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Chapter 1
Algebraic Geometry
Ivan Tomašić
School of Mathematical Sciences
Queen Mary University of London
London E1 4NS, UK
Some of the most beautiful chapters of pure mathematics of the 20th century were motivated by considerations around Weil conjectures for zeta functions of algebraic varieties over finite fields. The purpose of these notes is to introduce the reader to enough algebraic geometry to be able to understand some elementary aspects of their proof for the case of algebraic curves and appreciate their number-theoretic consequences.
1.Introduction
Intuitively speaking, algebraic geometry is the study of geometric shapes that can be locally (piecewise) described by polynomial equations.
The main advantage of restricting our attention to polynomial expressions is their versatility, given that they make sense in completely arbitrary rings and fields, including the ones which carry no intrinsic topology. This gives a ‘universal’ geometric intuition in areas where classical geometry and topology fail.
Consequently, methods of algebraic geometry apply in a range of mathematical disciplines, depending on the choice of rings or fields in which to solve our polynomial systems. In geometry we typically work over R or C, while in number theory we choose arithmetically significant structures, such as Z or Q (in Diophantine geometry), number fields, p-adic numbers, or even structures of positive characteristic and finite fields.
We focus on the problem of counting points on varieties over finite fields and the associated Weil conjectures in Section 4. We prove the rationality conjecture in an elementary way, as well as using étale cohomology as a ‘black-box’, and we outline the approach to the functional equation.
1.1.Solving polynomial equations
Let f(x, y) be a polynomial over a ring k, and let X be the ‘plane curve’ defined by the equation f(x, y) = 0. For any ring R extending k, we can consider the solution set
Example 1.1 (Diophantine geometry). Let X be a regular/smooth projective curve of genus g (Definitions 3.7 and 3.39) defined over k = Q. Intuitively, g is the number of ‘holes’ in X(C) conside...
