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Elliptic Curves

Welcome to the course website for the advanced course on elliptic curves at Ross/Ohio 2026. This course is intended as an introduction to the beautiful geometry and arithmetic of elliptic curves for second-year students (i.e., JCs) and counselors at Ross.1

Basic Information

  • Instructor: Dhruv Goel [dhruvgoel@princeton.edu]
  • Meeting Times and Locations: See the Ross Calendar for updated times and locations.
  • Topics Covered: The course topics will be adjusted based on the backgrounds and interests of the participants. A tentative list of topics can be found below.
  • Texts/References: I will type up my own lecture notes. For other references, see the section below.
  • Prerequisites: Prerequisites assumed will be adjusted based on the backgrouns and interests of the participants. The current plan is to assume familiarity with the material on the Ross first-year sets, up to and including topics such as the basics of rings and fields. We will also need the basics of group theory (up to the isomorphism theorems, and a little bit on the structure of finitely generated abelian groups). Some familiarity with number fields and with projective geometry will be helpful but is not strictly necessary.2
  • Assignments and Collaboration Policy: There are no formal assignments, although I will produce weekly exercise sheets. You do not need to turn in solutions, but you are welcome to discuss your solutions with me as well. You are highly encouraged to read the lecture notes carefully and to attempt each problem in the exercise sheets by yourself, having done which, collaboration and discussion with your peers is highly welcome and encouraged.
  • Software and Online Resources: We will primarily use SageMath to do computations, and we will also learn how to use the LMFDB.
Elliptic Curve

1. This course is NOT for first-year students at Ross, and first-year students are generally not encouraged to take this course; if you are a first-year student who really wants to sit in, let me and your counselor know and we can discuss your specific situation.↩
2. If you would like to come to this course but are unsure about whether you have enough background, you probably do. Even if you don't, I believe that you will certainly gain something from coming to the lectures, even if you don't understand them fully. You are also welcome to write to me to discuss this.↩

Topics Covered

The course topics will be adjusted based on the backgrounds and interests of the participants. The current (tentative, slightly ambitious) list of topics I plan to talk about is:
  1. Introduction and motivation: rational points on lines and conics, Pythagorean triples, Fermat's method of infinite descent
  2. Review of background material on projective geometry: homogeneous coordinates in the projective plane, intersections of projective curves and Bézout’s Theorem, singular points on curves, cubic curves and exceptional points on them
  3. Weierstrass forms of elliptic curves, and transformations between them
  4. Brief history detour and elliptic curves over \(\mathbb{C}\): Riemann surfaces, elliptic integrals, and basics of complex lattices
  5. Review of background material on curves: divisors, the Riemann-Roch Theorem, differentials, morphisms of curves, and ramification (as needed)
  6. The isomorphism classification of elliptic curves via the \(j\)-invariant, curves with special \(j\)-invariants and their automorphisms
  7. Nagell-Lutz theorem, outlook towards Mazur’s Theorem
  8. Isogenies, the invariant differential, and applications to torsion points
  9. Elliptic curves over finite fields: Hasse bound, the Weil conjectures including the Riemann hypothesis, and supersingular curves
  10. Review of local fields: nonarchimedean valuations, \(p\)-adic numbers, Hensel’s Lemma, Ostrowski's Theorem, local-to-global principles
  11. Elliptic curves over local fields: formal group laws, reduction, Tamagawa numbers, applications to torsion subgroups, Nagell-Lutz revisited
  12. Galois cohomology and applications to Kummer theory
  13. Elliptic curves over global fields I: local-to-global principles, Selmer and Tate-Shafarevich groups, and the weak Mordell-Weil Theorem
  14. Elliptic curves over global fields II: heights and the Mordell-Weil Theorem
  15. Descent by 2-isogeny and applications to rank computations and classical problems (e.g., 2 is not congruent, Euler's Theorem on four squares in AP, failure of the local-to-global principle)
Further topics (if time permits, and depending on the backgrounds and interests of the participants) could include:
  1. Special cases of Mazur’s Theorem: the Weil pairing, the advanced perspective on 3-torsion on elliptic curves over \(\mathbb{R}\), and the Levi-Lind Theorem on the 2-primary subgroup
  2. Explicit families of elliptic curves with specified torsion, modular curves
  3. Complex multiplication, Galois representations, and an outlook towards Fermat’s Last Theorem
  4. Arithmetic dynamics on elliptic curves: the dynamics of the duplication formula and Northcott’s Theorem on the finiteness of preperiodic points
  5. More on elliptic curves over finite fields: factorization algorithms and elliptic curve cryptography
  6. \(L\)-functions associated to elliptic curves and an outlook towards the BSD Conjecture
  7. Integral points on elliptic curves, towards theorems of Siegel and Roth
  8. More general descent and applications to rank computations, twisting
  9. Néron models and Tate's Algorithm, applications to Diophantine equations involving powerful numbers
We will try to keep the exposition as down-to-earth and example-heavy as possible, while still trying to develop enough of the general theory to see the basic theorems of the field.

