MIT OpenCourseWare

Browse materialsMIT 18.782 · Fall 2013

Introduction to Arithmetic Geometry

Subjects: Algebra · Geometry · Mathematics

Art: Anonymous painter,15th century, Westerner and Arab practicing geometry 15th century manuscript, 15th… · Public domain

Instructor: Andrew V. Sutherland
Term: Fall 2013
Level: Undergraduate
What's inside: 11 problem sets · 26 lecture notes · 1 reading · 3 other files · 4 web pages
Materials open: MIT OCW

Start here

Start course: Lecture 1: Introduction to Arithmetic Geometry Find a topic

Progress is saved in this browser.

Export or import progress

About this course

Source: MIT OpenCourseWare 18.782 Introduction to Arithmetic Geometry, Fall 2013, Andrew V. Sutherland — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: MIT's own listing order, section by section. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 23, 2026. How RedLectern is built.

Materials are grouped the way MIT organized them. A verified, course-wide order has not been established here.

Before you start

Prerequisites are not listed here. Check MIT's own syllabus for this course.

This course’s cited books, in our reading lists: The Arithmetic of Elliptic Curves · Elliptic Curves · A Course in Arithmetic

How this course was organized

What's here, and what isn't

MIT's own materials here did not support a verified, course-wide order, so they are grouped by section instead.

Not included here

33 items from the source could not be safely included:

MIT's page format for this item wasn't recognized clearly enough to place it safely. (32 items)
Referenced in a table, but without MIT's own verified order, so it isn't used to claim a sequence.

Materials by section

Grouped by section, as MIT organized them. Not sequenced or ordered.

Lecture Notes

28 files

Assignments

11 problem sets

2 images

MIT OpenCourseWare, Andrew V. Sutherland, 18.782 Introduction to Arithmetic Geometry, Fall 2013.

License: CC BY-NC-SA 4.0 · Edition: Fall 2013 · Open the original course on OCW ↗

Organized by RedLectern, independent of MIT and not endorsed by MIT.

Per-resource rights are recorded on each resource; All-Rights-Reserved items are link-only, never mirrored.

Introduction to Arithmetic Geometry

Before you start

What's here, and what isn't

Not included here

Materials by section

Lecture Notes

Assignments

Lecture 1: Introduction to Arithmetic Geometry

Lecture 2: Rational Points on Conics

Lecture 3: Finite Fields

Lecture 4: The Ring of p -adic Integers

Lecture 5: The Field of p -adic Numbers, Absolute Values, Ostrowski’s Theorem for Q

Lecture 6: Ostrowski’s Theorem for Number Fields

Lecture 7: Product Formula for Number Fields, Completions

Lecture 8: Hensel’s Lemma

Lecture 9: Quadratic Forms

Lecture 10: Hilbert Symbols

Lecture 11: Weak and Strong Approximation, Hasse-Minkowski Theorem for Q

Lecture 12: Field Extensions, Algebraic Sets

Lecture 13: Affine and Projective Varieties

Lecture 14: Zariski Topology, Morphisms of Affine Varieties and Affine Algebras

Lecture 15: Rational Maps and Function Fields

Lecture 16: Products of Varieties and Chevalley’s criterion for Completeness

Lecture 17: Tangent Spaces, Singular Points, Hypersurfaces

Lecture 18: Smooth Projective Curves

Lecture 19: Divisors, The Picard Group

Lecture 20: Degree Theorem for Morphisms of Curves

Lecture 21: Riemann-Roch Spaces

Lecture 22: Proof of the Riemann-Roch Theorem for Curves

Lecture 23: Elliptic Curves and Abelian Varieties

Lecture 24: Isogenies and Torsion Points, The Nagell-Lutz Theorem

Lecture 25: The Mordell-Weil Theorem

Lecture 26: Jacobians of Genus One Curves, The Weil-Chatelet and Tate-Shafarevich Groups