Mock Qualifying Exam

Elliptic Curves

Our curve is

E : y^{2} = x^{3} + 4x.

1. Draw a rough sketch of E.

2. Find Δ: Ans. 2^{12} Hints: The discriminant of y^{2} = x^{3} + ax + b is −16(4a^{3} + 27b^{2}).

3. Is the Weierstrass equation minimal? Ans. Yes, since any better change would make Δ = 1 which is impossible. Hints: By Silverman III.1, any change of variables changes Δ by a 12-th power. There are no elliptic curves over Q with everywhere good reduction (Silverman, Exercise 8.15, page 239.)

4. Find j: Ans. 1728 Hint:

j = \frac{2^{8} · 3^{3} · a^{3}}{4a^{3} + 27b^{2}}

5. (a) Find ω = ω_{E}: Ans. \frac{dx}{2y} Hint: In general it is

\frac{dx}{2y + a_{1}x + a_{3}}

(b) Compute explicitly the order of ω at the point P = (0, 0). Ans. 0 Hint: y^{2} = x^{3} + 4x implies ω = \frac{1}{3x^{2}+4}dy. A uniformizer at P is y.

6. Compute the ring End(E/C). Ans. Z[i].

Hint: Observe that x ↦ −x, y ↦ iy is an automorphism of order 4 to get Z[i] ⊂ End(E/C). The endormorphism ring is an order in Q(i).

7. Give a lattice Λ such that E_{C} ≅ E/Λ.

8. Let φ = 2 + i ∈ End(E/C).

(a) What is the degree of φ? Ans. 5.

(b) What is the isogeny dual to φ? Ans. 2 − i.

9. Compute the order of the group Ẽ(F_{3}). Ans. 4.

10. Compute the group E(Q)_{tor}. Ans. Cyclic of order 4, generated by (2, 4). Hint. The previous problem gives an upper bound. Compute the order of (2, 4) to get a lower bound.

11. Let E/Q be an elliptic curve and p > 2 be an odd prime. Show that E(Q) doesn’t contain a subgroup isomorphic to Z/p × Z/p. Hint: Algebraic: Let p be an odd prime and K = Q(E[p]) the field obtained by adjoining the coordinates of p-torsion points on E to Q. Use the Weil pairing to show that K contains Q(ζ_{p}). Hint: Geometric: Use the picture of E(R) and knowledge of finite subgroups of a S^{1}.

12. Let E/Q be an elliptic curve. Define � (E/Q). Sketch an argument that for any positive integer n, the group � [n] is finite. Hint: Follows from the Kummer descent sequence and finiteness of the Selmer group. Finiteness of the Selmer group is one of the steps in the proof of the Weak Mordell-Weil theorem.

Algebraic Geometry

1. Give a rigorous definition of a “scheme.”

Hint: A locally ringed space locally isomorphic to an affine scheme. Etc.

2. Give an example of a scheme which is a not a variety, and draw a picture. Hint: k[ε]/ε^{2}.

3. (a) Let X/C be a nonsingular projective algebraic variety. Define the geometrix genus of X. Ans: g = dim H^{0}(X, ω_{X}) = dim H^{1}(X, O_{X}).

(b) Is the surface

S : x^{4} + y^{4} + z^{4} + w^{4} = 0

nonsingular over the complex numbers?

(c) Compute the genus of S. Ans. g = 1. S is called a K3 surface.