18.782 | Fall 2013 | Undergraduate

# Introduction to Arithmetic Geometry

## Instructor Insights

### Course Overview

This page focuses on the course 18.782 Introduction to Arithmetic Geometry as it was taught by Dr. Andrew Sutherland in Fall 2013.

Arithmetic geometry lies at the intersection of number theory and algebraic geometry. One of its key motivations is the analysis of Diophantine problems: finding all integer solutions to a given set of polynomial equations. Many of these problems are very old (hundreds, or even thousands of years), but the techniques available for solving them have evolved dramatically in recent years and are currently an area of active research.

This is a first course in arithmetic geometry, and students are not required to enter with background in number theory or algebraic geometry. However, it is assumed that students have taken and mastered a full year of algebra (e.g., 18.701 Algebra I and 18.702 Algebra II).

Highlights of the course include an introduction to p-adic numbers, the Hasse-Minkowski theorem for quadratic forms, the Riemann-Roch theorem for curves, and the Mordell-Weil theorem.

18.782 Introduction to Arithmetic Geometry and 18.783 Elliptic Curves both cover material on elliptic curves, but there is essentially no overlap; the courses are complementary and may be taken in either order.

### Course Outcomes

#### Course Goals for Students

After completing 18.782 the student will have been introduced to some of the key tools of arithmetic geometry and should be well prepared for more advanced courses in the subject.

#### Possibilities for Further Study/Careers

Students interested in learning more about elliptic curves are encouraged to take 18.783 Elliptic Curves and may also want to consider graduate level course sequences such as:

### Instructor Interview

Below, Dr. Andrew Sutherland describes various aspects of how he taught 18.782 Introduction to Arithmetic Geometry.

The course includes an introduction to algebraic varieties and divisor class groups, working over fields of arbitrary characteristic that are perfect but not necessarily algebraically closed. Although we do not use the language of schemes (except in one problem set), a key goal is to prepare the student for more advanced courses that will use schemes.

In order to make the course as self-contained as possible, proofs of a few key results from commutative algebra that are not typically covered in a first year of algebra are included in the notes (e.g. a proof of Nakayama’s lemma and some standard facts about Dedekind domains and discrete valuation rings).

Each problem set included a brief survey at the end to collect the students’ feedback on the lectures and problem sets. This strategy is described in detail on the Instructor Insights page for 18.783 Elliptic Curves.

### Curriculum Information

#### Prerequisites

18.702 Algebra II

#### Requirements Satisfied

18.782 can be applied toward a Bachelor of Science in Mathematics.

#### Offered

This course is offered every other Fall semester.

### Student Information

#### Enrollment

Fewer than 10 students

#### Breakdown by Year

Roughly 3/4 undergraduates, most of whom were juniors and seniors; 1/4 graduate students.

#### Breakdown by Major

A mix of students majoring in mathematics and in electrical engineering and computer science.

### How Student Time Was Spent

During an average week, students were expected to spend 12 hours on the course, roughly divided as follows:

#### In Class

• Met 2 times per week for 1.5 hours per session; 26 sessions total.
• Several of the lectures included interactive sessions using Sage. The Sage worksheets are listed in the lecture notes section.

#### Out of Class

• There was no required textbook, but references to several books and articles were provided.
• Students completed 11 problem sets. The course grade was determined by the student’s average problem set score, after dropping the lowest score. There were no midterm exams and no final.

## Course Info

Fall 2013
##### Learning Resource Types
Problem Sets
Lecture Notes
Online Textbook
Instructor Insights