2011/09/30 by Clayton Petsche · 7 citations
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Algebraic number #Conjecture #Discrete mathematics #Discriminant #Elliptic curve #Isomorphism (crystallography) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Multiplier (economics) #Polynomial and algebraic computation #Pure mathematics #Rational point #Separable space #math.NT #msc:11G05 #msc:37P15 #msc:37P45
paper · pdf · doi:10.1112/s0010437x12000346
published in Compositio Mathematica 148(6), 1880-1896 (Cambridge University Press) · In this version, some notation and terminology has changed. In particular, this results in a slight change in the title of the paper. Many small expository changes have been made, a reference has been added, and a remark/example has been added to the end of section 3
arxiv created 2012/02/02 · openalex publication_date 2012/10/12 · arxiv updated 2019/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract Given a number field K , we consider families of critically separable rational maps of degree d over K possessing a certain fixed-point and multiplier structure. With suitable notions of isomorphism and good reduction between rational maps in these families, we prove a finiteness theorem which is analogous to Shafarevich’s theorem for elliptic curves. We also define the minimal critical discriminant, a global object which can be viewed as a measure of arithmetic complexity of a rational map. We formulate a conjectural bound on the minimal critical discriminant, which is analogous to Szpiro’s conjecture for elliptic curves, and we prove that a special case of our conjecture implies Szpiro’s conjecture in the semistable case.