Education, Master Class, Topics 1999/2000

Master Class 1999/2000


Arithmetic Algebraic Geometry

The geometry of diophantine equations, the arithmetic of Riemann surfaces.
Arithmetic algebraic geometry provides methods for studying questions in number theory and in geometry. It lies at the heart of several spectacular breakthroughs in recent times. A family of varieties over a base and a set of equations over a number field provide examples of situations that are studied.

In this Master Class we present essential techniques in this field. Our presentation will centre around questions such as ``Fermat's Last Theorem" and ``Galois groups of number fields". These will be the catalysts for our presentation (although we realize that a fullfledged treatment would require more time). The 1999/2000 Master Class is at a crossroads of fascinating developments in mathematics in our century. We hope to prepare the students for understanding these rich aspects of the past and of the future.

Students are expected to take part in an active way in the courses, and in working sessions. Concrete examples and abstract theory will both be studied. Participants enhance their knowledge by working at exercises and research problems and by giving a seminar.

In the first semester foundational material will be covered. Topics such as basic algebraic geometry, the theory of sheaves and the arithmetic of elliptic curves will be studied extensively. This will be used and extended in the second semester in more specialized directions.


Prerequisites:

Students are expected to have a basic knowledge of geometry, algebra, and analysis. In particular the following topics are assumed known:

  • linear algebra,
  • theory of rings and fields,
  • function theory,
  • basic topology.

Some knowledge of the following topics is useful:

  • Galois theory,
  • projective geometry,
  • commutative algebra.

What is needed from these theories shall be reviewed in the beginning of the courses. It is advisable to study these topics a little before entering this Master Class.


1st semester

  • Introduction to algebraic geometry: algebraic varieties (Ben Moonen)
  • Introduction to algebraic geometry: schemes (Frans Oort)
  • The arithmetic of elliptic curves (Jaap Top)
  • Seminar: Riemann surfaces (Marius van der Put, Jaap Top)


2nd semester

  • Modular curves (Gerard van der Geer)
  • Etale cohomology (Eduard Looijenga)
  • Fundamental groups of algebraic curves (Frans Oort, Frits Beukers)
  • Modular forms (Jozef Steenbrink)