The Minimal Resultant and Conductor for Self Maps of the Projective Line
Item
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Title
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The Minimal Resultant and Conductor for Self Maps of the Projective Line
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Identifier
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d_2009_2013:b48d188c1d78:10949
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identifier
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11167
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Creator
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Williams, Phillip,
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Contributor
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Lucien Szpiro
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Date
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2011
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics | Applied mathematics | Algebraic Dynamics | Conductor | Function Fields | Geometric Invariant Theory | Minimal Resultant | Number Theory
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Abstract
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We develop and study the minimal resultant divisor of self-maps of the projective line over a number field or a function field and its relation to the conductor. The guiding focus is the exploration of a dynamical analog to Szpiro's Theorem (Theorem 5.1), which bounds the degree of the minimal discriminant of an elliptic curve over a function field in terms of the conductor. We also explore a question about Lattes maps: given an elliptic curve over a function field with semi-stable bad reduction, we show the associated Lattes maps for multiplication by n have unstable bad reduction for each n. We then study minimality and semi-stability, considering what conditions imply minimality and whether semi-stable models and presentations are minimal, proving results in the degree two case. We prove the singular reduction of a semi-stable presentation coincides with the bad reduction. Degree 2 maps in normal form with semi-stable bad reduction are used to construct a counterexample to a natural dynamical analog to Szpiro's Theorem. Finally, we consider the notion of "critical bad reduction," and show that a dynamical analog may still be possible using the locus of critical bad reduction to define the conductor.
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Type
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dissertation
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Source
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2009_2013.csv
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degree
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Ph.D.
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Program
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Mathematics
