EVENT DETAILS AND ABSTRACT

Title: Metric aspects of the tropicalization of a Berkovich curve
Speaker: Joseph Rabinoff
Speaker Info: Harvard University
Brief Description:
Special Note: The speaker will give an introductory talk for graduate students at 11 am in Lunt 104.
Abstract:

The Berkovich analytification X^Berk of a smooth curve X over a non-Archimedean field is a locally contractible and path connected topological space which is well suited to studying analytic properties of X. It is naturally endowed with a metric as it is canonically homeomorphic to an inverse limit of finite metric graphs, and this inverse system essentially encodes the semistable reduction theory of X. On the other hand, Payne has shown that X^Berk is canonically homeomorphic to the inverse limit of all tropicalizations of X relative to embeddings in toric varieties. Here the tropicalization of a subspace X^Berk of a torus (G_m^n)^Berk is the image of X^Berk in R^n under the map trop: (x_1,...,x_n) \mapsto (-log|x_1|,...,-log|x_n|), and is denoted Trop(X). The set Trop(X) is a graph whose edges have rational slope, and is endowed with a metric defined in terms of lengths along primitive vectors in the lattice Z^n\subset R^n. It is extremely well suited for calculations (by hand or with a computer), and has applications to arithmetic geometry besides. I'll discuss in what sense the tropicalization map trop: X^Berk -> Trop(X) respects the natural metrics on both sides. One theorem along these lines is that trop becomes an isometry in the inverse limit -- that is, any piece of X^Berk is faithfully and isometrically represented by a suitable "large enough" tropicalization. As another application I'll give a conceptual understanding of the result of E. Katz-Markwig-Markwig relating the valuation of the j-invariant of a Tate curve with the lattice length of a loop in its tropicalization.