Reduction of dynatomic curves
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Authors
Doyle, John R
Obus, Andrew
Pries, Rachel
Rubinstein-Salzedo, Simon
West, Lloyd
Publication Date
2019-10Journal Title
ERGODIC THEORY AND DYNAMICAL SYSTEMS
ISSN
0143-3857
Publisher
Cambridge University Press (CUP)
Volume
39
Pages
2717-2768
Type
Article
This Version
AM
Metadata
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Doyle, J. R., Krieger, H., Obus, A., Pries, R., Rubinstein-Salzedo, S., & West, L. (2019). Reduction of dynatomic curves. ERGODIC THEORY AND DYNAMICAL SYSTEMS, 39 2717-2768. https://doi.org/10.1017/etds.2017.140
Abstract
The dynatomic modular curves parametrize polynomial maps together with a
point of period $n$. It is known that the dynatomic curves $Y_1(n)$ are smooth
and irreducible in characteristic 0 for families of polynomial maps of the form
$f_c(z) = z^m +c$ where $m\geq 2$. In the present paper, we build on the work
of Morton to partially characterize the primes $p$ for which the reduction
modulo $p$ of $Y_1(n)$ remains smooth and/or irreducible. As an application, we
give new examples of good reduction of $Y_1(n)$ 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.
Identifiers
External DOI: https://doi.org/10.1017/etds.2017.140
This record's URL: https://www.repository.cam.ac.uk/handle/1810/273802
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