Introduction to Number Theory, Math 104, Spring quarter 2012:

Course material: We plan to study elliptic curves in number theory. We use the book `A friendly introduction to number theory' by Joseph Silverman, at least for the first part of the course. I will use the second edition, but other editions should work as well. This is a very good book, but, unfortunately, also rather expensive. I put it on reserve in the library. So if you attend the course, you should be able to get by without having to buy it.

We also use material from the book Rational Points on Elliptic Curves, by Joseph Silverman and John Tate. Material about factorization algorithms and RSA schemes can also be found in the book An Introduction To Mathematical Cryptography / [Electronic Resource] / By Jeffrey Hoffstein, Jill Pipher, J.H. Silverman, in particular chapter 3. You can download any chapter of this book onto your laptop, or print it out.

Office hours: MW3-4 and by appt. (just talk to me after class, or send me an email)

Office: APM 5256, tel. 534-2734, email:

Teaching assistant: ,

Computation of grade: homework: 25%, midterm 25% final: 50%

Dates of exams:

midterm: Friday, May 11 in class

final: see course listings

Homework assignments: You can turn in homework in class, at my office (APM 5256, please slide under door if I am not in) or at my mailbox in the APM mailroom, APM 7132. Deadline 5pm on Tuesdays. The numbers for the exercise refer to the second edition of Silverman's "Friendly Introduction".

for 4/10: Ch 40(Cubic curves and elliptic curves): 1(a)(c), 2, 6 and one additional problem:

As mentioned in class, one can define a formal addition for rational solutions of an elliptic curve E, where for two rational points P_1 and P_2 on an elliptic curve, the point P_1+P_2 is defined to be the reflection at the x-axis of the third intersection of E with the line through P_1 and P_2. We add a formal point \infty to E. A line between a point P and \infty is defined to be the line through P perpendicular to the x-axis. Show: \infty + P=P for any point P on E, i.e. \infty is the zero element for our addition. What is the inverse of a point P with respect to our addition? (Recall that Q is the inverse of P if P + Q = \infty).

for 4/17: Chapter 41 (Elliptic curves with few rational points): 41.2(b),(c) and: part (c) shows that we have a group of torsion points (together with \infty) of order 4. Is that group isomorphic to Z/4, or to Z/2 x Z/2? (please ask me if you are not familiar with the concept of isomorphism).

Find all torsion points of y^2=x^3+4x.

Ex. 41.4

for 4/24: click here for third homework assignment

for 5/1: Ch. 42 (points on elliptic curves mod p): 42.3, 42.5 (see right lower corner of hand-out; for (e), the primes are

p=541, (A,B)= (46,4), (29,21), (17,25)

p=2029, (A,B)= (79, 25), (77, 27), (2,52)

p=8623, (A,B)=(173,39), (145,67), (28,106)

Additional Problem: Let t be the number of torsion points for the elliptic curve E: y^2 = x^3 + c.

(a) Prove that (t+1) must be a divisor of 6 if c is NOT divisible by 5.

(b) Prove the same statement for all c with 0 < c < 100, except, possibly c= 55 and c = 85.

for 5/22: See at problems at the end of the following notes: click here for course notes and homework

for 5/30:(you can wait with turning in the homework until Wednesday. I will be around on Tuesday afternoon, but will go to a talk 2-3pm. Leave a note at my office or email me if you can not find me right away) See at problems at the end of the following notes: click here for course notes and homework 6

for 6/5: Please do Problems 44.1 and 44.2(a),(b) in the Friendly Introduction. Hint for 44.1(c): Try to relate c_pc_{p^r}-c_{p^{r+1}} with some of the other coefficients.

for midterm: click here for practice problems

Remarks for Final: The material for the final will go over the whole term. You can expect most of the problems to be similar to homework problems, or problems posted here. Some indications about what type of problems can be expected will be given in the last lecture on Friday. We plan to post some useful material on this web page in the next few days, so check it regularly. As usual, you will be allowed to use one cheat sheet.

click here for solutions of midterm problems

Here are some notes about material which does not appear explicitly in the Friendly Introduction. I plan to post some more information late this weekend or early next week.

course notes

A few more remarks: Look at notes, the homework problems, review problems for midterm and the midterm itself. I can not ask very complicated questions about lattices and modular forms. But make sure you know the following:

For lattices: The main result for abstract lattices is that if w_1, w_2 is a Z-basis for a lattice, then aw_1+bw_2, cw_1+dw_2 is a basis for L if and only if ad-bc= plusminus 1, and a,b,c,d are integers.

For modular forms, you should know the basic definitions (see the notes), the meaning of the coefficients c_n in the power series expansion (at least for n=p prime), and very basic knowledge about complex numbers.