43 research outputs found
The Minimal Resultant Locus
Let K be a complete, algebraically closed nonarchimedean valued field, and
let f(z) in K(z) be a rational function of degree d at least 2. We give an
algorithm to determine whether f(z) has potential good reduction over K, based
on a geometric reformulation of the problem using the Berkovich Projective
Line. We show the minimal resultant is is either achieved at a single point in
the Berkovich line, or on a segment, and that minimal resultant locus is
contained in the tree in spanned by the fixed points and the poles of f(z).
When f(z) is defined over the rationals, the algorithm runs in probabilistic
polynomial time. If f(z) has potential good reduction, and is defined over a
subfield H of K, we show there is an extension L/H in K with degree at most (d
+ 1)^2 such that f(z) achieves good reduction over L.Comment: 37 page
Equidistribution of small points, rational dynamics, and potential theory
If phi(z) is a rational function on P^1 of degree at least 2 with
coefficients in a number field k, we compute the homogeneous transfinite
diameter of the v-adic filled Julia sets of phi for all places v of k by
introducing a new quantity called the homogeneous sectional capacity. In
particular, we show that the product over all places of these homogeneous
transfinite diameters is 1. We apply this product formula and some new
potential-theoretic results concerning Green's functions on Riemann surfaces
and Berkovich spaces to prove an adelic equidistribution theorem for dynamical
systems on the projective line. This theorem, which generalizes the results of
Baker-Hsia, says that for each place v of k, there is a canonical probability
measure on the Berkovich space P^1_{Berk,v} over C_v such that if z_n is a
sequence of algebraic points in P^1 whose canonical heights with respect to phi
tend to zero, then the z_n's and their Galois conjugates are equidistributed
with respect to mu_{phi,v} for all places v of k. For archimedean v,
P^1_{Berk,v} is just the Riemann sphere, mu_{phi,v} is Lyubich's invariant
measure, and our result is closely related to a theorem of Lyubich and
Freire-Lopes-Mane.Comment: 50 pages; v2 contains additional references, exposition has been
modified, and Sections 7 and 8 from v1 have been removed to shorten the
paper's lengt
The Fekete-Szego theorem with Local Rationality Conditions on Curves
Let be a number field or a function field in one variable over a finite
field, and let be a separable closure of . Let be a smooth,
complete, connected curve. We prove a strong theorem of Fekete-Szego type for
adelic sets on , showing that under appropriate conditions
there are infinitely many points in whose conjugates all belong to
at each place of . We give several variants of the theorem,
including two for Berkovich curves, and provide examples illustrating the
theorem on the projective line, and on elliptic curves, Fermat curves, and
modular curves
Configuration of the Crucial Set for a Quadratic Rational Map
Let be a complete, algebraically closed non-archimedean valued field, and
let have degree two. We describe the crucial set of
in terms of the multipliers of at the classical fixed
points, and use this to show that the crucial set determines a stratification
of the moduli space related to the reduction type of
. We apply this to settle a special case of a conjecture of Hsia
regarding the density of repelling periodic points in the non-archimedean Julia
set