Lecture Notes

Lecture notes will be updated after the corresponding lecture. The full file, updated on 26/07/10, can be found here. Here are the notes sorted by lecture:
  • Lecture 1; posted on 26/06/15; Diophantine equations, Rational Root Theorem, plane curves, Bachet's example
  • Lecture 2; posted on 26/06/16; Pythagorean triples and Fermat for exponent \(n=3\)
  • Lecture 3; posted on 26/06/17; Fermat for exponent \(n=4\) and infinite descent, “review” of projective geometry basics
  • Lecture 4; posted on 26/06/18; continue of “review” of projective geometry, projective linear changes of coordinates
  • Lecture 5; posted on 26/06/22; continue of “review” of projective geometry: smoothness
  • Lecture 6; posted on 26/06/23; continue of “review” of projective geometry: wrapping up smoothness, introduction to Bézout's theorem
  • Lecture 7; posted on 26/06/24; wrapping up “review” of projective geometry: proving Bézout's theorem for intersection with a line; the geometry of cubic curves: the group law on the points of a smooth cubic curve
  • Lecture 8; posted on 26/06/25; the geometry of cubic curves: proof sketch of associativity of the group law
  • Lecture 9; posted on 26/06/29; intro to Weierstrass equations of elliptic curves
  • Lecture 10; posted on 26/06/30; Weierstrass equations of elliptic curves and the origins of \(\Delta\); admissible changes of coordinates; working with Sage
  • Lecture 11; posted on 26/07/01; Weierstrass equations of elliptic curves: automorphism groups, the \(j\)-invariant, and the isomorphism classification of elliptic curves, and the case of characteristics \(2\) and \(3\)
  • Lecture 12; posted on 26/07/02; the origin of the moniker “elliptic curves”, two and three torsion on elliptic curves, and an introduction to supersingularity
  • Lecture 13; posted on 26/07/06; the story over \(K=\mathbb{C}\) and the real reason for the structure of the \(n\)-torsion subgroup; introduction to the Nagell-Lutz theorem
  • Lecture 14; posted on 26/07/07; the proof of the Nagell-Lutz theorem via the \(p\)-adic filtration
  • Lecture 15; posted on 26/07/08; finishing the proof of the Nagell-Lutz theorem, and introduction to the Mordell-Weil theorem and the abstract descent machinery
  • Lecture 16; posted on 26/07/09; the height machine, and reducing the proof of the Mordell/finite basis theorem to the weak Mordell/finite basis theorem
  • Lecture 17; posted on 26/07/10; the Néron-Tate canonical height
  • Lecture 18; posted on 26/07/13; the case of the weak Mordell Theorem when there is no nontrivial rational 2-torsion, and an introduction to the case of the completely split rational 2-torsion
  • Lecture 19; posted on 26/07/14; wrapping up the proof of weak Mordell for completely rational split 2-torsion
  • Lecture 20; posted on 26/07/15; isogenies and descent via two-isogeny
  • Lecture 21; posted on 26/07/16; finishing the proof of Mordell's theorem, and the structure of finitely generated abelian groups via the Smith Normal Form

Exercise Sheets

There will be six of these; one for each week. The exercise sheet for a week will be posted at the beginning of that week, so you can start engaging with the problems as soon as possible. Here are the exercise sheets:

Textbooks and References

The following are standard textbooks at a level similar to (or probably slightly easier than) that of this course:
  1. Lectures on Elliptic Curves by Cassels, and
  2. Rational Points on Elliptic Curves by Silverman and Tate.
The following are standard textbooks at a level similar to (or possibly slightly harder than) that of this course:
  1. The Arithmetic of Elliptic Curves by Silverman,
  2. Elliptic Curves by Husemöller, and
  3. Elliptic Curves, Modular Forms, and Their \(L\)-functions by Álvaro Lozano-Robledo.
Some lectures notes by Zhiyun Bai covering similar material can be found here.

These resources do not cover everything we will talk about in the course, but cover other topics. All the course materials for this course will be available freely via the lecture notes above.

If you need assistance in accessing any of these references, please do not hesitate to reach out to me.

Notices

  • 2026/06/12: If you are interested in being a part of this course, please fill out this Google form as soon as possible. This will help me plan the course.
  • 2026/07/05: The Google form for mid-course feedback can be found here. We will not have class on 26/07/22 or 26/07/23; instead, we will use the dates 26/07/10 and 26/07/17 instead. Discussion sessions will be moved to the evening. See the Discord channel for updates.