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Critically Separable Rational Maps in Families

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https://ir.library.oregonstate.edu/concern/articles/k930bx56p

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  • 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.
  • Keywords: Arithmetic dynamics, Szpiro’s conjecture, Elliptic curves, Critical discriminant, Critically separable rational maps
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  • Clayton Petsche (2012). Critically separable rational maps in families. Compositio Mathematica, 148, pp 1880-1896. doi:10.1112/S0010437X12000346.
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  • 148
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  • 6
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  • This research is supported by grant DMS-0901147 of the National Science Foundation.
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