Elliptic curves are, depending on who you ask, either breakfast item or solutions to equations of the form \[ y^2 = x^3 + ax + b. \] The focus of this seminar is the rich arithmetic theory of these curves, which means that we are interested in finding solutions in which \(x\) and \(y\) are rational numbers. For instance, a first question would be whether or not there is at least one solution of the above equation in rational numbers? If this is answered in the affirmative, then maybe we want to know how many solutions are there. Perhaps there are infinitely many solutions, in which case we might want to have a measure of the size. From another tack, we might wish to know how to construction rational solutions to the above equation. And we can go on!

The goal of this seminar is to get familiar with the basic notions around elliptic curves and to play with their arithmetic, after which there are many topics we can jump to: modular forms, cryptography, a Millennium Problem… It will be fun!

### Prerequisites, Format, Expectations, etc.

This seminar should be accessible to one with a good background in linear algebra, some analysis, and basics of modern algebra.

Beyond the mathematical content, these seminars are really about communicating mathematics. In particular, I think that the small seminar environment is a perfect place to develop skills and a sense of comfort in speaking and presenting mathematics. Thus the seminar consists mostly of lectures by the participants. Moreover, this experience is a great place to think carefully about how to explain mathematics to an audience.

## References

The following are some references for the content of this seminar:

Joseph H. Silverman and John Tate (1992), Rational Points on Elliptic Curves.

Avner Ash and Robert Gross (2012), Elliptic Tales: Curves, Counting, and Number Theory.

Álvaro Lozano-Robledo (2011), Elliptic Curves, Modular Forms, and Their L-functions.

Thomas R. Shemanske (2017), Modern Cryptography and Elliptic Curves: A Beginner’s Guide.

Finally, here is a short, recent survey on some of the topics around the arithmetic of elliptic curves:

- Harris B. Daniels and Álvaro Lozano-Robledo (2017), What is… an Elliptic Curve?.

## Schedule

We meet Tuesdays and Thursdays between 3:00 PM and 4:00 PM in Mathematics Room 507.

- 01/23
- Organization.
- 02/06
- Raymond Cheng,
- Introduction.
- [ST, Introduction], [L-R, Chapter 1], [Sh, Chapter 1].
- 02/11
- Julian Goldberg,
- Conics & Pythagorean Triples. Cubic Curves.
- [ST, §§I.1, I.2], [Sh, §§2.1–2.3].
- 02/13
- Carly Roth,
- Projective Curves. Weierstrass Form.
- [ST, §§A.1, A.2, I.3], [L-R, §2.2], [Sh, §§7.1–7.3],
- 02/18
- Prakruth Adari,
- Explicit Group Law. 2- and 3-Torsion.
- [ST, §§I.4, II.1], [L-R, §2.4], [Sh, §§7.4, 7.5].
- 02/20
- Arad Lev Ari & Jonathan Schwartz,
- Nagel–Lutz Theorem.
- [ST, §§II.2–II.5], [L-R, §2.5].
- 02/25
- No Seminar.
- Raymond is away.
- 02/27
- No Seminar.
- Raymond is still away.
- 03/03
- Eli Goldin,
- Finite Fields. Nagel–Lutz Redux.
- [ST, §§IV.1, IV.3], [L-R, §§2.5, 2.6].
- 03/05
- Miles Van Tongeren,
- Hasse–Weil Theorem. A Theorem of Gauss.
- [ST, §IV.2], [L-R, §2.6].
- 03/10
- Michael Meng,
- Statement of Mordell’s Theorem. Heights and Descent.
- [ST, §III.1], [L-R, §§2.7, 2.9].
- 03/12
- Garrett Gregor-Splaver,
- Bounds on Heights. Canonical Heights.
- [ST, §§III.2, III.3], [L-R, §§2.7, 2.8].
- 03/17
- No Seminar.
- Spring Break.
- 03/19
- No Seminar.
- Spring Break.
- 03/24
- TBA,
- Weak Mordell–Weil Theorem.
- [L-R, §2.9], [ST, §§III.4, III.5].
- 03/26
- TBA,
- Mordell’s Theorem. Selmer and Sha.
- [ST, §§III.5, III.6], [L-R, §§2.10, 2.11].
- 03/31
- TBA,
- \(L\)-Function of an Elliptic Curve. Birch–Swinnerton-Dyer Conjecture.
- [L-R, §§5.1–5.2].
- 04/02
- TBA,
- Statement of Taniyama–Shimura–Weil.
- [L-R, §§5.3–5.4].
- 04/07
- TBA.
- 04/09
- TBA.
- 04/14
- TBA.
- 04/15
- TBA.
- 04/21
- TBA.
- 04/23
- TBA.
- 04/28
- TBA.
- 04/30
- TBA.
- 05/05
- TBA.
- 05/07
- TBA.