Reduction of dynatomic curves


Type
Article
Change log
Authors
Doyle, John R 
Obus, Andrew 
Pries, Rachel 
Rubinstein-Salzedo, Simon 
Abstract

The dynatomic modular curves parametrize polynomial maps together with a point of period n. It is known that the dynatomic curves Y1(n) are smooth and irreducible in characteristic 0 for families of polynomial maps of the form fc(z)=zm+c where m≥2. In the present paper, we build on the work of Morton to partially characterize the primes p for which the reduction modulo p of Y1(n) remains smooth and/or irreducible. As an application, we give new examples of good reduction of Y1(n) for several primes dividing the ramification discriminant when n=7,8,11. The proofs involve arithmetic and complex dynamics, reduction theory for curves, ramification theory, and the combinatorics of the Mandelbrot set.

Description
Keywords
math.DS, math.DS, math.AG, math.NT, 37F45, 37P05, 37P35, 37P45, 11G20, 11S15, 14H30
Journal Title
ERGODIC THEORY AND DYNAMICAL SYSTEMS
Conference Name
Journal ISSN
0143-3857
1469-4417
Volume Title
39
Publisher
Cambridge University Press (CUP)