**
Number Theory – Mathematics 2540
Brown University – Spring, 2017
Professor Joseph Silverman
**

Topic | We will study elliptic curves, primarily from a number theoretic viewpoint. |
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Text |
I'll be covering material from:The Arithmetic of Elliptic Curves
,
Joseph H. Silverman,
Springer-Verlag,
ISBN: 978-0-387-09493-9 – 2nd edition, © 2009.
($59.95 hardcover, $47.36 electronic)If time permits, we'll do some additional topics from the second volume. |

Office | Mathematics Department, Kassar House, Room 202 |

Phone | 863-1124 |

jhs@math.brown.edu | |

Web Site | www.math.brown.edu/~jhs/MA0254/MA0254HomePage.html |

Office Hours |
By appointment, send me an email and I'll find a time we can
meet. I'm generally on campus on MWF, seldom on TTh.) |

Course Time | MWF 10:00–10:50am (C hour) |

Course Location | Kassar 105 |

Homework | Homework assignments are posted below. |

Note on Using Computers in Math 254 | I tend to use a computer program called PARI-GP to do number theory and elliptic curve calculations. The good news about PARI is that it is free and very fast and powerful at doing number theoretic computations. The bad news is that it's not tremendouly user friendly. You can download PARI by clicking here. Another way to use PARI to do short calculations is to use the SAGE web site. You'll need to create a (free) account. Then you'll be able to type one or more PARI commands and type Shift-Return to perform the computation. Or you can use SAGE itself, which has a large collection of algorithms for working with number fields, local fields, and elliptic curves. |

Week | Chapter | Sections | Topic | HW to turn in / Due date | Problems to look at | |
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1 | Jan 25–27 | I | §§1.1–1.3 | Algebraic Varieties | # 1.1(b), 1.4, 1.6. 1.7, 1.8, 1.10 / Mon Jan 30 | # 1.2, 1.3, 1.11, 1.12 |

II | §§2.1–2.2 | Algebraic Curves | — | — | ||

2 | Jan 30–Feb 3 | II | §§2.3–2.5 | Algebraic Curves | # 2.3, 2.4, 2.8 / Mon Feb 6 | # 2.1, 2.6, 2.7, 2.10, 2.11, 2.14 |

III | §3.3 | Geometry of Elliptic Curves | — | — | ||

3 | Feb 6–Feb 10 | III | §§3.1, 3.2 | Geometry of Elliptic Curves | # 3.4, 3.5, 3.6(a,b), 3.8 / Weds Feb 22 | # 3.3, 3.7, 3.9, 3.10 |

4 | Feb 13 | VI | §6.1, §6.2 (start) | Elliptic Curves over C Lecture by Minsik Han |
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Feb 15 | VI | §6.2 (finish), §6.3 | Elliptic Curves over C Lecture by Thomas Silverman |
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Feb 17 | VI | §6.4, §6.5 | Elliptic Curves over C Lecture by Seoyoung Kim |
# 6.1, 6.2, 6.3(a,b), 6.6 / Weds Feb 22 | ||

5 | Feb 22–Feb 24 | III | §§3.4, 3.5 | Geometry of Elliptic Curves | ||

6 | Feb 27–Mar 3 | III | §§3.6, 3.7, 3.8 | Geometry of Elliptic Curves | # 3.12, 3.14, 3.15, 3.20, 3.24 / Mon Mar 6 | # 3.16, 3.23 |

7 | Mar 6–Mar 10 | V | §§5.1, 5.2 | Elliptic Curves over Finite Fields | # 5.1, 5.3, 5.4(a), 5.12, 5.13 / Mon Mar 13 | # 5.2, 5.14 |

8 | Mar 13–Mar 17 | VI | §§6.1–6.4 | Elliptic Curves over Local Fields | # 7.2, 7.5(c), 7.7(a,b), 7.8, 7.15(a,b) / Mon Mar 20 | |

9 | Mar 20–Mar 24 | VIII | §§8.1–8.3, 8.5 | Elliptic Curves over Global Fields | ||

10 | Apr 3–Apr 7 | VIII | §§8.5, 8.6, 8.9 | Elliptic Curves over Global Fields | # 8.1, 8.2, 8.6, 8.7(b), 8.8(c,d) / Mon Apr 10 | |

11 | Apr 10–Apr 14 | VIII, C, X | §§ 8.9, C.16, 10.1 | Elliptic Curves over Global Fields | # 8.10, 8.13(a,b), 8.17, 8.19(a,c) / Mon Apr 17 | |

11 | Apr 17–Apr 21 | X, B | §§ 10.4, 10.5, B.1, B.2 | Computing the Mordell-Weil Group | # 10.13, 10.16, 10.19, 10.20 / Mon Apr 24 | |

12 | Apr 24–Apr 28 | ATAEC^{*} I | §§ 1.1, 1.2 | Modular Curves Forms | ||

13 | May 1–May 5 & May 8 | ATAEC^{*} I | §§ 1.3, 1.9, 1.10, 1.11 | Modular Curves and Forms | ATAEC # 1.2, 1.5, 1.6, 1.7, 1.19, 1.20, 1.24(a), 1.25(a) / Mon May 8 | 1.27, 1.28 |

**Course Goals**:
To learn the fundamentals of the arithmetic of elliptic curves, including
their properties over finite fields, local fields, and algebraic number fields.

**Learning Activities and Time Allocation**:
Learning activities include class attendance, frequent problem sets,
and a take-home final exam. The time to complete these activities are
(1) attending lectures, approximately 3 hours/week; (2) working on the
problem sets and the final exam, approximately 9 hours/week.

**Assessment**:
Course grades will be determined by the quantity and quality of
problem sets submitted (80% of grade) and by their grade on the takehome final exam (20% of grade).

**Expectations of Students**:
It is expected that students will attend all lectures and participate in class discussion
in an appropriate manner.
Assignments are due on the listed dates.
All students are expected to abide by Brown's academic code, which may
found here

**Tentative Syllabus**:

- Algebraic Geometry Overview
- The Geometry of Elliptic Curves
- The Formal Group of an Elliptic Curve
- Elliptic Curves over Finite Fields
- Elliptic Curves over Local Fields
- Elliptic Curves over Global Fields
- Introduction to Group Cohomology: H
^{0}and H^{1} - Computing the Mordell–Weil Group

- Integral Points on Elliptic Curves
- Elliptic Curves and Elliptic Functions over
**C** - Elliptic Modular Functions and Modular Curves
- Tate Curves
- Elliptic Surfaces and Neron Models