Course 428a

Elliptic Curves II

Dr Timothy Murphy

Seminar Room Friday, 14 April 2000 16:15–17:45

Attempt 5 questions. (If you attempt more, only the best 5 will be counted.) All questions carry the same number of marks.

1. What is meant by saying that an elliptic curve has good reduction at the prime p?

Sketch the proof that if p is a good prime then the map

E(ℚ) → E(𝔽_{p})

is a homomorphism.

Determine the good primes for the elliptic curve

y^{2} + y = x^{3} + x.

2. Determine the group E(𝔽_{p}) for the curve

y^{2} = x^{3} + x + 1

and the primes p = 3, 5, 7. Hence or otherwise determine the torsion group of E(ℚ).

3. Prove that a finitely-generated abelian group A can be expressed as a

direct sum

A = F ⊕ ℤ^{r},

where F is finite; and show that r depends only on A.

4. What is a lattice Λ ⊂ ℂ; and what is meant by saying that f(z) is elliptic with respect to Λ? Explain what you mean by a fundamental parallelogram Π for Λ?

Show that the numbers of poles and zeros of f(z) in Π are equal.

Show also that if the zeros are a_{1}, …, a_{r} and the poles are b_{1}, …, b_{r} then

a_{1} + ⋯ + a_{r} ≡ b_{1} + ⋯ + b_{r} mod Λ.

5. Let s, u be the elements of the modular group Γ corresponding to the

matrices

S = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) and U = (\begin{matrix}0 & −1 \\ 1 & 1 \end{matrix}) .

Show that

s^{2} = u^{3} = 1;

and prove that Γ is generated by s and u.

6. Explain how a lattice Λ ⊂ ℂ gives rise to an elliptic curve

E(ℂ) : y^{2} = x^{3} + bx + c;

and sketch the proof that every elliptic curve of this form arises in this way from a unique lattice Λ.

7. Prove Fermat’s Last Theorem in the case n = 4.