Homework Assignment 9 Due Wednesday December 11

William Stein

Math 124  Harvard University  Fall 2002

This is the last homework assignment. Your projects are due on December 13 in print in class. You will presents your projects

on December 16 and 18, and your presentations will (hopefully) be videotaped. Your presentations will not affect your project grade. Incidentally, if you’re excited about your project and feel it could be developed in more depth, consider doing a 91r (Supervised Reading and Research) with me next semester.

1. (1 point) Let S_{2}(Γ_{0}(N)) denote the set of cuspidal modular forms of level N. Prove that S_{2}(Γ_{0}(N)) forms a ℂ-vector space under addition.

2. (3 points) Suppose y^{2} = x^{3} + ax + b with a, b ∈ ℚ defines an elliptic curve. Show that there is another equation Y^{2} = X^{3} + AX + B with A, B ∈ ℤ whose solutions are in bijection with the solutions to y^{2} = x^{3} + ax + b. (Hint: Multiply both sides of y^{2} = x^{3} + ax + b by a power of a common denominator, and “absorb” powers into x and y.)

3. (a) (3 points) Use the modularity and dimension theorems from the notes to deduce that there is no elliptic curve y^{2} = x^{3} + ax + b (with a, b ∈ ℤ) that has discriminant ±16.

(b) (2 points) The point (12, 36) lies on the elliptic curve y^{2} = x^{3} − 432. Use this fact and elementary algebra to find a rational solution (a, b) to 4a^{3}+27b^{2} = −1, and hence exhibit an elliptic curve over ℚ with discriminant 16.

4. (5 points) One can prove that the function

f = q ∏_{n=1}^{∞}(1 − q^{n})^{2}(1 − q^{11n})^{2} = ∑_{n=1}^{∞} a_{n}q^{n}

spans S_{2}(Γ_{0}(11)), and that the following three matrices generate the subgroup Γ_{0}(11) of SL_{2}(ℤ):

S = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix})  T = (\begin{matrix}3 & −2 \\ 11 & −7 \end{matrix})  U = (\begin{matrix}4 & −3 \\ 11 & −8 \end{matrix}) .

Using the above product expression for f, compute f to some large precision then give numerical evidence that f(z) satisfies the defining equation for an element of S_{2}(Γ_{0}(11)).

5. (2 points) Show that if Fermat’s last theorem is true for prime exponents, then it is true for all exponents.

6. (3 points) Let R be a ring. Say that Fermat’s last theorem is false in R if there exists x, y, z ∈ R and n ∈ ℤ with n ≥ 3 such that x^{n} + y^{n} = z^{n} and xyz ≠ 0. For which prime numbers p is Fermat’s last theorem false in the ring ℤ/p?