arXiv subjects

Laura DeMarco

Publications and source records attributed to Laura DeMarco.

At least 19 recordsLinked to original sources

arXiv2024

The geometry of preperiodic points in families of maps on $\mathbb{P}^N$

We study the dynamics of algebraic families of maps on $\mathbb{P}^N$, over the field $\mathbb{C}$ of complex numbers, and the geometry of their preperiodic points. The goal of this note is to formulate a conjectural characterization of the subvarieties of $S \times\mathbb{P}^N$ containing a Zariski-dense set of preperiodic points, where the parameter space $S$ is a quasiprojective complex algebraic variety; the characterization is given in terms of the non-vanishing of a power of the invariant Green current associated to the family of maps. This conjectural characterization is inspired by and generalizes the Relative Manin-Mumford Conjecture for families of abelian varieties, recently proved by Gao and Habegger, and it includes as special cases the Manin-Mumford Conjecture (theorem of Raynaud) and the Dynamical Manin-Mumford Conjecture (posed by Ghioca, Tucker, and Zhang). We provide examples where the equivalence is known to hold, and we show that several recent results can be viewed as special cases. Finally, we give the proof of one implication in the conjectural characterization.

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arXiv2024

Bounded geometry for PCF-special subvarieties

For each integer $d\geq 2$, let $M_d$ denote the moduli space of maps $f: \mathbb{P}^1\to \mathbb{P}^1$ of degree $d$. We study the geometric configurations of subsets of postcritically finite (or PCF) maps in $M_d$. A complex-algebraic subvariety $Y \subset M_d$ is said to be PCF-special if it contains a Zariski-dense set of PCF maps. Here we prove that there are only finitely many positive-dimensional irreducible PCF-special subvarieties in $M_d$ with degree $\leq D$. In addition, there exist constants $N = N(D,d)$ and $B = B(D,d)$ so that for any complex algebraic subvariety $X \subset M_d$ of degree $\leq D$, the Zariski closure $\overline{X\cap\mathrm{PCF}}~$ has at most $N$ irreducible components, each with degree $\leq B$. We also prove generalizations of these results for points with small critical height in $M_d(\bar{\mathbb{Q}})$.

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arXiv2023

Geometry of PCF parameters in spaces of quadratic polynomials

We study algebraic relations among postcritically finite (PCF) parameters in the family $f_c(z) = z^2 + c$. Ghioca, Krieger, Nguyen and Ye proved that an algebraic curve in $\mathbb{C}^2$ contains infinitely many PCF pairs $(c_1, c_2)$ if and only if the curve is special (i.e., the curve is a vertical or horizontal line through a PCF parameter, or the curve is the diagonal). Here we extend this result to subvarieties of $\mathbb{C}^n$ for any $n\geq 2$. Consequently, we obtain uniform bounds on the number of PCF pairs on non-special curves in $\mathbb{C}^2$ and the number of PCF parameters in real algebraic curves in $\mathbb{C}$, depending only on the degree of the curve. We also compute the optimal bound for the general curve of degree $d$.

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arXiv2023

Dynamics on $\mathbb{P}^1$: preperiodic points and pairwise stability

In [DKY], it was conjectured that there is a uniform bound $B$, depending only on the degree $d$, so that any pair of holomorphic maps $f, g :\mathbb{P}^1\to\mathbb{P}^1$ with degree $d$ will either share all of their preperiodic points or have at most $B$ in common. Here we show that this uniform bound holds for a Zariski open and dense set in the space of all pairs, $\mathrm{Rat}_d \times \mathrm{Rat}_d$, for each degree $d\geq 2$. The proof involves a combination of arithmetic intersection theory and complex-dynamical results, especially as developed recently by Gauthier-Vigny, Yuan-Zhang, and Mavraki-Schmidt. In addition, we present alternate proofs of recent results of DeMarco-Krieger-Ye and of Poineau. In fact we prove a generalization of a conjecture of Bogomolov-Fu-Tschinkel in a mixed setting of dynamical systems and elliptic curves.

