Affine homogeneous spaces and finite subgroups of arithmetic groups over function fields

1977 ◽  
Vol 11 (1) ◽  
pp. 64-66 ◽  
Author(s):  
E. A. Nisnevich
Keyword(s):  
Homogeneous Spaces ◽  
Function Fields ◽  
Arithmetic Groups ◽  
Finite Subgroups

scholarly journals Local-global principles for zero-cycles on homogeneous spaces over arithmetic function fields

2019 ◽  
Vol 372 (8) ◽  
pp. 5263-5286
Author(s):  
J.-L. Colliot-Thélène ◽  
D. Harbater ◽  
J. Hartmann ◽  
D. Krashen ◽  
R. Parimala ◽  
...  
Keyword(s):  
Homogeneous Spaces ◽  
Arithmetic Function ◽  
Function Fields ◽  
Global Principles

scholarly journals Finite presentability of arithmetic groups over global function fields

1987 ◽  
Vol 30 (1) ◽  
pp. 23-39 ◽  
Author(s):  
Helmut Behr
Keyword(s):  
Small Groups ◽  
Function Fields ◽  
Algebraic Groups ◽  
Number Fields ◽  
Arithmetic Groups ◽  
Finitely Generated ◽  
Global Function ◽  
Global Function Fields ◽  
Finite Presentability ◽  
Finitely Presented

Arithmetic subgroups of reductive algebraic groups over number fields are finitely presentable, but over global function fields this is not always true. All known exceptions are “small” groups, which means that either the rank of the algebraic group or the set S of the underlying S-arithmetic ring has to be small. There exists now a complete list of all such groups which are not finitely generated, whereas we onlyhave a conjecture which groups are finitely generated but not finitely presented.

scholarly journals ON GENERATORS OF ARITHMETIC GROUPS OVER FUNCTION FIELDS

2011 ◽  
Vol 07 (06) ◽  
pp. 1573-1587
Author(s):  
MIHRAN PAPIKIAN
Keyword(s):  
Quaternion Algebra ◽  
Function Fields ◽  
Rational Functions ◽  
Arithmetic Groups ◽  
Finite Sets

Let F = 𝔽q(T) be the field of rational functions with 𝔽q-coefficients, and A = 𝔽q[T] be the subring of polynomials. Let D be a division quaternion algebra over F which is split at 1/T. For certain A-orders in D we find explicit finite sets generating their groups of units.

scholarly journals Patching and local-global principles for homogeneous spaces over function fields of p-adic curves

2012 ◽  
Vol 87 (4) ◽  
pp. 1011-1033 ◽  
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Raman Parimala ◽  
Venapally Suresh
Keyword(s):  
Homogeneous Spaces ◽  
Function Fields ◽  
Global Principles

scholarly journals Local-global principles for homogeneous spaces over some two-dimensional geometric global fields

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Diego Izquierdo ◽  
Giancarlo Lucchini Arteche
Keyword(s):  
Laurent Series ◽  
Homogeneous Spaces ◽  
Function Fields ◽  
Independent Interest ◽  
Complex Numbers ◽  
Brauer Group ◽  
Two Dimensional ◽  
Global Fields ◽  
Global Principles ◽  
Global Principle

Abstract In this article, we study the obstructions to the local-global principle for homogeneous spaces with connected or abelian stabilizers over finite extensions of the field ℂ ⁢ ( ( x , y ) ) {\mathbb{C}((x,y))} of Laurent series in two variables over the complex numbers and over function fields of curves over ℂ ⁢ ( ( t ) ) {\mathbb{C}((t))} . We give examples that prove that the Brauer–Manin obstruction with respect to the whole Brauer group is not enough to explain the failure of the local-global principle, and we then construct a variant of this obstruction using torsors under quasi-trivial tori which turns out to work. In the end of the article, we compare this new obstruction to the descent obstruction with respect to torsors under tori. For that purpose, we use a result on towers of torsors, that is of independent interest and therefore is proved in a separate appendix.

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