## Math 115: Elliptic Curves

### Spring 2016

Course Info:

Syllabus:

[PDF] Syllabus

An elliptic curve is a cubic plane curve with the structure of a group; the group law is defined by geometric formulas.

Elliptic curves are ubiquitous in mathematics, with deep connections between number theory, algebra, geometry, and complex analysis. Their study is rich and they remain a topic of significant ongoing research. They first made an (implicit) appearance in the problem of giving the arc length of an ellipse, hence their name. In number theory, the set of solutions to a cubic equation in two variables with rational solutions is often understood as the set of rational points of an elliptic curve. From the point of view of manifolds, elliptic curves as Riemann surfaces are (flat) complex tori. In algebraic geometry, elliptic curves are perhaps the simplest nontrivial algebraic varieties. Finally, there are important applications of elliptic curves to cryptography.

In this course, we will survey elliptic curves from an arithmetic point of view. Topics may include: plane curves, basic theory of elliptic curves (Weierstrass equations), elliptic curves over the complex numbers, arithmetic of elliptic curves, and some relationships to modular forms.

 1 28 Mar (M) Introduction, I.1: Affine Varieties 2 30 Mar (W) I.2: Projective Varieties HW 1: 1.2, 1.3, 1.7, 1.8 3 31 Mar (R) I.3: Maps Between Varieties 4 1 Apr (F) III.1: Weierstrass Equations 5 4 Apr (M) III.2: The Group Law 6 6 Apr (W) II.1: Curves HW 2: 3.3, 3.4, 3.5 7 7 Apr (R) II.2: Maps Between Curves - 8 Apr (F) No class, JV at CUNY 8 11 Apr (M) Number field analogy 9 13 Apr (W) II.3: Divisors HW 3: 2.1, 2.2, 2.4, 2.10 10 14 Apr (R) II.4: Differentials 11 15 Apr (F) II.4 12 18 Apr (M) II.5: The Riemann-Roch Theorem 13 20 Apr (W) III.3: Elliptic Curves HW 4: 2.3, 2.7, 2.8 * 21 Apr (R) Exercise 2.10(a) 14 22 Apr (F) III.3 15 25 Apr (M) III.4: Isogenies 16 27 Apr (W) III.4 HW 5: 3.8, 3.9 17 29 Apr (F) III.5: The Invariant Differential 18 2 May (M) III.5, III.6: The Dual Isogeny 19 4 May (W) III.6, III.7: The Tate Module HW 6: 3.12, 3.14, 3.30 20 5 May (R) III.7 Final projects 21 6 May (F) III.8: The Weil Pairing 22 9 May (M) III.9: The Endomorphism Ring, III.10: The Automorphism Group 23 11 May (W) V.1: Number of Rational Points, V.2: The Weil Conjectures HW 7: 5.2, 5.4, 5.10, 5.6 24 13 May (F) V.3: The Endomorphism Ring, V.4: Calculating the Hasse Invariant 25 16 May (M) VI.1: Elliptic Integrals, VI.2: Elliptic Functions 26 18 May (W) VI.3: Construction of Elliptic Functions * 19 May (R) Office hour to discuss projects 27 20 May (F) VI.4: Maps Analytic and Maps Algebraic, VII.1: Minimal Weierstrass Equations 28 23 May (M) VII.2: Reduction Modulo pi, VII.3: Points of Finite Order VII.5: Good and Bad Reduction 29 25 May (W) VIII.1: The Weak Mordell-Weil Theorem 30 26 May (R) VIII.3: The Descent Procedure, VIII.4: The Mordell-Weil Theorem over Q 31 27 May (F) VIII.7: Torsion Points, VIII.10: The Rank of an Elliptic Curve - 30 May (M) No class, Memorial Day * 8 Jun (W) Final project due by e-mail

Homework:

The homework assignments will be posted above. Late homework will be accepted with a penalty. Standard weekly homework assignments, counting for 65% of the grade, will be typically due on Wednesdays.

Cooperation on homework is permitted (and encouraged), but if you work together, do not take any paper away with you--in other words, you can share your thoughts (say on a blackboard), but you have to walk away with only your understanding. In particular, write the solution up on your own. Please write on your assignment the names of any other collaborators you worked with.

Certain problems will be computational in nature and can be solved using a computer algebra package; please print out and attach complete code and output.

Plagiarism, collusion, or other violations of the Academic Honor Principle, after consultation, will be referred to the The Committee on Standards.

Final project:

A final research project will be assigned in place of a final exam. You may work individually or in groups. Choose a chapter or research article on the topic of elliptic curves, write an article summarizing (an interesting part of) its contents, pose a research question naturally arising in this work, and then try to answer it. The approximate length should be 3-20 pages per person, and the intended audience is your peers. Depending on your choice of topic, you may get quite far into this sequence or you may have to stop after the summary itself.