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Let $f_1,...,f_gin {mathbb C}(z)$ be rational functions, let $Phi=(f_1,...,f_g)$ denote their coordinatewise action on $({mathbb P}^1)^g$, let $Vsubset ({mathbb P}^1)^g$ be a proper subvariety, and let $P=(x_1,...,x_g)in ({mathbb P}^1)^g({mathbb C})$ be a nonpreperiodic point for $Phi$. We show that if $V$ does not contain any periodic subvarieties of positive dimension, then the set of $n$ such that $Phi^n(P) in V({mathbb C})$ must be very sparse. In particular, for any $k$ and any sufficiently large $N$, the number of $n leq N$ such that $Phi^n(P) in V({mathbb C})$ is less than $log^k N$, where $log^k$ denotes the $k$-th iterate of the $log$ function. This can be interpreted as an analog of the gap principle of Davenport-Roth and Mumford.
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We formulate a general question regarding the size of the iterated Galois groups associated to an algebraic dynamical system and then we discuss some special cases of our question.
Let K be a non-archimedean field, and let f in K(z) be a polynomial or rational function of degree at least 2. We present a necessary and sufficient condition, involving only the fixed points of f and their preimages, that determines whether or not the dynamical system f on P^1 has potentially good reduction.
A rational function of degree at least two with coefficients in an algebraically closed field is post-critically finite (PCF) if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers, once the well-understood family known as flexible Lattes maps is excluded. As a consequence, there are only finitely many conjugacy classes of non-Lattes PCF rational maps of a given degree defined over any given number field. The key ingredient of the proof is a non-archimedean version of Fatous classical result that every attracting cycle of a rational function over the complex numbers attracts a critical point.
Let K be a non-archimedean field, and let f in K(z) be a rational function of degree d>1. If f has potentially good reduction, we give an upper bound, depending only on d, for the minimal degree of an extension L/K such that f is conjugate over L to a map of good reduction. In particular, if d=2 or d is greater than the residue characteristic of K, the bound is d+1. If K is discretely valued, we give examples to show that our bound is sharp.
We consider a general class of Fourier coefficients for an automorphic form on a finite cover of a reductive adelic group ${bf G}(mathbb{A}_{mathbb{K}})$, associated to the data of a `Whittaker pair. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are `Levi-distinguished Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a $mathbb{K}$-distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In follow-up papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of their top Fourier coefficients.
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