Number theory meets tropical geometry:

the semistable reduction of curves

April 17-21, 2023 at MPI MiS Leipzig

    

The week will consist of daily talks to learn about relevant work in number theory and tropical geometry, as well as daily computational working sessions to further our understanding. The week will bring together people working in number theory, tropical geometry, and related fields to learn from each other.

Resources

• Arithmetic of hyperelliptic curves over local fields by Tim Dokchitser, Vladimir Dokchitser, Céline Maistret, Adam Morgan
• Analytification is the limit of all tropicalizations by Sam Payne
• Diophantine and tropical geometry, and uniformity of rational points on curves by Eric Katz, Joseph Rabinoff, David Zureick-Brown
• Lectures on tropical curves and their moduli spaces by Melody Chan
• Nonarchimedean geometry, tropicalization, and metrics on curves by Matthew Baker, Sam Payne, Joseph Rabinoff, (arXiv:1104.0320v2)

Schedule

Location: G3 10

There will also be a preliminary lecture on background and introductory material on Friday, April 14 from 14:00 - 15:00 in G3 10.

*Other interesting events going on at MPI / University of Leipzig, not part of this workshop. Check links for details (like location, speaker, abstract).

Contact: sachi.hashimoto [at] mis [dot] mpg [dot] de

Some instructions on how to get here.

Abstracts

Sachi Hashimoto: Introductory Lecture

In this introductory lecture, we will study curves over valuated fields and their combinatorial structure. Tropical geometry and number theory study the degeneration of curves in this setting. We give definitions from both fields, discuss semistable models, and give motivation for the following week's lectures.

Paul Helminck: Recovering a covering of schemes from Galois orbits of power series

In this talk, I will give a general technique to find poset substructures of schemes using discrete Galois-data associated to coverings. In particular, this gives a fast algorithm to calculate dual intersection complexes of semistable models of varieties, provided a suitable covering is given. I will show how tropical geometry and schön compactifications of branch loci can be used to generate these coverings.

Ole Ossen: Semistable reduction of plane quartics

I will explain how to describe models of curves using valuations. This will be applied to studying models of the projective line and models of coverings of curves. Specifically, we will discuss the problem of determining the semistable reduction of plane quartics via suitable maps to the projective line, including the wild case of reduction at p=3.

Emeryck Marie: Mirror symmetry and tropical geometry

This short talk is meant to advertise a seminar to be hold this spring/summer about tropical aspects of mirror symmetry. We will mainly be interested in the case of projective spaces (or even only P^2), and follow the book of Mark Gross "Tropical geometry and mirror symmetry". This talk will briefly review classical mirror symmetry of P^2 (expressed as an isomorphism of certain systems of differential equations), and then comment on the program for the seminar.

Bernd Sturmfels: Tropical Plane Curves

Tropical geometry is a combinatorial shadow of classical geometry. Algebraic curves in the tropical plane are dual to triangulations of convex polygons. We discuss the intrinsic geometry of these objects, with focus on the moduli space of metric graphs that represent tropical plane curves. This lectures is based on work with Sarah Brodsky, Michael Joswig and Ralph Morrision from nearly a decade ago (arXiv:1409.4395).

Adam Morgan: Invariants of hyperelliptic curves over local fields

Let C:y^2=f(x) be a hyperelliptic curve over a local field of odd residue characteristic. I will discuss joint work with Tim and Vladimir Dokchitser, and Celine Maistret, demonstrating how several arithmetic invariants of the curve and its Jacobian, most notably its potential stable reduction, can be described in terms of simple combinatorial data involving the p-adic distances between the roots of f(x).

Stefano Mereta: Tropical geometry and Berkovich analytification

I will introduce the concept of Berkovich analytification and discuss briefly the motivations that led to its definition. We will go through the interesting example of the affine line, state the inverse limit theorem proved by Sam Payne, linking the tropical and the Berkovich world, and sketch its proof.