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

2001 ◽  
Vol 106 (3) ◽  
pp. 305-314 ◽  
Author(s):  
Julie Tzu-Yueh Wang
Keyword(s):  
Function Fields ◽  
Drinfeld Modules ◽  
Finitely Generated

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 .

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.

The u-invariant of p-adic function fields

2013 ◽  
Vol 2013 (679) ◽  
pp. 65-73 ◽  
Author(s):  
David B. Leep
Keyword(s):  
Quadratic Form ◽  
Recent Result ◽  
Quadratic Forms ◽  
Function Fields ◽  
Field Extension ◽  
Finitely Generated ◽  
Nontrivial Zero

Abstract Over a finitely generated field extension in m variables over a p-adic field, any quadratic form in more than 2m + 2 variables has a nontrivial zero. This bound is sharp. We extend this result to a wider class of fields. A key ingredient to our proofs is a recent result of Heath-Brown on systems of quadratic forms over p-adic fields.

scholarly journals Arithmetic of positive characteristic -series values in Tate algebras

2015 ◽  
Vol 152 (1) ◽  
pp. 1-61 ◽  
Author(s):  
B. Anglès ◽  
F. Pellarin ◽  
F. Tavares Ribeiro ◽  
F. Demeslay
Keyword(s):  
Finite Fields ◽  
Positive Characteristic ◽  
Function Fields ◽  
Drinfeld Modules ◽  
New Class ◽  
Arithmetic Theory

The second author has recently introduced a new class of$L$-series in the arithmetic theory of function fields over finite fields. We show that the values at one of these$L$-series encode arithmetic information of a generalization of Drinfeld modules defined over Tate algebras that we introduce (the coefficients can be chosen in a Tate algebra). This enables us to generalize Anderson’s log-algebraicity theorem and an analogue of the Herbrand–Ribet theorem recently obtained by Taelman.

scholarly journals Local Shtukas and Divisible Local Anderson Modules

2019 ◽  
Vol 71 (5) ◽  
pp. 1163-1207 ◽  
Author(s):  
Urs Hartl ◽  
Rajneesh Kumar Singh
Keyword(s):  
Lie Groups ◽  
Function Fields ◽  
Drinfeld Modules ◽  
Arbitrary Base ◽  
Divisible Groups ◽  
Dieudonne Theory

AbstractWe develop the analog of crystalline Dieudonné theory for$p$-divisible groups in the arithmetic of function fields. In our theory$p$-divisible groups are replaced by divisible local Anderson modules, and Dieudonné modules are replaced by local shtukas. We show that the categories of divisible local Anderson modules and of effective local shtukas are anti-equivalent over arbitrary base schemes. We also clarify their relation with formal Lie groups and with global objects like Drinfeld modules, Anderson’s abelian$t$-modules and$t$-motives, and Drinfeld shtukas. Moreover, we discuss the existence of a Verschiebung map and apply it to deformations of local shtukas and divisible local Anderson modules. As a tool we use Faltings’s and Abrashkin’s theories of strict modules, which we review briefly.

scholarly journals On the arithmetic of abelian varieties

2020 ◽  
Vol 2020 (762) ◽  
pp. 1-33
Author(s):  
Mohamed Saïdi ◽  
Akio Tamagawa
Keyword(s):  
Function Fields ◽  
Abelian Varieties ◽  
Galois Cohomology ◽  
Rational Points ◽  
Selmer Groups ◽  
Selmer Group ◽  
Cohomology Groups ◽  
Finitely Generated ◽  
Natural Objects ◽  
Weil Group

AbstractWe prove some new results on the arithmetic of abelian varieties over function fields of one variable over finitely generated (infinite) fields. Among other things, we introduce certain new natural objects “discrete Selmer groups” and “discrete Shafarevich–Tate groups”, and prove that they are finitely generated {\mathbb{Z}}-modules. Further, we prove that in the isotrivial case, the discrete Shafarevich–Tate group vanishes and the discrete Selmer group coincides with the Mordell–Weil group. One of the key ingredients to prove these results is a new specialisation theorem for first Galois cohomology groups, which generalises Néron’s specialisation theorem for rational points of abelian varieties.

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