Algebraic curves are one of the most classical objects of mathematics, Their study led to the concept of Jacobian and more generally that of an Abelian variety. The goal of this session is to focus on the arithmetic aspects of theory.
We will explore minimal models of curves, rational points on curves, Abelian varieties, isogenies, Honda-Tate theory, Weil descent, applications to isogeny based cryptography, etc. The area is a very active area of research and we expect that the session will be well attended.
We intend to invite many younger mathematicians, graduate students, and recent PhD’s.
Equations of curves over their minimal field of definition
Field of moduli versus the field of definition
Models of curves with minimal height
Moduli height of curves
Rational points on curves
Rational points in the moduli space of curves
Jacobians of curves and their decompositions
Neron-Tate models of algebraic curves
Neron-Tate heights on Jacobians
Minimal discriminants and conductors
Selmer groups in Jacobians
Arithmetic invariant theory
Pairings and Weil descent
Abelian varieties with complex multiplication
Lectures will be all in English. There will be 5 days of lectures, Monday through Friday. This is equivalent to an 8-weeks course.
- Morning lecture 9:30-12:00.
- Afternoon lecture 2:00-4:00.
This will be a crash course on algebraic curves. Lectures will be based on the book "From hyperelliptic to superelliptic curves", to be published by the American Math Society.
- Function fields and coverings of the projective line
- Riemann Existence Theroem
- Automorphisms of Curves
- Hurwitz spaces and the braid action
- Moduli space of curves (Hurwitz, Severi, Grothendick, Fulton, Fried, Mumford, Deligne)
- Equations of curves
- Invariant theory
- Field of moduli versus the minimal fields of definition
- Weighted moduli space and weighted heights
- Minimal models, Neron heights
If you are interested to attend, please contact Dorisa Tabaku at [email protected]