Resultant and conductor of geometrically semi-stable self maps of the projective line over a number field or function field

Lucien Szpiro, Michael Tepper, Phillip Williams

Research output: Contribution to journalArticle

Abstract

We 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 Theorem 1.1, which bounds the degree of the minimal discriminant of an elliptic surface in terms of the conductor. The main theorems of this paper (5.5 and 5.6) establish that, for a degree 2 map, semi-stability in the Geometric Invariant Theory sense on the space of self maps, implies minimality of the resultant. We prove the singular reduction of a semi-stable presentation coincides with the simple bad reduction (Theorem 4.1). Given an elliptic curve over a function field with semi-stable bad reduction, we show the associated Lattès map has unstable bad reduction (Proposition 4.6). Degree 2 maps in normal form with semi-stable bad reduction are used to construct a counterexample (Example 3.1) to a simple dynamical analog to Theorem 1.1.

Original languageEnglish (US)
Pages (from-to)295-329
Number of pages35
JournalPublicacions Matematiques
Volume58
Issue number2
DOIs
StatePublished - 2014

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Function Fields
Conductor
Number field
Line
Theorem
Geometric Invariant Theory
Semistability
Analogue
Elliptic Surfaces
Minimality
Discriminant
Proposition
Elliptic Curves
Divisor
Normal Form
Counterexample
Unstable
Imply

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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Resultant and conductor of geometrically semi-stable self maps of the projective line over a number field or function field. / Szpiro, Lucien; Tepper, Michael; Williams, Phillip.

In: Publicacions Matematiques, Vol. 58, No. 2, 2014, p. 295-329.

Research output: Contribution to journalArticle

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