SYLLABUS
A rough outline of the course is available here. Lecture notes from the 2021 edition can be found on OCW. The 2023 edition will be similar but may include some additional material on elliptic curves over ℚ that was not covered in 2021.
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 free access). 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 12 weekly problem sets, each of which typically contain three to five multipart problems. You are not expected to solve all of the problems, you be given the option to choose a subset to turn in. Some problems are purely theoretical in nature, while others are more computationally focused; 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 15.
Problem sets are to be prepared in typeset form (typically via latex) and submitted electronically as pdf-file to Gradescope. Collaboration is permitted/encouraged, but you must write up your own solutions and explicitly identify any collaborators, or simply give the name of your pset group on pset partners, as well as any resources you used 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. Your lowest problem set score is dropped when computing your grade, so you can skip one with no penalty. Please do not ask to turn in a problem set late. If you find yourself short on time it is better to skip a problem set and get an early start on the next one rather than putting additional stress on yourself by trying to finish two problem sets in the same week.
COVID-related Contingency Plans
Missing lectures: If you are unable to attend class because you are in isolation, you can read the detailed lecture notes that will posted within 24 hours of each class on the course website.
Missing assignments: As noted above, you can skip one problem set without penalty. If you are unable to complete more than one problem set due to illness you should contact S3 to request an accomodation. In situations where more than 2 problem sets are missed this will likely mean an incomplete grade with the opportunity to finish the course over IAP; it is not realistic to make up multiple missed problem sets while simultaneously completing the new problem sets that are assigned each week and keep up with your other classes.
Missing instructor: If I need to isolate due to a positive COVID test I will deliver lectures over Zoom if I am asymptomatic, or find a substitute instructor to teach in person if I am not.
Grading
Your grade will be determined by your average problem set score, after dropping your lowest/missing score, plus bonus points you may earn by finding typos/errors in the lecture notes or problem sets.
There is no curve, your grade will be computed using a standard fixed scale (anything over 97.5 is an A+, 92.5 to 97.5 is an A, 90 to 92.5 is an A-, etc...), so you will always know exactly where you stand. This applies to graduate students as well as undergraduates, there is not a separate scale for graduate students.
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.