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arXiv2022

Variation of Canonical Height for Fatou points on $\mathbb{P}^1$

Let $f: \mathbb{P}^1\to \mathbb{P}^1$ be a map of degree $>1$ defined over a function field $k = K(X)$, where $K$ is a number field and $X$ is a projective curve over $K$. For each point $a \in \mathbb{P}^1(k)$ satisfying a dynamical stability condition, we prove that the Call-Silverman canonical height for specialization $f_t$ at point $a_t$, for $t \in X(\bar{\mathbb{Q}})$ outside a finite set, induces a Weil height on the curve $X$; i.e., we prove the existence of a $\mathbb{Q}$-divisor $D = D_{f,a}$ on $X$ so that the function $t\mapsto \hat{h}_{f_t}(a_t) - h_D(t)$ is bounded on $X(\bar{\mathbb{Q}})$ for any choice of Weil height associated to $D$. We also prove a local version, that the local canonical heights $t\mapsto \hatλ_{f_t, v}(a_t)$ differ from a Weil function for $D$ by a continuous function on $X(\mathbb{C}_v)$, at each place $v$ of the number field $K$. These results were known for polynomial maps $f$ and all points $a \in \mathbb{P}^1(k)$ without the stability hypothesis, and for maps $f$ that are quotients of endomorphisms of elliptic curves $E$ over $k$. Finally, we characterize our stability condition in terms of the geometry of the induced map $\tilde{f}: X\times \mathbb{P}^1 \rightarrow X\times \mathbb{P}^1$ over $K$; and we prove the existence of relative Néron models for the pair $(f,a)$, when $a$ is a Fatou point at a place $γ$ of $k$, where the local canonical height $\hatλ_{f,γ}(a)$ can be computed as an intersection number.

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arXiv2022

Elliptic surfaces and intersections of adelic $\mathbb{R}$-divisors

Suppose $\mathcal{E} \to B$ is a non-isotrivial elliptic surface defined over a number field, for smooth projective curve $B$. Let $k$ denote the function field $\overline{\mathbb{Q}}(B)$ and $E$ the associated elliptic curve over $k$. In this article, we construct adelically metrized $\mathbb{R}$-divisors $\overline{D}_X$ on the base curve $B$ over a number field, for each $X \in E(k)\otimes \mathbb{R}$. We prove non-degeneracy of the Arakelov-Zhang intersection numbers $\overline{D}_X\cdot \overline{D}_Y$, as a biquadratic form on $E(k)\otimes \mathbb{R}$. As a consequence, we have the following Bogomolov-type statement for the Néron-Tate height functions on the fibers $E_t(\overline{\mathbb{Q}})$ of $\mathcal{E}$ over $t \in B(\overline{\mathbb{Q}})$: given points $P_1, \ldots, P_m \in E(k)$ with $m\geq 2$, there exist an infinite sequence $t_n\in B(\overline{\mathbb{Q}})$ and small-height perturbations $P_{i,t_n}' \in E_{t_n}(\overline{\mathbb{Q}})$ of specializations $P_{i,t_n}$ so that the set $\{P_{1, t_n}', \ldots, P_{m,t_n}'\}$ satisfies at least two independent linear relations for all $n$, if and only if the points $P_1, \ldots, P_m$ are linearly dependent in $E(k)$. This gives a new proof of results of Masser and Zannier and of Barroero and Capuano and extends our earlier results. In the Appendix, we prove an equidistribution theorem for adelically metrized $\mathbb{R}$-divisors on projective varieties (over a number field) using results of Moriwaki, extending the equidistribution theorem of Yuan.

