3:00pm - 5:00pmRiemann Surfaces
Chair(s): Daniele Agostini (Humboldt-Universität), Türkü Özlüm Çelik (Max Planck Institute for Mathematics in the Sciences), Christian Klein (Institut de Mathématiques de Bourgogne), Emre Can Sertöz (Max Planck Institute for Mathematics in the Sciences)
In the past decades, the central role played by Riemann surfaces in pure mathematics has been strengthened with their surprising appearance in string theory, cryptography and material science. This minisymposium is intended for the curve theorists and the avant-garde applied mathematician. Our emphasis will be on the computational aspects of Riemann surfaces that are prominent in pure mathematics but are not yet part of the canon of applied mathematics. Some of the subjects that will be touched upon by our speakers are integrable systems, Teichmüller curves, Arakelov geometry, tropical geometry, arithmetic geometry and cryptography of curves.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
Computing endomorphism rings of Jacobians
Jeroen Sijsling
Universität Ulm
Let C be a curve over a number field, with Jacobian J, and let End(J) be the endomorphism ring of J. The ring End(J) is typically isomorphic to ZZ, but the cases where it is larger are interesting for many reasons, most of all because the corresponding curves can then often be matched with relatively simple modular forms.
We give a provably correct algorithm to verify the existence of additional endomorphisms on a Jacobian, which to our knowledge is the first such algorithm. Conversely, we also describe how to get upper bounds on the rank of End(J). Together, these methods make it possible to completely and explicitly determine the endomorphism ring End(J) starting from an equation for C, with acceptable running time when the genus of C is small.
This is joint work with Edgar Costa, Nicolas Mascot, and John Voight.
Inverse Jacobian problem for cyclic plane quintic curves
Anna Somoza
Universitat Politècnica de Catalunya, Universiteit Leiden
We consider the problem of computing the equation of a curve with given analytic Jacobian, that is, with a certain period matrix. In the case of genus one, this can be done by using the classical Weierstrass function, and it is a key step if one wants to write down equations of elliptic curves with complex multiplication (CM). Also in higher genus, the theory of CM gives us all period matrices of principally polarized abelian varieties with CM, among which the periods of the curves whose Jacobian has CM, and computing curve equations is the hardest part. Beyond the classical case of elliptic curves, efficient solutions to this problem are now known for both genus~2 and genus~3. In this talk I will give a method that deals with the case y5 = a5x5 + ... + a1x + a0, inspired by some of the ideas present in the method for the genus-3 family of Picard curves y3 = x(x-1)(x-λ)(x-μ).
Teichmüller curves, Kobayashi geodesics and Hilbert modular forms
David Torres-Teigell
Goethe-Universität
Teichmüller curves are totally geodesic curves inside the moduli space of Riemann surfaces. By results of Möller, they can always be seen as Kobayashi geodesics inside a Hilbert modular variety parametrising abelian varieties with real multiplication. Our main objective is to cut Teichmüller curves out as the vanishing locus of a Hilbert modular form in order to calculate their Euler characteristics. The building blocks of these modular forms turn out to be certain theta functions and their derivatives, which can be made very precise.
In this talk we will focus on how the interplay between the theoretical approach and the computational methods allows us to overcome several difficulties that arise while trying to solve this problem.
Counting special points on teichmüller curves
Jonathan Zachhuber
Goethe-Universität
A flat surface is a Riemann surface together with the choice of a non-zero holomorphic differential. The moduli space of flat surfaces admits a natural SL2(R) action and the closed orbits are Teichmüller curves in the moduli space of Riemann surfaces. While a lot of the original motivation stems from dynamical systems, the known examples of families of such Teichmüller curves carry a surprising amount of arithmetic information. This permits explicit formulas for the genus, the number of cusps and the number and types of orbifold points as well as, in many cases, precise asymptotic behavior of these numbers.