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Lucien Szpiro

Publications and source records attributed to Lucien Szpiro.

13 recordsLinked to original sources
arXiv2017

A dynamical Shafarevich theorem for rational maps over number fields and function fields

We prove a dynamical Shafarevich theorem on the finiteness of the set of isomorphism classes of rational maps with fixed degeneracies. More precisely, fix an integer d at least 2 and let K be either a number field or the function field of a curve X over a field k, where k is of characteristic zero or p>2d-2 that is either algebraically closed or finite. Let S be a finite set of places of K. We prove the finiteness of the set of isomorphism classes of rational maps over K with a natural kind of good reduction outside of S. We also prove auxiliary results on finiteness of reduced effective divisors in $\mathbb{P}^1_K$ with good reduction outside of S and on the existence of global models for rational maps.

math.AG↗
arXiv2012

Semi-stable Reduction Implies Minimality of the Resultant

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.

math.DS↗
arXiv2012

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

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).

math.DS↗
arXiv2011

Very ample polarized self maps extend to projective space

Let $X$ be a projective variety defined over an infinite field, equipped with a line bundle $L$, giving an embedding of $X$ into $\mb{P}^m$ and let $ϕ: X \to X$ be a morphism such that $ϕ^*L \cong L^{\otimes q}, q\geq 2$. Then there exists an integer $r>0$ extending $ϕ^r$ to $\mb{P}^m$.

math.DS↗
arXiv2011

Kunz Regularity Criterion via algebraic entropy

In arXiv:1109.6438v1 [math.AG] we introduced and studied a notion of algebraic entropy. In this paper we will give an application of algebraic entropy in proving Kunz Regularity Criterion for all contracting self-maps of finite length of Noetherian local rings in arbitrary characteristic. Some conditions of Kunz Criterion have already been extended to the general case by Avramov, Iyengar and Miller in arXiv:math/0312412v2 [math.AC], using different methods.

math.AC↗
arXiv2011

Dynamics and entropy in local algebra

We introduce and study a notion of algebraic entropy for self-maps of finite length of Noetherian local rings, and develop its properties. We show that it shares the standard properties of topological entropy. For finite self-maps we explore the connection between the degree of the map and its algebraic entropy, when the ring is a Cohen-Macaulay domain. As an application of algebraic entropy, we give a characteristic-free interpretation of the definition of Hilbert-Kunz multiplicity.

math.AG↗
arXiv2011

Lifting finite self maps of equicharacteristic complete local rings

In the spirit of Fakhruddin (arXiv:math/0212208v1) and Szpiro-Bhatnagar (arXiv:1010.2715v1), we show that for an equicharacteristic complete local ring A, with a given embedding of Spec(A) in the prime spectrum Spec(R) of some complete regular local ring R, any finite self map of A can be lifted to a finite self map of R, keeping the given embedding.

math.AC↗
arXiv2011

Almost Newton, sometimes Lattès

Self-maps everywhere defined on the projective space $¶^N$ over a number field or a function field are the basic objects of study in the arithmetic of dynamical systems. One reason is a theorem of Fakkruddin \cite{Fakhruddin} (with complements in \cite{Bhatnagar}) that asserts that a "polarized" self-map of a projective variety is essentially the restriction of a self-map of the projective space given by the polarization. In this paper we study the natural self-maps defined the following way: $F$ is a homogeneous polynomial of degree $d$ in $(N+1)$ variables $X_i$ defining a smooth hypersurface. Suppose the characteristic of the field does not divide $d$ and define the map of partial derivatives $ϕ_F = (F_{X_0},...,F_{X_N})$. The map $ϕ_F$ is defined everywhere due to the following formula of Euler: $\sum X_i F_{X_i} = d F$, which implies that a point where all the partial derivatives vanish is a non-smooth point of the hypersuface F=0. One can also compose such a map with an element of $\PGL_{N+1}$. In the particular case addressed in this article, N=1, the smoothness condition means that $F$ has only simple zeroes. In this manner, fixed points and their multipliers are easy to describe and, moreover, with a few modifications we recover classical dynamical systems like the Newton method for finding roots of polynomials or the Lattès map corresponding to the multiplication by 2 on an elliptic curve.

math.NT↗
arXiv2009

A dynamical pairing between two rational maps

Given two rational maps $φ$ and $ψ$ on $\PP^1$ of degree at least two, we study a symmetric, nonnegative-real-valued pairing $<φ,ψ>$ which is closely related to the canonical height functions $h_φ$ and $h_ψ$ associated to these maps. Our main results show a strong connection between the value of $<φ,ψ>$ and the canonical heights of points which are small with respect to at least one of the two maps $φ$ and $ψ$. Several necessary and sufficient conditions are given for the vanishing of $<φ,ψ>$. We give an explicit upper bound on the difference between the canonical height $h_ψ$ and the standard height $h_\st$ in terms of $<σ,ψ>$, where $σ(x)=x^2$ denotes the squaring map. The pairing $<σ,ψ>$ is computed or approximated for several families of rational maps $ψ$.

math.NT↗
arXiv2008

Isotriviality is equivalent to potential good reduction for endomorphisms of ${\mathbb P}^N$ over function fields

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 ${\mathcal L}$.

math.AG↗
arXiv2007

Equidistribution and generalized Mahler measures

Let g be a nonconstant rational map from the projective line to itself that has degree greater than one and is defined over a number field. The map g gives rise to generalized Mahler measures for polynomials in one variable. We use diophantine approximation to show that the generalized Mahler measure of a polynomial F at a place v can be computed by averaging the log of the v-adic absolute value of F over the periodic points of g. This allows us to compute canonical heights of algebraic points via equidistribution on periodic points.

math.NT↗
arXiv2006

One half log discriminant

We give a geometric proof that one may compute a particular generalized Mahler integral using equidistribution of preperiodic points of a dynamical system on the sphere. The dynamical system is associated to the multiplication by 2 map on an elliptic curve over a number field K with Weierstrass equation y^2 = P(x) (a Lattes dynamical system).

math.NT↗