Prof. Paul Gunnells, LGRT 1115L, 413–545–6009, gunnells at umass dot edu. The best way to contact me is by email. Please don’t leave a message on my office phone or through any other way; I won’t get it.
Office hours are by appointment.
Number theory concerns the integer and rational solutions of polynomial equations. By exploiting the geometric properties of the solution sets of these equations over various fields, such as the reals, complex numbers, finite fields, number fields, and local fields, we can often extract non-trivial arithemtic information about the original equations. In this course we will carry out a systematic study of an important class of equations, those whose solution sets over the complex numbers are topologically a torus (donut with one hole). They form the basis of much of modern research in number theory, they are simultaneously concrete and tangible, and the tools and results we cover in this class provide a very accessible entry point to the very exciting field of arithmetic geometry.
The class will consist of three hours of face-to-face lecture per week.
The textbook is
Some additional resources:
James Milne, Elliptic Curves. Available from his website. You can also purchase a hardcopy for $17.
Alain Robert, Elliptic Curves, Springer.
Larry Washington, Elliptic Curves: Number Theory and Cryptography, Second Edition (Discrete Mathematics and Its Applications).
Anthony Knapp, Elliptic Curves, Princeton University Press.
Neal Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer.
Joseph H. Silverman, John Tate, Rational Points on Elliptic Curves, Springer UTM. This is an excellent undergraduate textbook.
There is no required software for the course, but it is recommended that you use software to explore material from the course. Our personal favorite is gp-pari, but of course you can use anything you like. More information about gp-pari, along with some sample code, can be found here. Other options are Sage and Magma.
The primary learning objective of this course is for students to learn about arithmetic of elliptic curves, The goal of the course is to prepare students to do research in mathematical disciplines which rely on arithmetic geometry as well as to prepare them for more advanced courses in this area. Some specific learning objectives include:
Students will learn the language of arithmetic of elliptic curves.
Students will become familiar with basic examples and theorems, such as the group law on elliptic curves.
Students will learn advanced theorems, such as the Mordell-Weil theorem.
In addition, students will learn how to write an expository mathematics paper on an advanced topic.
In addition, despite their reputation, mathematics courses involve a large amount of very careful and detailed writing. Proof writing is a specialized writing skill, but at its core it is a form of persuasive technical writing.
The following schedule is tentative and may change according to the interests of the students.
Overview/Summary: Geometry of Algebraic Curves (chap. 1 and 2)
Geometry of elliptic curves (chap. 3)
Elliptic curves over finite fields (chap. 5)
Elliptic curves over local fields (chap. 7)
Elliptic curves over global fields (chap. 8 and 10)
Students will be expected to actively participate in the course by asking questions during lecture and discussing material with the instructor and their peers.
Some exercises will be assigned during the course. These will be graded and returned. Details will follow once a grader has been assigned.
The assignments can be found here.
Together with the instructor, each student will pick a topic of interest to them that is related to the course material. The student will then prepare an expository paper of 5–10 pages on this topic. More information can be found here.
Attendance at all classes is expected. However, formal attendance will not be taken.
The final grade will be based equally on course participation, homework, and the final paper. Letter grades will be assigned as follows:
| A | A– | B+ | B | B– | C+ | C | C– | D+ | D | F |
|---|---|---|---|---|---|---|---|---|---|---|
| 90 | 87 | 83 | 79 | 75 | 71 | 67 | 63 | 59 | 55 | <55 |
The University of Massachusetts Amherst is committed to providing an equal educational opportunity for all students. If you have a documented physical, psychological, or learning disability on file with Disability Services (DS), you may be eligible for reasonable academic accommodations to help you succeed in this course. If you have a documented disability that requires an accommodation, please notify me within the first two weeks of the semester so that we may make appropriate arrangements. For further information, please visit the DS website.
Since the integrity of the academic enterprise of any institution of higher education requires honesty in scholarship and research, academic honesty is required of all students at the University of Massachusetts Amherst. Academic dishonesty is prohibited in all programs of the University.
Academic dishonesty includes but is not limited to: cheating, fabrication, plagiarism, and facilitating dishonesty. Appropriate sanctions may be imposed on any student who has committed an act of academic dishonesty. Instructors should take reasonable steps to address academic misconduct. Any person who has reason to believe that a student has committed academic dishonesty should bring such information to the attention of the appropriate course instructor as soon as possible. Instances of academic dishonesty not related to a specific course should be brought to the attention of the appropriate department Head or Chair.
Since students are expected to be familiar with this policy and the commonly accepted standards of academic integrity, ignorance of such standards is not normally sufficient evidence of lack of intent. The policy can be found here.
In accordance with Title IX of the Education Amendments of 1972 that prohibits gender-based discrimination in educational settings that receive federal funds, the University of Massachusetts Amherst is committed to providing a safe learning environment for all students, free from all forms of discrimination, including sexual assault, sexual harassment, domestic violence, dating violence, stalking, and retaliation. This includes interactions in person or online through digital platforms and social media. Title IX also protects against discrimination on the basis of pregnancy, childbirth, false pregnancy, miscarriage, abortion, or related conditions, including recovery. There are resources here on campus to support you. A summary of the available Title IX resources (confidential and non-confidential) can be found here. You do not need to make a formal report to access them. If you need immediate support, you are not alone. Free and confidential support is available 24 hours a day / 7 days a week / 365 days a year at the SASA Hotline 413–545–0800.