Dynamics of quadratic polynomials over quadratic fields
Institution: | University of Georgia |
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Department: | Mathematics |
Degree: | PhD |
Year: | 2014 |
Keywords: | Arithmetic dynamics |
Record ID: | 2024508 |
Full text PDF: | http://purl.galileo.usg.edu/uga_etd/doyle_john_r_201405_phd |
In 1998, Poonen gave a conjecturally complete classification of the possible preperiodic structures for quadratic polynomials defined over Q. In this thesis, we prove a number of results toward a similar classification, but over quadratic extensions of Q. In order to do so, we formalize a more general notion of emph{dynamical modular curve} in the setting of quadratic polynomial dynamics. We also discuss the geometry of these curves, proving in some cases that they are irreducible over C, extending a result of Bousch.