211 research outputs found
Algebraic dynamics of the lifts of Frobenius
We study the algbraic dynamics for endomorphisms of projective spaces with
coefficients in a p-adic field whose reduction in positive characteritic is the
Frobenius. In particular, we prove a version of the dynamical Manin-Mumford
conjecture and the dynamical Mordell-Lang conjecture for the coherent backward
orbits for such endomorphisms. We also give a new proof of a dynamical version
of the Tate-Voloch conjecture in this case. Our method is based on the theory
of perfectoid spaces introduced by P. Scholze. In the appendix, we prove that
under some technical condition on the field of definition, a dynamical system
for a polarized lift of Frobenius on a projective variety can be embedding into
a dynamical system for some endomorphism of a projective space.Comment: 37 page
Intersection of valuation rings in
We associate to any given finite set of valuations on the polynomial ring in
two variables over an algebraically closed field a numerical invariant whose
positivity characterizes the case when the intersection of their valuation
rings has maximal transcendence degree over the base fields.
As an application, we give a criterion for when an analytic branch at
infinity in the affine plane that is defined over a number field in a suitable
sense is the branch of an algebraic curve
The existence of Zariski dense orbits for polynomial endomorphisms of the affine plane
In this paper we prove the following theorem. Let be a dominate polynomial endomorphisms defined over an
algebraically closed field of characteristic . If there are no
nonconstant rational function
satisfying , then there exists a point whose
orbit under is Zariski dense in .
This result gives us a positive answer to a conjecture of Amerik, Bogomolov
and Rovinsky ( and Zhang) for polynomial endomorphisms on the affine plane.Comment: arXiv admin note: substantial text overlap with arXiv:1503.0077
Surjective endomorphisms of projective surfaces -- the existence of infinitely many dense orbits
Let be a surjective endomorphism of a normal projective
surface. When , applying an (iteration of)
-equivariant minimal model program (EMMP), we determine the geometric
structure of . Using this, we extend the second author's result to singular
surfaces to the extent that either has an -invariant non-constant
rational function, or has infinitely many Zariski-dense forward orbits;
this result is also extended to Adelic topology (which is finer than Zariski
topology)
The Geometric Bombieri-Lang Conjecture for Ramified Covers of Abelian Varieties
In this paper, we prove the geometric Bombieri-Lang conjecture for projective
varieties which have finite morphisms to abelian varieties of trivial traces
over function fields of characteristic 0. The proof is based on the idea of
constructing entire curves in the pre-sequel "Partial heights, entire curves,
and the geometric Bombieri-Lang conjecture." A new ingredient is an explicit
description of the entire curves in terms of Lie algebras of abelian varieties.Comment: The original paper arXiv:2305.14789v1 is split into two papers:
arXiv:2305.14789v2 and the current pape
Partial Heights and the Geometric Bombieri-Lang Conjecture
We prove the geometric Bombieri-Lang conjecture for projective varieties
which have finite morphisms to abelian varieties over function fields of
characteristic 0. Our proof is complex analytic, which applies the classical
Brody lemma to construct entire curves on complex varieties. Our key
ingredients includes a new notion of partial height and its non-degeneracy in a
suitable sense. The non-degeneracy is required in the application of the Brody
lemma.Comment: 61 page
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