Arithmetic of elliptic curves
Math 714 - Fall 2026

Santiago Arango-Piñeros, Ph.D. (He/Him).

My office is in LGRT 1238, and you can contact me via email at:

• sarangopiner@umass.edu.

TBD.

• [S] J. H. Silverman, The Arithmetic of Elliptic Curves, 2nd edition, Graduate Texts in Mathematics 106, Springer, 2009. (errata)

• Milne, Elliptic Curves.
• Diamond & Shurman, A first course in modular forms.
• Poonen, Lectures on rational points on curves.
• Poonen, Rational points on varieties. (errata)
• Poonen, Elliptic curves.
• Siksek, Explicit arithmetic of modular curves.
• Silverman, Advanced topics in the arithmetic of elliptic curves. (errata)
• Sutherland, MIT Math 18.783 - Elliptic Curves.
• LMFDB, Elliptic curve database (Over \(\mathbb{Q}\), over number fields).
• Magma, the handbook and calculator.

Your grade will be computed as a weighted average of the following work:

• Reading, participation, collaboration, and engagement (20%)
• Homework (42%)
• Final Project (38%)

I will have a one-hour meeting with each student before the end of classes. For this meeting, the student will bring their solved problems and research report. We will talk about your homework solutions, and discuss possible future research directions based on your report.

• You are responsible for scheduling this meeting with me (in person) before <2026-12-16 Wed>.

I will compute your grade as the weighted average \[ 0.20\cdot(\text{participation}) + 0.42\cdot(\text{homework}) + 0.38\cdot(\text{final project}), \] of the number of points \( \in [0,100]\) in each category, and then use the following tables to convert it into a letter grade.

• Cremona & Sutherland, On a theorem of Mestre and Schoof.
• Dokchitser & Dokchitser, Surjectivity of mod \(2^n\) representations of elliptic curves.
• Lenstra, Factoring integers with elliptic curves.
• Klagsbrun & Lemke Oliver, The distribution of the Tamagawa ratio in the family of elliptic curves with a two-torsion point.
• Sutherland, Identifying supersingular elliptic curves.
• Schoof, Elliptic curves over finite fields and the computation of square roots mod p.
• Schoof & Tzanakis, Integral points on a modular curve of level 11.
• Wong, Quadratic twists of pairs of elliptic curves.
• Wong, On the density of elliptic curves.
• Zywina, An elliptic surface with infinitely many fibers for which the rank does not jump.
• Zywina, There are infinitely many elliptic curves over the rationals of rank 2.

Your effort and attitude determine your abilities. Embrace challenges and failure as an opportunity to grow. Find inspiration in other people's success.

Paraphrasing Galileo:

``You cannot teach a person anything; you can only help them find it within themselves.''

We are all here to understand. My job as a more experienced learner is to assist you on your journey. But you are responsible for investing the time and effort necessary to learn.

This is hard work, and it will be frustrating at times. In my opinion, the reward is well worth the investment, as is often the case with challenging endeavors. In the words of JFK:

``We choose to go to the Moon in this decade and do the other things, not because they are easy, but because they are hard; because that goal will serve to organize and measure the best of our energies and skills, because that challenge is one that we are willing to accept, one we are unwilling to postpone, and one we intend to win, and the others, too.''

We will subscribe to Federico's axioms.

• Axiom 1. Mathematical potential is equally present in different groups, irrespective of geographic, demographic, and economic boundaries.
• Axiom 2. Everyone can have joyful, meaningful, and empowering mathematical experiences.
• Axiom 3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
• Axiom 4. Every student deserves to be treated with dignity and respect.

The dates below are the graduate student deadlines for Fall 2026.

• Last day to add or drop with no record: <2026-09-21 Mon>.
• Last day to drop with DR: <2026-11-04 Wed>.
• See the academic calendar for other important dates.
• Incomplete grades are warranted only if a student is passing the course at the time of the request and if the course requirements can be completed by the end of the following semester. Read more here.

By UMass policy, students are expected to attend all regularly scheduled classes at the University for which they are registered. This is a small, discussion-based graduate course: the lectures are the course, and they do not work if you are not in the room. For this reason I do keep track of attendance, and attending at least 80% of the lectures is one of the conditions for passing the participation component.

If you cannot attend a lecture, let me know in advance when possible, read the corresponding sections yourself, and talk to your classmates to check whether you missed anything important. If something in your life is making regular attendance difficult, come talk to me early rather than late.

• I expect you to be present and refrain from using your phone.
• Arrive on time. If you arrive late, try to minimize your disruption.
• Laptops and tablets are allowed during lectures, provided that you do not disrupt your fellow classmates or the lecture.

Academic dishonesty includes but is not limited to:

• Cheating: intentional use, and/or attempted use of trickery, artifice, deception, breach of confidence, fraud, and/or misrepresentation of one's academic work.
• Fabrication: intentional and unauthorized falsification and/or invention of any information or citation in any academic exercise.
• Plagiarism: knowingly representing the words or ideas of another as one's own work in any academic exercise. This includes submitting without citation, in whole or in part, prewritten term papers of another or the research of another, including but not limited to commercial vendors who sell or distribute such materials.
• Facilitating dishonesty: knowingly helping or attempting to help another commit an act of academic dishonesty, including substituting for another in an examination, or allowing others to represent as their own one's papers, reports, or academic works.

Formal definitions of academic dishonesty, examples of various forms of dishonesty, and the procedures which faculty must follow to penalize dishonesty are detailed on the Academic Honesty website. Appeals must be filed within ten days of notification by the Academic Honesty Office that a formal charge has been filed by an instructor who suspects dishonesty. Contact the Academic Honesty Office for more information on the process. The Ombuds Office is also available to support individuals engaging with the Academic Honesty process. The Provost’s Office is where appeals are processed and filed.

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 (http://www.umass.edu/dean_students/codeofconduct/acadhonesty/).

UMass Amherst is strongly committed to academic integrity, which is defined as completing all academic work without cheating, lying, stealing, or receiving unauthorized assistance from any other person, or using any source of information not appropriately authorized or attributed. As a community, we hold each other accountable and support each other’s knowledge and understanding of academic integrity. Academic dishonesty is prohibited in all programs of the University and includes but is not limited to: cheating, fabrication, plagiarism, lying, and facilitating dishonesty, via analogue and digital means. Sanctions may be imposed on any student who has committed or participated in an academic integrity infraction. Any person who has reason to believe that a student has committed an academic integrity infraction should bring such information to the attention of the appropriate course instructor as soon as possible. All students at the University of Massachusetts Amherst have read and acknowledged the Commitment to Academic Integrity and are knowingly responsible for completing all work with integrity and in accordance with the policy: (https://www.umass.edu/senate/book/academic-regulations-academic-integrity-policy).

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 Disability Services (https://www.umass.edu/disability/).

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 on campus to support you. A summary of the available Title IX resources (confidential and non-confidential) can be found at the following link: https://www.umass.edu/titleix/resources. 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.