Universiteit Leiden Mathematisch Instituut
Geometry Seminar

Wednesday, 20 November 2002, 16:00 - 17:00

Theo van den Bogaart: The Tate curve

Abstract:

The Tate curve is a particular curve defined over Z[[q]]. It allows one to describe elliptic curves in a way which is reminiscent of 'C modulo a lattice' in the complex analytic setting. In particular: it explains why the j-invariant has integer coefficients and it gives a description of the boundary of the moduli space.

After an informal description, an explicit geometric construction of the Tate curve via an infinite number of blow-ups is given. We describe an ample line bundle on the Tate curve: this will give an embedding in projective space; however, for the explicit formulas we refer to the literature. The last section contains a brief description of some important properties of the Tate curve.

The exposition is based on explanations by Bas Edixhoven.