Course 428

Elliptic Curves II

Dr Timothy Murphy

Exam Hall  Friday, 7 April 2000  9:00–11:00

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.

What happens if p is bad?

Determine the good primes for the elliptic curve

y^{2} + 2xy = x^{3} − 1.

2. If the elliptic curve

y^{2} = x^{3} + ax^{2} + bx + c

has good reduction at p, show that the torsion subgroup of E(ℚ) is mapped injectively into E(𝔽_{p}).

Let E(ℚ) be the curve

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

Determine the group E(𝔽_{p}) for two good primes p, and hence or otherwise determine the torsion group of E.

3. Prove that a finite abelian group A is a direct sum of cyclic subgroups of prime-power order; and show that the set of prime-powers 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?

Define the Weierstrass elliptic function φ(z) associated to Λ, and show that φ(z) is indeed elliptic with respect to Λ.

Show that every even elliptic function f(z) with respect to Λ is a rational function of φ(z), ie

f(z) = \frac{P (φ(z))}{Q(φ(z))}

where P (w), Q(w) are polynomials.

5. Show that the group SL(2, ℤ) is generated by

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

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 either for the case n = 3 or for the case n = 4.