Math 115: Elliptic Curves
Spring 2016
Course Info:
- Lectures: Monday, Wednesday, Friday, block 10 (10:00 - 11:05 a.m.)
- x-period: Thursday, 12:00 noon - 12:50 p.m.
- Dates: 28 March 2016 - 31 May 2016
- Room: 004 Kemeny Hall
- Instructor: John Voight
- Office: 341 Kemeny Hall
- E-mail: jvoight@gmail.com
- Office hours: Monday 3:00 - 4:30 p.m., Tuesday 10:00 - 11:30 a.m., or by appointment
- Course Web Page: http://www.math.dartmouth.edu/~m115s16/
- Prerequisites: Math 101 and 111
- Required Texts: Joseph H. Silverman, The Arithmetic of Elliptic Curves, Second edition, 2009; Errata.
- Recommended Texts:
- Joseph H. Silverman and John Tate, Rational Points on Elliptic Curves, 1994.
- Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography, Second edition, 2010; Errata.
- J.S. Milne, Elliptic Curves, 2006; Errata.
- Grading: Grade will be based on weekly homework (65%) and a final project (35%).
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 |
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| 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.