### Abstract

Let K = k (C) be the function field of a complete non-singular curve C over an arbitrary field k. The main result of this paper states that a morphism φ : P_{K}^{N} → P_{K}^{N} is isotrivial if and only if it has potential good reduction at all places v of K; this generalizes results of Benedetto for polynomial maps on P_{K}^{1} and Baker for arbitrary rational maps on P_{K}^{1}. We offer two proofs: the first uses algebraic geometry and geometric invariant theory, and it is new even in the case N = 1. The second proof uses non-archimedean analysis and dynamics, and it more directly generalizes the proofs of Benedetto and Baker. We will also give two applications. The first states that an endomorphism of P_{K}^{N} of degree at least two is isotrivial if and only if it has an isotrivial iterate. The second gives a dynamical criterion for whether (after base change) a locally free coherent sheaf E of rank N + 1 on C decomposes as a direct sum L ⊕ ⋯ ⊕ L of N + 1 copies of the same invertible sheaf L.

Original language | English (US) |
---|---|

Pages (from-to) | 3345-3365 |

Number of pages | 21 |

Journal | Journal of Algebra |

Volume | 322 |

Issue number | 9 |

DOIs | |

State | Published - Nov 1 2009 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

^{N}over function fields.

*Journal of Algebra*,

*322*(9), 3345-3365. https://doi.org/10.1016/j.jalgebra.2008.11.027

}

^{N}over function fields',

*Journal of Algebra*, vol. 322, no. 9, pp. 3345-3365. https://doi.org/10.1016/j.jalgebra.2008.11.027

**Isotriviality is equivalent to potential good reduction for endomorphisms of P ^{N} over function fields.** / Petsche, Clayton; Szpiro, Lucien; Tepper, Michael.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Isotriviality is equivalent to potential good reduction for endomorphisms of PN over function fields

AU - Petsche, Clayton

AU - Szpiro, Lucien

AU - Tepper, Michael

PY - 2009/11/1

Y1 - 2009/11/1

N2 - Let K = k (C) be the function field of a complete non-singular curve C over an arbitrary field k. The main result of this paper states that a morphism φ : PKN → PKN is isotrivial if and only if it has potential good reduction at all places v of K; this generalizes results of Benedetto for polynomial maps on PK1 and Baker for arbitrary rational maps on PK1. We offer two proofs: the first uses algebraic geometry and geometric invariant theory, and it is new even in the case N = 1. The second proof uses non-archimedean analysis and dynamics, and it more directly generalizes the proofs of Benedetto and Baker. We will also give two applications. The first states that an endomorphism of PKN of degree at least two is isotrivial if and only if it has an isotrivial iterate. The second gives a dynamical criterion for whether (after base change) a locally free coherent sheaf E of rank N + 1 on C decomposes as a direct sum L ⊕ ⋯ ⊕ L of N + 1 copies of the same invertible sheaf L.

AB - Let K = k (C) be the function field of a complete non-singular curve C over an arbitrary field k. The main result of this paper states that a morphism φ : PKN → PKN is isotrivial if and only if it has potential good reduction at all places v of K; this generalizes results of Benedetto for polynomial maps on PK1 and Baker for arbitrary rational maps on PK1. We offer two proofs: the first uses algebraic geometry and geometric invariant theory, and it is new even in the case N = 1. The second proof uses non-archimedean analysis and dynamics, and it more directly generalizes the proofs of Benedetto and Baker. We will also give two applications. The first states that an endomorphism of PKN of degree at least two is isotrivial if and only if it has an isotrivial iterate. The second gives a dynamical criterion for whether (after base change) a locally free coherent sheaf E of rank N + 1 on C decomposes as a direct sum L ⊕ ⋯ ⊕ L of N + 1 copies of the same invertible sheaf L.

UR - http://www.scopus.com/inward/record.url?scp=70349386342&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70349386342&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2008.11.027

DO - 10.1016/j.jalgebra.2008.11.027

M3 - Article

AN - SCOPUS:70349386342

VL - 322

SP - 3345

EP - 3365

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 9

ER -

^{N}over function fields. Journal of Algebra. 2009 Nov 1;322(9):3345-3365. https://doi.org/10.1016/j.jalgebra.2008.11.027