arXiv subjects
Publications and source records attributed to Michael Tepper.
For a dynamical system on n-dimensional projective space over a number field or a function field, we show that semi-stable reduction implies the minimality of the resultant. We use this to show that every such dynamical system over a number field admits a globally minimal presentation.
The moduli space of degree $d$ morphisms on $\mathbb{P}^1$ has received much study. McMullen showed that, except for certain families of Latt\`es maps, there is a finite-to-one correspondence (over $\mathbb{C}$) between classes of morphisms in the moduli space and the multipliers of the periodic points. For degree 2 morphisms Milnor (over $\mathbb{C}$) and Silverman (over $\mathbb{Z}$) showed that the correspondence is an isomorphism. In this article we address two cases: polynomial maps of any degree and rational maps of degree 3.
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. We study minimality and semi-stability, considering what conditions imply minimality (Theorem 4.4) and whether semi-stable models and presentations are minimal, proving results in the degree two case (Theorems 4.6, 4.7). We prove the singular reduction of a semi-stable presentation coincides with the bad reduction (Theorem 3.1). Given an elliptic curve over a function field with semi-stable bad reduction, we show the associated Lattes map has unstable bad reduction (Theorem 3.6). Degree 2 maps in normal form with semi-stable bad reduction are used to construct a counterexample (Theorem 2.1) to a natural dynamical analog to Theorem 1.1. Finally, we consider the notion of "critical bad reduction," and show that a dynamical analog to Theorem 1.1 may still be possible using the locus of critical bad reduction to define the conductor (Theorem 4.10, Theorem 4.13).
Let $K=k(C)$ be the function field of a complete nonsingular curve $C$ over an arbitrary field $k$. The main result of this paper states that a morphism $ϕ:{\mathbb P}^N_K\to{\mathbb P}^N_K$ 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 ${\mathbb P}^1_K$ and Baker for arbitrary rational maps on ${\mathbb P}^1_K$. 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 ${\mathbb P}^N_K$ 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 ${\mathcal E}$ of rank $N+1$ on $C$ decomposes as a direct sum ${\mathcal L}\oplus...\oplus{\mathcal L}$ of $N+1$ copies of the same invertible sheaf ${\mathcal L}$.