Manuscripta Mathematica

, Volume 137, Issue 3–4, pp 273–286 | Cite as

Ramification in iterated towers for rational functions

  • John CullinanEmail author
  • Farshid Hajir


Let \({\phi(x)}\) be a rational function of degree > 1 defined over a number field K and let \({\Phi_{n}(x,t) = \phi^{(n)}(x)-t \in K(x,t)}\) where \({\phi^{(n)}(x)}\) is the nth iterate of \({\phi(x)}\). We give a formula for the discriminant of the numerator of Φ n (x, t) and show that, if \({\phi(x)}\) is postcritically finite, for each specialization t 0 of t to K, there exists a finite set \({S_{t_0}}\) of primes of K such that for all n, the primes dividing the discriminant are contained in \({S_{t_0}}\).


Rational Function Branch Point Elliptic Curve Galois Group Prime Divisor 
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  1. 1.
    Aitken, W., Hajir, F., Maire, C.: Finitely ramified iterated extensions. Int. Math. Res. Not. 2005(14), 855–880Google Scholar
  2. 2.
    Beckmann S.: Ramified primes in the field of moduli of branched coverings of curves. J. Algebra 125(1), 236–255 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Beckmann S.: On extensions of number fields obtained by specializing branched coverings. J. Reine Angew. Math. 419, 27–53 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Silverman J.: The arithmetic of dynamical systems, Graduate Texts in Mathematics 241. Springer, New York (2007)Google Scholar

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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsBard CollegeAnnandale-On-HudsonUSA
  2. 2.Department of MathematicsUniversity of Massachusetts AmherstAmherstUSA

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