Local-global problem for Drinfeld modules (original) (raw)
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On a reduction map for Drinfeld modules
Acta Arithmetica, 2020
In this paper we investigate a local to global principle for Mordell-Weil group defined over a ring of integers O K of t-modules that are products of the Drinfeld modules ϕ = φ e1 1 × • • • × φ et t. Here K is a finite extension of the field of fractions of A = F q [t]. We assume that the rank(φ i) = d i and endomorphism rings of the involved Drinfeld modules of generic characteristic are the simplest possible, i.e. End K sep (φ i) = A for i = 1,. .. , t. Our main result is the following numeric criterion. Let N = N e1 1 ו • •×N et t be a finitely generated A submodule of the Mordell-Weil group ϕ(O K) = φ 1 (O K) e1 ו • •×φ t (O K) et , and let Λ ⊂ N be an A-submodule. If we assume d i ≥ e i and P ∈ N such that red W (P) ∈ red W (Λ) for almost all primes W of O K , then P ∈ Λ + N tor. We also build on the recent results of S.Barańczuk [B17] concerning the dynamical local to global principle in Mordell-Weil type groups and the solvability of certain dynamical equations to the aforementioned t-modules. This theorem is in fact the detecting linear dependence problem for number fields and of the kind considered in Question 1.1. The reduction map is the usual reduction modulo a non Archimedean prime in a number field. It is well known that some questions concerning number fields can be translated to the context of abelian varieties. An analogous question
The Erdos and Halberstam theorems for Drinfeld modules of any rank
2007
Abstract Let Fq be the finite field with q elements, A:= Fq [T] and F:= Fq (T). Let φ be a Drinfeld A-module over F with trivial endomorphism ring. We prove analogues of the Erdös and Halberstam Theorems for φ. If φ has rank≥ 3, we assume the validity of the Mumford-Tate Conjecture for φ. 1
Journal of Number Theory, 2021
For q an odd prime power, A = Fq[T ], and F = Fq(T), let ψ : A → F {τ } be a Drinfeld A-module over F of rank 2 and without complex multiplication, and let p = pA be a prime of A of good reduction for ψ, with residue field Fp. We study the growth of the absolute value |∆p| of the discriminant of the Fp-endomorphism ring of the reduction of ψ modulo p and prove that, for all p, |∆p| grows with |p|. Moreover, we prove that, for a density 1 of primes p, |∆p| is as close as possible to its upper bound |a 2 p − 4µpp|, where X 2 + apX + µpp ∈ A[X] is the characteristic polynomial of τ deg p .
On the Grothendieck rings of generalized Drinfeld doubles
Journal of Algebra, 2017
In this paper it is shown that any irreducible representation of a Drinfeld double D(A) of a semisimple Hopf algebra A can be obtained as an induced representation from a certain Hopf subalgebra of D(A). This generalizes a well known result concerning the irreducible representations of Drinfeld doubles of finite groups [11]. Using this description we also give a formula for the fusion rules of semisimple Drinfeld doubles. This shows that the Grothendieck rings of these Drinfeld doubles have a ring structure similar to the Grothendieck rings of Drinfeld doubles of finite groups.
On the torsion points of Drinfeld modules in abelian extensions
Journal of Pure and Applied Algebra, 2002
Let be a Drinfeld module deÿned over a ÿnite extension K of the rational function ÿeld Fq(T), we show that the submodule (K ab)tors of all torsion points in the maximal abelian extension K ab is inÿnite if and only if is of complex multiplication type over K.
Kummer theory of division points over Drinfeld modules of rank one
Journal of Pure and Applied Algebra, 2001
A Kummer theory of division points over rank one Drinfeld A = Fq[T ]-modules defined over global function fields was given. The results are in complete analogy with the classical Kummer theory of division points over the multiplicative algebraic group Gm defined over number fields. 1991 Mathematics Subject Classification. 11G09.
Cogalois Theory and Drinfeld Modules
Journal of Algebra and Its Applications, 2018
In this paper, we generalize the results of [M. Sánchez-Mirafuentes and G. Villa–Salvador, Radical extensions for the Carlitz module, J. Algebra 398 (2014) 284–302] to rank one Drinfeld modules with class number one. We show that, in the present form, there does not exist a cogalois theory for Drinfeld modules of rank or class number larger than one. We also consider the torsion of the Carlitz module for the extension [Formula: see text].