午夜探花

The research at the Institute of Pure Mathematics is centered around and , especially in their fruitful synthesis called arithmetic geometry.

The research is arithmetic geometry in the institute is centered around over number fields. Here you can think of hyperelliptic curves, such as that defined by

y² = x⁶ + x + 1,

but also of more general curves of higher genus such as plane quartic curves. An example of such a plane quartic is the , which is defined by the homogeneous equation

x³ y + y³ z + z³ x = 0.

A complex of this curve is depicted on the right.

Describing the geometric and number-theoretic properties of algebraic curves is a main goal of our research. Here are its main themes.

Moduli of curves
This theme is centered around the of curves of a given genus, and the reconstruction of curves from their invariants. Other subjects include generalizations of these moduli spaces, such as Hurwitz spaces and Teichm眉ller curves, as well as the question of descent, that is, when a curve can be defined over the field generated by its invariants.

p-adic aspects and the reduction of curves and covers
This includes the study of , as well as the calculation of the conductor exponent of an algebraic curve at a given prime, which encodes rather fine arithmetic information concerning the reduction properties of that curve.

of curves and their endomorphisms
This includes the study of curves with real multiplication. At present a main topic of study is the explicit decomposition of general Jacobians into simple parts, as well as, conversely, to glue given Jacobians of curves together via torsion.

Explicit aspects and
Explicit aspects play an important role in all of these works. The goal is not only to give theoretical answers and construction, but also to show how these can be implemented, and often to write these implementations in computer algebra systems such as SageMath. For example, the animation next to this text shows a package by Bouw and Wewers at work, which builds upon work of Julian R眉th. It studies the reduction properties of an algebraic curve given by explicit equations.

• Arithmetic geometry (Bouw, Sijsling, Wewers)
• Computer Algebra (Sijsling)

• : "Endomorphismen algebraischer Kurven"
• PICS JADERE (with Reynald Lercier, Elisa Lorenzo Garc铆a and Christophe Ritzenthaler; sponsored by the French )
• Sachbeihilfe "Abstieg algebraischer Kurven"