## Publicacions Matemàtiques

- Publ. Mat.
- Volume 58, Number 2 (2014), 295-329.

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

Lucien Szpiro, Michael Tepper, and Phillip Williams

#### 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.

#### Article information

**Source**

Publ. Mat., Volume 58, Number 2 (2014), 295-329.

**Dates**

First available in Project Euclid: 21 July 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.pm/1405949320

**Mathematical Reviews number (MathSciNet)**

MR3264499

**Zentralblatt MATH identifier**

06333797

**Subjects**

Primary: 37P05: Polynomial and rational maps 14L24: Geometric invariant theory [See also 13A50] 11G05: Elliptic curves over global fields [See also 14H52] 14G99: None of the above, but in this section 37P45: Families and moduli spaces

**Keywords**

Resultant conductor discriminant bad reduction critical bad reduction Lattèes map minimality self map of the projective line

#### Citation

Szpiro, Lucien; Tepper, Michael; Williams, Phillip. Resultant and conductor of geometrically semi-stable self maps of the projective line over a number field or function field. Publ. Mat. 58 (2014), no. 2, 295--329. https://projecteuclid.org/euclid.pm/1405949320