Finite Group

Let II be a finite group and denote by MII the class of all (finitely generated Z-free) II-modules. In the previous paper [3] we defined an equivalence relation in MII and constructed the abelian semigroup T(II) by giving an addition to the set of all equivalence classes in MII. The investigation of the semigroup T(II) seems interesting and important, because this gives a classification of the function fields of algebraic tori defined over a field k which split over a Galois extension of k with group II.

Function fields of algebraic tori revisited

2017 ◽  
Vol 21 (2) ◽  
pp. 197-224 ◽  
Author(s):  
Shizuo Endo ◽  
Ming-Chang Kang
Keyword(s):  
Function Fields ◽  
Algebraic Tori

scholarly journals ON THE GROWTH OF LINEAR RECURRENCES IN FUNCTION FIELDS

2020 ◽  
pp. 1-10
Author(s):  
CLEMENS FUCHS ◽  
SEBASTIAN HEINTZE
Keyword(s):  
Number Field ◽  
Function Field ◽  
Function Fields ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Linear Recurrences ◽  
Field Case ◽  
Power Sum ◽  
Linear Recurrence Sequence

Abstract Let $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $ . We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.

On geometric Z p -extensions of function fields

1988 ◽  
Vol 62 (2) ◽  
pp. 145-161 ◽  
Author(s):  
R. Gold ◽  
H. Kisilevsky
Keyword(s):  
Function Fields
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