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arXiv2021

Common preperiodic points for quadratic polynomials

Let $f_c(z) = z^2+c$ for $c \in \mathbb{C}$. We show there exists a uniform bound on the number of points in $\mathbb{P}^1(\mathbb{C})$ that can be preperiodic for both $f_{c_1}$ and $f_{c_2}$ with $c_1\not= c_2$ in $\mathbb{C}$. The proof combines arithmetic ingredients with complex-analytic; we estimate an adelic energy pairing when the parameters lie in $\bar{\mathbb{Q}}$, building on the quantitative arithmetic equidistribution theorem of Favre and Rivera-Letelier, and we use distortion theorems in complex analysis to control the size of the intersection of distinct Julia sets. The proof is effective, and we provide explicit constants for each of the results.

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arXiv2019

Uniform Manin-Mumford for a family of genus 2 curves

We introduce a general strategy for proving quantitative and uniform bounds on the number of common points of height zero for a pair of inequivalent height functions on $\mathbb{P}^1(\overline{\mathbb{Q}}).$ We apply this strategy to prove a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds on the number of common torsion points of elliptic curves in the case of two Legendre curves over $\mathbb{C}$. As a consequence, we obtain two uniform bounds for a two-dimensional family of genus 2 curves: a uniform Manin-Mumford bound for the family over $\mathbb{C}$, and a uniform Bogomolov bound for the family over $\overline{\mathbb{Q}}.$

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arXiv2018

Current Trends and Open Problems in Arithmetic Dynamics

Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from $p$-adic analogues of theorems and conjectures in classical complex dynamics. In this article we survey some of the motivating problems and some of the recent progress in the field of arithmetic dynamics.

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arXiv2018

Discontinuity of a degenerating escape rate

We look at degenerating meromorphic families of rational maps on $\mathbb{P}^1$ -- holomorphically parameterized by a punctured disk -- and we provide examples where the bifurcation current fails to have a bounded potential in a neighborhood of the puncture. This is in contrast to the recent result of Favre-Gauthier that we always have continuity across the puncture for families of polynomials; and it provides a counterexample to a conjecture posed by Favre in 2016. We explain why our construction fails for polynomial families and for families of rational maps defined over finite extensions of the rationals $\mathbb{Q}$.

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arXiv2017

Rationality of dynamical canonical height

We present a dynamical proof of the well-known fact that the Neron-Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field k of transcendence degree 1 over an algebraically closed field K of characteristic 0. More generally, we investigate the mechanism for which the local canonical height for a rational function f defined over k can take irrational values (at points in a local completion of k), providing examples in all degrees greater than 1. Building on Kiwi's classification of non-archimedean Julia sets for quadratic maps, we give a complete answer in degree 2 characterizing the existence of points with irrational local canonical heights. As an application of our results, we prove that if the canonical heights of two points a and b under the action of two rational functions f and g (defined over k) are positive rational numbers, and if the degrees of f and g are multiplicatively independent, then the orbit of a under f intersects the orbit of b under g in at most finitely many points, complementing the results of Ghioca-Tucker-Zieve.

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arXiv2017

Bounded height in families of dynamical systems

Let a and b be algebraic numbers such that exactly one of a and b is an algebraic integer, and let f_t(z):=z^2+t be a family of polynomials parametrized by t. We prove that the set of all algebraic numbers t for which there exist positive integers m and n such that f_t^m(a)=f_t^n(b) has bounded Weil height. This is a special case of a more general result supporting a new bounded height conjecture in dynamics. Our results fit into the general setting of the principle of unlikely intersections in arithmetic dynamics.

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arXiv2017

Variation of canonical height and equidistribution

Let $π: E\to B$ be an elliptic surface defined over a number field $K$, where $B$ is a smooth projective curve, and let $P: B \to E$ be a section defined over $K$ with canonical height $\hat{h}_E(P)\not=0$. In this article, we show that the function $t \mapsto \hat{h}_{E_t}(P_t)$ on $B(\overline{K})$ is the height induced from an adelically metrized line bundle with non-negative curvature on $B$. Applying theorems of Thuillier and Yuan, we obtain the equidistribution of points $t \in B(\overline{K})$ where $P_t$ is torsion, and we give an explicit description of the limiting distribution on $B(\mathbb{C})$. Finally, combined with results of Masser and Zannier, we show there is a positive lower bound on the height $\hat{h}_{A_t}(P_t)$, after excluding finitely many points $t \in B$, for any "non-special" section $P$ of a family of abelian varieties $A \to B$ that split as a product of elliptic curves.

