# MATH 5020 – The Arithmetic of Elliptic Curves

### Course Description:

This course will be an introduction to elliptic curves which, roughly speaking, are smooth cubic curves in the projective plane with at least one rational point (turns out they have a simple model of the form y^2=x^3+ax+b). The surprising feature of elliptic curves is that their points can be made into an abelian group, and this group is finitely generated when we focus on points with coordinates in the rational numbers lying on an elliptic curve with rational coefficients. Elliptic curves are central in modern number theory, e.g., they were essential in the proof of Fermat’s Last Theorem. The goal of the course will be to understand and calculate the group of all rational points on an elliptic curve (i.e., calculate its torsion and rank), and a number of more refined invariants (such as the order of the Shafarevich-Tate group).

The prerequisites for this course are the abstract algebra sequence (Math 5210 and 5211) and a basic understanding of algebraic number theory and algebraic geometry, although I will adjust the material to the audience background as much as I can. Our textbook will be “The Arithmetic of Elliptic Curves,” by J. H. Silverman, which is the standard graduate-level textbook for the subject.

Prerequisites: MATH 5210 and MATH 5211 (i.e., a year of abstract algebra).
Recommended Preparation: A semester of Algebraic Number Theory, and a semester of Algebraic Geometry.
Credits: 3
Meets (Spring 2024): Tuesdays, and Thursdays, 12:15 – 1:45 at MONT 113.

### About the Instructor: Álvaro Lozano-Robledo

I’m a Professor of mathematics at the University of Connecticut.

Office hours: TBA.

### About the Book and Other Resources

I will be following the main reference: Joseph H. Silverman, “The Arithmetic of Elliptic Curves,” Springer, 2nd Edition. However, there are other references that may be very useful.

1. H. Daniels and Á. Lozano-Robledo, “What is… an elliptic curve?”, in the Notices of the American Mathematical Society, Vol 64, Issue 3, March 2017, pp. 241-243.

2. J. H. Silverman, J. Tate, “Rational Points on Elliptic Curves,” Springer – this book is an elementary version of the main reference.

3. Á. Lozano-Robledo, “Elliptic Curves, Modular Forms, and their L-Functions” is another elementary introduction to elliptic curves.

4. J. H. Silverman, “Advanced Topics in the Arithmetic of Elliptic Curves,” Springer.

5. J. S. Milne, “Elliptic Curves,” published by Kea Books – this book is freely available at Milne’s website.

6. N. Koblitz, “Introduction to Elliptic Curves and Modular Forms,” Springer.

7. J. Tate,The arithmetic of elliptic curves,” Invent. Math. 23 (1974), 179-206 – The main reference was heavily influenced by this key survey article.

8. D. Husemöller, “Elliptic Curves,” Second Edition, Springer.

9. L. Washington, “Elliptic Curves: Number Theory and Cryptography,” Second Edition (Discrete Mathematics and Its Applications) 2nd Edition, CRC Press.

10. E. Brown and A. Rice, Math. Mag. 85 (2012), no. 3, 163–176 – For those interested in the history of elliptic curves.

During the course we will discuss computability aspects of the theory, so it is helpful for students to familiarize themselves with computer packages that can handle computations on elliptic curves:

### Homework

Homework sets will be posted here periodically. The sets are a small selection of problems that should help students in understanding the theory, but of course there are many other problems available in Silverman’s book that are also a great resource for the learner. This is a topics class, so the homework sets are not mandatory. I’ll be happy to look at sets if you want to send me your solutions, but in general I expect that if you are interested in the material, you will be trying the problems on your own and reaching out to me when you need help or hints. For the sake of the grade, though, I do expect that you will submit some problems during the course (at least one per problem set).. If you have difficulties, do not waste time — get help! Please come to office hours!

### Tentative Schedule

The videos of the lectures can be found (as they become available) in this YouTube playlist.
Note: the PDF files below do not display correctly in some browsers. I recommend downloading the files and opening them locally in your machine.

 Lecture Topics (links to videos) Notes -1 What is… an elliptic curve? An introduction Slides from the talk. 0 Class logistics and description of the course Lecture 0 1 Introduction to diophantine equations Lecture 1 2 Number Fields… VS … Elliptic Curves Lecture 2 3 Intro. to algebraic varieties and curves Lecture 3 4 Varieties, function fields, dimension (part 1), and singularities, morphisms (part 2) Lecture 4 5 Order of vanishing (part 1), an example (part 2), and ramification (part 3) Lecture 5 6 More on ramification (part 1) and divisors, and differentials (part 2) Lecture 6 7 Riemann-Roch, Hurwitz, Weierstrass models Lecture 7 8 Models of elliptic curves, discriminant and j-invariant (part 1), and the group law (part 2) Lecture 8 9 The (geometric) group law (part 1), and the (Picard) group law (part 2) Lecture 9 10 Isogenies (part 1) and (part 2), and the dual isogeny (part 3) Lecture 10 11 The Tate module. Lecture 11 (Optional lecture on adelic Galois representations) Slides 12 The Weil pairing (construction) and (properties) Lecture 12 13 Consequences of the Weil pairing (part 1),  and elliptic curves over finite fields (Hasse’s bound, part 2) Lecture 13 14 An example of an elliptic curve over a finite field (part 1), and elliptic curves over C (part 2) Lecture 14 15 An introduction to the formal group of an elliptic curve. Lecture 15 16 Formal groups, homomorphisms (part 1), groups associated to formal groups, the formal logarithm, and formal exponential (part 2). Lecture 16 Happy Pi Day! (Extra lecture about pi and elliptic curves.) Pi Day 17 Torsion points in the groups associated to formal groups over DVR’s (part 1), and local fields, minimal models (part 2) Lecture 17 18 Elliptic curves over local fields: the fundamental exact sequence (part 1), and points of finite order over local fields: an example (part 2). Lecture 18 19 More on points of finite order (part 1), and the action of inertia (part 2) Lecture 19 20 Criterion of Néron-Ogg-Shafarevich, and good and bad reduction (part 1), potential good reduction and the j-invariant (part 2), and the group E/E_0 (part 3). Lecture 20 and a bit more 21 The weak Mordell–Weil theorem: the Kummer pairing Lecture 21 22 Properties of the Kummer pairing (part 1) and the finiteness of the maximal abelian extension of exponent m unramified outside a finite set of primes (part 2) Lecture 22 23 The proof of the weak Mordell-Weil theorem (part 1), the descent procedure and heights of elliptic curves (part 2) Lecture 23 24 Torsion points (Nagell-Lutz) (part 1), and the canonical height (part 2) Lecture 24 25 The canonical height pairing, and Z-linear independence of rational points (part 1); computing the Mordell-Weil group (part 2) Lecture 25 (Extra lectures) An introduction to group (and Galois) cohomology (part 1) and (part 2) Cohomology notes 26 Complete 2-descent (part 1), and an example (part 2). Lecture 26 27 Example of 2-descent (part 1), and definitions of Selmer and Sha (part 2) Lecture 27 28 Selmer and Sha: the fundamental sequence (part 1), twists, homogeneous spaces, and descent (part 2), and an example of descent via 2-isogeny (part 3). Lecture 28

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