The Number of Points on a Curve over a Finite Field
Lectures:
 Lauter: Introduction to curves over finite fields with many
rational points
 Zieve: Curves with many points II
 Schoof: Curves over F_{2} with many points
 Lauter: Class field theory constructions of curves with many
points
 Zieve: Curves of large genus with many points
 Schoof: Class field towers
Prerequisites:
The prerequisite for our talks is an understanding of basic properties of
curves and their function fields. A convenient reference is the book
`Algebraic Function Fields and Codes' by Henning Stichtenoth; below we
give specific references from this book.

Definitions: function field, place, valuation, zero/pole, degree (Section I.1)

Example of rational function field (Section I.2)

Some understanding of the genus of a function field and of RiemannRoch:
it would be enough to understand the statement of Thm.I.5.17 on p.29
(after looking up the definitions of degree and dimension, one could define
the genus to be the unique integer `g' for which this theorem is true)

Behavior of places in extensions of function fields  SUM e_i f_i = n
(read enough of Section III.1 to understand the statements of III.1.6,
III.1.7, and III.1.11)

RiemannHurwitzZeuthen genus formula (Thm.III.4.12, Def.III.4.3)
and basic properties of the different (III.4.11, III.5.1),
notions of wild/tame ramification (III.5.4,III.5.5,III.5.7)

Basic properties of Galois extensions (III.7.1,III.7.2),
decomposition and inertia groups (III.8.1III.8.4)

Correspondence between curves and function fields (Appendix B.9,B.10,B.12)
Suggested reading:
René Schoof has provided a
preliminary version of notes on "Algebraic curves and coding theory"
(in dvi, ps, and pdf formats). He emphasizes that
they are preliminary and not for distribution.
