Preface Preface
Last modified: November 22, 2025.
This document is a revised compilation of the lecture notes from a course given at UC San Diego during the fall 2019 quarter. The goal of the course was to give a high-level survey of the mathematics surrounding Weil’s highly influential conjectures on the number of solutions of polynomial equations over finite fields, as encapsulated in the properties of the Hasse–Weil zeta functions associated to algebraic varieties over finite fields. The course was intended for graduate or advanced undergraduate students in mathematics (or mathematically-minded students in nearby disciplines) with some prior exposure to algebraic geometry; however, within this constraint the audience ended up being quite diverse.
While giving the lectures and preparing this compilation, I have imagined my role as that of a safari guide, narrating a tour of a wild and wondrous landscape. This landscape is so rich that one could easily spend an entire course (or an entire lifetime) exploring a small corner of it without exhausting it fully, so a tour such as this one is necessarily quite superficial. My hope is to have provided enough of a roadmap, including indications of key results and pointers deeper into the literature, so that the reader can return to some favorite topics later to make a much deeper acquaintance.
In preparing this compilation from the original notes taken by students in the course (see below), I have largely maintained the structure of the original lectures, each of which represents one chapter in the final text. To improve the narrative flow, I have shifted a small number of topics and filled in a few details that were missing (or incorrect) in the original lectures.
I distributed five problem sets over the course of the term. I have included these as well as a few supplemental exercises that came up during the revision process.
There exist many excellent expositions about different parts of this material, some of which I surely do not yet know about. We particularly recommend the lecture notes of Mustața [104], which take a similar approach to this document. Suggestions for additional readings would be greatly appreciated.
Thanks to the following students in the course for compiling the original draft of the notes: Samir Canning, Mingjie Chen, Patrick Girardet, Thomas Grubb, Jacob Keller, Bochao Kong, Woonam Lim, Zeyu Liu, Alex Mathers, Baiming Qiao, Nandagopal Ramachandran, Sankeerth Rao, Peter Wear, Wei Yin. (Special thanks to Peter Wear for coordinating this effort.) Thanks also to David Hansen for additional feedback.
