Math 346/646 - Modern Number Theory
Course Syllabus
Course Location
TR 11:00-12:15 pm, Manchester Hall, Room 018
Contact information
Name: Jeremy Rouse
E-mail: rouseja@wfu.edu
Office: Room 340, Manchester Hall.
Office Phone: (336) 758-4159.
Office Hours: 4-5 pm on Tuesdays, 3-4 pm on Fridays.
Course Information
This course will be about elliptic curves. Roughly speaking, an elliptic curve is a curve of the
form y 2 = x3 + ax2 + bx + c. Elliptic curves are fundamental objects in modern number theory,
and with them, one can answer a surprising number of questions (that, taken at face value,
have nothing to do with elliptic curves). There is much we do not yet know about elliptic
curves.
The only official pre-requisite for this course is Math 117 - discrete mathematics. The course
will use tools from number theory and abstract algebra, and these will be introduced as they
are needed. This is, however, a 300-level course, and if you have more exposure to proofs and
theoretical mathematics, it will go more smoothly. Prior background with abstract algebra or
number theory is also likely to be helpful.
The course will start with an overview, presentation of the facts we’ll need from number theory,
a discussion of “genus zero curves”, and a crash course in group theory. Next, we’ll develop
the basic theory of elliptic curves and rational points on them. We’ll discuss integral points on
elliptic curves, elliptic curves modulo a prime p and applications (to integer factorization, and
cryptography). We’ll conclude with the Birch and Swinnerton-Dyer conjecture, an unsolved
problem with a 1 million dollar prize.
I expect that this course will show you some exciting applications of abstract algebraic thinking,
and give you a taste of what modern number theory is about. A more thorough list of the
course learning outcomes is posted at the website below.
Website: http://www.wfu.edu/∼rouseja/teaching/F2015/346/
Textbook: Joseph Silverman and John Tate’s Rational Points on Elliptic Curves, ISBN:
978-1-4419-3101-6.
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Homework
After class on Thursdays, I will post a list of homework problems on the class website that
will be due the following Thursday. Some of these will be more straightforward, and some will
be more challenging. Each week one of these problems will be a “challenge problem.” This
problem is required for graduate students, and extra-credit for undergraduate students.
Homework will contribute 30 percent of the course grade. I encourage working on homework
in groups. I ask, however, that each student write up their own solutions without reference to
their friends, or their friends’ work.
Some of the homework problems will require the use of SAGE, an open-source computer software package with built-in routines for working with elliptic curves and modular forms. The
software and documentation is available at http://www.sagemath.org and can be used in your
browser at http://cloud.sagemath.com.
Exams
There will be one midterm exam and one final exam. The midterm exam will be a take-home
exam that will be given out on Thursday, October 8 and will be due at the beginning of class
on Tuesday, October 13. The midterm exam will contribute 30 percent of the final grade.
The final exam will also be a take-home exam that will be given out at the end of class on
Thursday, December 3. It will be due on at noon on Thursday, December 10. The final exam
will contribute 40 percent of the course grade.
Grades
The function which maps percentages to letter grades will be chosen at the end of the course.
However, if you earn at least 90 percent of the points, your grade will be at least an A-. If you
earn at least 80 percent of the points, your grade will be at least a B-, etc.
Getting help
• Do the homework. The only way to really learn the material is to practice it and
struggle with it. This is your chance.
• Attend Class. The material in the course is quite challenging, and I will keep that in
mind when presenting my lectures. Some necessary background material, and a few
advanced topics will be covered in lecture, but aren’t in the text.
• Read the book. The book has more information and context than I will be able to
provide in lecture. Reading the book will also reinforce the material and your understanding of it.
• Office hours. If you have questions, please ask! You are encouraged to ask questions in
class, but office hours are another good time to get your questions answered.
• Other students in the class. You should talk to other students, ask them questions,
answer their questions, and work with them on homework.