Questions on self maps of algebraic varieties

By N. Fakhruddin. 2002

183 citations recorded by the source corpus.

In this note we discuss some arithmetic and geometric questions concerning self maps of projective algebraic varieties.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/bb5c4c618e0271fd73f17619dda23f75f61e24a5

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In Algebra and Number Theory

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Preperiodic points of polynomials over global fields (2005, background): “More generally, Fakhruddin has shown [12] that the full Morton and Silverman Conjecture for D = 1 would imply uniform boundedness of torsion for abelian varieties.”
Distributions in algebraic dynamics (2005, background): “1 is equivalent to the following two statements: (1) The preperiodic points on any preperiodic subvariety are Zariski dense; when X is projective, this is actually a Fakhruddin [23], Theorem 5.”
Current trends and open problems in arithmetic dynamics (2018, background): “Zhang [246, 247] formulated a version of the Manin–Mumford conjecture for polarizable endomorphisms,(15) which are endomorphisms f : X → X of a projective variety that extend to noninvertible endomorphisms of the projective space in which X sits; see [66].”
Periods of rational maps modulo primes (2011, background): “By a result of Fakhruddin [6], a positive answer to Question 2 for maps : PN K −→ P N K would give a positive answer for any polarizable self-map f : X −→ X of projective varieties.”
Building blocks of polarized endomorphisms of normal projective varieties (2016, background): “[10]), the set of g-periodic points Per(Y, g) := {y ∈ Y | g(y) = y for some s > 0} is Zariski dense in Y .”
Polarized endomorphisms of complex normal varieties (2009, background): “The following conjecture is proved in [14], Theorem 4.”
Periodic points, linearizing maps, and the dynamical Mordell–Lang problem (2008, background): “We note that Fakhruddin ([ Fak03 ]) has shown that periodic points are Zariski dense for a wide class of morphisms of varieties; moreover, his methods show that in many cases over number elds, there is a periodic point within any periodic residue class at a nite prime.”