scholarly journals Torsion points of Drinfeld modules over large algebraic extensions of finitely generated function fields

2021 ◽  
Author(s):  
Takuya Asayama
Keyword(s):  
Function Fields ◽  
Drinfeld Modules ◽  
Finitely Generated ◽  
Torsion Points

scholarly journals Galois criterion for torsion points of Drinfeld modules

2021 ◽  
pp. 1-12
Author(s):  
Chien-Hua Chen
Keyword(s):  
Maximal Ideal ◽  
Abelian Varieties ◽  
Number Fields ◽  
Rational Points ◽  
Drinfeld Module ◽  
Torsion Point ◽  
Drinfeld Modules ◽  
Torsion Points ◽  
Ideal Formula ◽  
Almost All

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal [Formula: see text] of [Formula: see text], the question essentially asks whether, up to isogeny, a Drinfeld module [Formula: see text] over [Formula: see text] contains a rational [Formula: see text]-torsion point if the reduction of [Formula: see text] at almost all primes of [Formula: see text] contains a rational [Formula: see text]-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is 2, but negative if the rank is 3. Moreover, for rank 3 Drinfeld modules we classify those cases where the answer is positive.

Torsion Points of Drinfeld Modules

1995 ◽  
Vol 38 (1) ◽  
pp. 3-10
Author(s):  
Sunghan Bae ◽  
Jakyung Koo
Keyword(s):  
Drinfeld Module ◽  
Drinfeld Modules ◽  
Locally Compact ◽  
Torsion Points ◽  
Complete Field ◽  
Rank 2

AbstractThe finiteness of K-rational torsion points of a Drinfeld module of rank 2 over a locally compact complete field K with a discrete valuation is proved.

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 residually transcendental valued function fields of conics

1996 ◽  
Vol 38 (2) ◽  
pp. 137-145
Author(s):  
Sudesh K. Khanduja
Keyword(s):  
Valuation Ring ◽  
Function Fields ◽  
Residue Field ◽  
Field Extension ◽  
Transcendence Degree ◽  
Finitely Generated ◽  
Uniqueness Property ◽  
Transcendental Extension

Let K/Kobe a finitely generated field extension of transcendence degree 1. Let u0 be a valuation of Koand v a valuation of Kextending v0such that the residue field of vis a transcendental extension ofthe residue field k0of vo/such a prolongation vwill be called a residually transcendental prolongation of v0. Byan element with the uniqueness propertyfor (K, v)/(K0, v0) (or more briefly for v/v0)we mean an element / of Khaving u-valuation 0 which satisfies (i) the image of tunder the canonicalhomomorphism from the valuation ring of vonto the residue field of v(henceforth referred to as the v-residue ot t) is transcendental over ko; that is vcoincides with the Gaussian valuation on the subfield K0(t) defined by (ii) vis the only valuation of K (up to equivalence) extending the valuation .

On the distribution of torsion points modulo primes: the case of function fields

2015 ◽  
Vol 148 (3-4) ◽  
pp. 435-445
Author(s):  
Yen-Mei J. Chen ◽  
Yen-Liang Kuan
Keyword(s):  
Function Fields ◽  
Torsion Points

Primitive submodules for Drinfeld modules

2015 ◽  
Vol 159 (2) ◽  
pp. 275-302 ◽  
Author(s):  
WENTANG KUO ◽  
DAVID TWEEDLE
Keyword(s):  
Residue Field ◽  
Finite Extension ◽  
Rational Numbers ◽  
Module Structure ◽  
Drinfeld Modules ◽  
Finitely Generated ◽  
Natural Density

AbstractThe ring A = $\mathbb{F}$r[T] and its fraction field k, where r is a power of a prime p, are considered as analogues of the integers and rational numbers respectively. Let K/k be a finite extension and let φ be a Drinfeld A-module over K of rank d and Γ ⊂ K be a finitely generated free A-submodule of K, the A-module structure coming from the action of φ. We consider the problem of determining the number of primes ℘ of K for which the reduction of Γ modulo ℘ is equal to $\mathbb{F}$℘ (the residue field of the prime ℘). We can show that there is a natural density of primes ℘ for which Γ mod ℘ is equal to $\mathbb{F}$℘. In certain cases, this density can be seen to be positive.

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