Finiteness Properties of Certain Metabelian Arithmetic Groups in the Function Field Case

1997 ◽  
Vol 75 (2) ◽  
pp. 308-322 ◽  
Author(s):  
K-U Bux
Keyword(s):  
Function Field ◽  
Arithmetic Groups ◽  
Field Case ◽  
Finiteness Properties ◽  
Function Field Case

scholarly journals Squarefree doubly primitive divisors in dynamical sequences

2017 ◽  
Vol 164 (3) ◽  
pp. 551-572 ◽  
Author(s):  
DRAGOS GHIOCA ◽  
KHOA D. NGUYEN ◽  
THOMAS J. TUCKER
Keyword(s):  
Number Field ◽  
Function Field ◽  
Field Case ◽  
Primitive Divisors ◽  
Finite Set ◽  
Function Field Case

AbstractLetKbe a number field or a function field of characteristic 0, letφ∈K(z) with deg(φ) ⩾ 2, and letα∈ ℙ1(K). LetSbe a finite set of places ofKcontaining all the archimedean ones and the primes whereφhas bad reduction. After excluding all the natural counterexamples, we define a subsetA(φ,α) of ℤ⩾0× ℤ>0and show that for all but finitely many (m,n) ∈A(φ,α) there is a prime 𝔭 ∉Ssuch that ord𝔭(φm+n(α)−φm(α)) = 1 andαhas portrait (m,n) under the action ofφmodulo 𝔭. This latter condition implies ord𝔭(φu+v(α)−φu(α)) ⩽ 0 for (u,v) ∈ ℤ⩾0× ℤ>0satisfyingu<morv<n. Our proof assumes a conjecture of Vojta for ℙ1× ℙ1in the number field case and is unconditional in the function field case thanks to a deep theorem of Yamanoi. This paper extends earlier work of Ingram–Silverman, Faber–Granville and the authors.

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