Tag Archives: Berkovich spaces

The Jacobian of the skeleton and the skeleton of the Jacobian

My new Georgia Tech colleague Joe Rabinoff and I recently posted this paper to the arXiv, entitled The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves.  In this post I will attempt to give some context and background for that paper.

Suppose X is an algebraic variety over a complete non-Archimedean field K.  For simplicity of exposition, I will assume throughout that K is algebraically closed and that the value group \Lambda = {\rm val} (K^\times) is nontrivial.  The set of points X(K) possesses a natural (Hausdorff) analytic topology, but X(K) is totally disconnected and not even locally compact in this topology.  This is bad for all sorts of reasons.  Over the years there have been many attempts at “fixing” this problem, beginning with the Tate-Grothendieck theory of rigid analytic spaces.  One of the most successful and elegant solutions to the problem is Berkovich’s theory of non-Archimedean analytic spaces.  One can associate to X, in a natural and functorial way, an analytic space X^{\rm an} which contains X(K) as a dense subspace but has much nicer topological properties.  For example, X^{\rm an} is a locally compact Hausdorff space which is locally contractible, and is arcwise connected if X is connected.  As in the theory of complex analytic spaces, X^{\rm an} is compact if and only if X^{\rm an} is proper.  For much more information about Berkovich analytic spaces, see for example these course notes of mine.

One of the nice features of Berkovich’s theory (established in full generality only very recently, by Hrushovski and Loeser) is that the space X^{\rm an} always admits a deformation retraction onto a finite polyhedral complex \Sigma.  In general, there is no canonical choice for \Sigma.  However, in certain special cases there is, e.g. when X is a curve of positive genus or X is an abelian variety.  (More generally, the existence of canonical skeleta is closely tied in with the Minimal Model Program in birational algebraic geometry, see for example this recent paper of Mustata and Nicaise.)  For the rest of this post, I will focus on the special cases of curves and abelian varieties. Continue reading →

Attracting cycles and post-critically finite maps

A theme which I hope to pursue in multiple posts on this blog is that even if one is interested mainly in complex dynamics, it is often useful to employ p-adic (or more generally non-Archimedean) methods.  There are a number of relatively recent research papers which illustrate and support this thesis.  The one I want to talk about today is this beautiful paper of Benedetto, Ingram, Jones, and Levy, henceforth referred to as [BIJL].

From a complex dynamicist’s point of view, the main result of [BIJL] is the following theorem:

Theorem 1: For any fixed integers d ≥ 2 and B ≥ 1, there are only finitely many conjugacy classes of post-critically finite rational maps of degree d which can be defined over a number field of degree at most B, except (when d is a perfect square) for the infinite family of flexible Lattès maps.

To explain the terms used in the statement, suppose f is a rational map of degree d with complex coefficients.  We say that f is post-critically finite (henceforth denoted PCF) if the orbit of the critical points of f under iteration is finite.  PCF maps play a fundamental role in complex dynamics, roughly speaking because many dynamical features of f can be read off from the behavior of the critical points under iteration.  One source of examples are the flexible Lattès maps, which can be defined whenever d=m2 is a perfect square: these consist of all degree m rational maps obtained as the map on x-coordinates induced by multiplication-by-m on some elliptic curve.  (McMullen calls such maps affine in the paper cited below.)

As an illustration of the theorem, the only quadratic polynomials of the form z2 + c with c a rational number which are PCF are z2, z2-1, and z2-2, and every quadratic polynomial is conjugate to a polynomial of this form. Continue reading →