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arXiv2016

Convex shapes and harmonic caps

Any planar shape $P\subset \mathbb{C}$ can be embedded isometrically as part of the boundary surface $S$ of a convex subset of $\mathbb{R}^3$ such that $\partial P$ supports the positive curvature of $S$. The complement $Q = S \setminus P$ is the associated {\em cap}. We study the cap construction when the curvature is harmonic measure on the boundary of $(\hat{\mathbb{C}}\setminus P, \infty)$. Of particular interest is the case when $P$ is a filled polynomial Julia set and the curvature is proportional to the measure of maximal entropy.

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arXiv2016

Bifurcations, intersections, and heights

In this article, we prove the equivalence of dynamical stability, preperiodicity, and canonical height 0, for algebraic families of rational maps $f_t: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})$, parameterized by $t$ in a quasi-projective complex variety. We use this to prove one implication in the if-and-only-if statement of Conjecture 1.10 in [Baker-DeMarco, 2013] on unlikely intersections in the moduli space of rational maps; we present the conjecture here in a more general form.

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arXiv2016

KAWA 2015: Dynamical moduli spaces and elliptic curves

In these lecture notes, we present a connection between the complex dynamics of a family of rational functions $f_t: \mathbb{P}^1\to \mathbb{P}^1$, parameterized by $t$ in a Riemann surface $X$, and the arithmetic dynamics of $f_t$ on rational points $\mathbb{P}^1(k)$ where $k = \mathbb{C}(X)$ or $\bar{\mathbb{Q}}(X)$. An explicit relation between stability and canonical height is explained, with a proof that contains a piece of the Mordell-Weil theorem for elliptic curves over function fields. Our main goal is to pose some questions and conjectures about these families, guided by the principle of "unlikely intersections" from arithmetic geometry, as in [Zannier 2012]. We also include a proof that the hyperbolic postcritically-finite maps are Zariski dense in the moduli space of rational maps of any given degree $d>1$. These notes are based on four lectures at KAWA 2015, in Pisa, Italy, designed for an audience specializing in complex analysis, expanding upon the main results of [Baker-DeMarco 2013, DeMarco 2016, DeMarco-Wang-Ye 2016].

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arXiv2016

Degenerations of Complex Dynamical Systems II: Analytic and Algebraic Stability

We study pairs $(f, Γ)$ consisting of a non-Archimedean rational function $f$ and a finite set of vertices $Γ$ in the Berkovich projective line, under a certain stability hypothesis. We prove that stability can always be attained by enlarging the vertex set $Γ$. As a byproduct, we deduce that meromorphic maps preserving the fibers of a rationally-fibered complex surface are algebraically stable after a proper modification. The first article in this series examined the limit of the equilibrium measures for a degenerating 1-parameter family of rational functions on the Riemann sphere. Here we construct a convergent countable-state Markov chain that computes the limit measure. A classification of the periodic Fatou components for non-Archimedean rational functions, due to Rivera-Letelier, plays a key role in the proofs of our main theorems. The appendix contains a proof of this classification for all tame rational functions.

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arXiv2015

Bifurcation measures and quadratic rational maps

We study critical orbits and bifurcations within the moduli space of quadratic rational maps on $\mathbb{P}^1$. We focus on the family of curves, $Per_1(λ)$ for $λ$ in $\mathbb{C}$, defined by the condition that each $f\in Per_1(λ)$ has a fixed point of multiplier $λ$. We prove that the curve $Per_1(λ)$ contains infinitely many postcritically-finite maps if and only if $λ= 0$; addressing a special case of [BD2, Conjecture 1.4]. We also show that the two critical points of a map $f$ define distinct bifurcation measures along $Per_1(λ)$.

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