SYLLABUS
A rough outline of the course is available here. Lecture notes from the 2023 edition can be found here. The 2025 edition will be similar, but there will be some changes
Text Book
There is no required text; lecture notes will be provided. We will make reference to material in the following books, all of which can be accessed electronically from MIT (see the MIT Libraries web page for offsite access).
Elliptic curves: Number theory and cryptography, second edition, Lawrence C. Washington. (errata)
Elliptic curves, J.S. Milne. (errata)
The arithmetic of elliptic curves, Joseph H. Silverman. (errata)
Advanced topics in the arithmetic of elliptic curves, Joseph H. Silverman. (errata)
Primes of the form x2+ny2: Fermat, class field theory, and complex multiplication, David A. Cox. (errata)
The following two books give quite accessible introductions to elliptic curves from different perspectives. You may find them useful as supplemental reading, but we will not use of them in the course.
Elliptic curves in cryptography, Blake, Seroussi, and Smart.
Rational points on elliptic curves, Joseph H. Silverman and John Tate.
The following references provide introductions to algebraic number theory and complex analysis; neither of these topics is an official prerequisites for this course, but we will occasionally need to make use of their results.
Algebraic number theory, J.S. Milne.
Complex analysis, Serge Lang
Software
Some of the theorems and algorithms presented in lecture will be demonstrated using Sage, a python-based computer algebra system, hosted on CoCalc (all 18.783 students will be provided access to a CoCalc project for the class). Most of the problem sets will contain at least one computationally-focused problem, which you will likely want to use Sage to solve, but you are free to use other packages, or roll your own code from scratch. You will be graded on your results and your mathematical explanation and analysis of your algorithm, not your code.
Problem Sets
There will be weekly problem sets, each of which typically contain three to five multipart problems. You are not expected to solve all of the problems on each problem set, you will choose a subset to turn in. Some problems are theoretical in nature, while others are more computational; those who prefer proofs to programming (or vice versa) can choose problems that appeal to their interests. The first problem set will be due on Friday, Sep 10.
Problem sets are to be prepared in typeset form (typically via latex) and submitted electronically as pdf-file to Gradescope. Collaboration (with humans and LLMs) is permitted/encouraged, but you must write up your own solutions and explicitly identify any collaborators (including LLMs such as chat-GPT, Claude, or Gemini, if applicable), and/or give the name of your pset group on pset partners, as well as any resources you consulted that are not listed above. There will be computational problems for which the correct answer will be different for every student, based on a unique identifier derived from your student ID.
Late Policy
Late problem sets will not be accepted. I understand that there may be circumstances beyond your control that make it impossible for you to submit a problem set on time. If you find yourself in this situation it is better to skip the problem set (for which you will incur no penalty, see the grading section below) and shift your focus to the next problem set, rather than putting additional stress on yourself by trying to finish two problem sets in the same week.
Similarly, there are will be no makeup exams. You can miss one of the two in-class quizzes without any penalty.
Grading
There will be in-class mid-term exams on Oct 16 and Nov 6, plus a 3-hour final exam on on Tuesday Dec 16 at 1:30 pm. Sixty percent of your grade will be determined by your problem set scores, and forty percent will be determined by exams. When computing your scores I will do whichever of the following gives you the best grade: (a) drop your two worst pset scores, (b) drop your worst pset score and your worst mid-term score, (c) drop your grade on the final. This means you can skip the final exam if you are happy with your grade at that point.
Your grade will be computed using a standard scale (over 97.5 is an A+, 92.5 to 97.5 is an A, 90 to 92.5 is an A-, etc...).
Disability Accommodations
Please contact Disability and Access Services as early in the term as possible, if you have not already done so. If you already have an accommodation letter, please be sure to submit a copy to Mathematics Academic Services. Even if you do not plan to use any accommodations, if there is anything I can do to facilitate your learning, please let me know.
