ELL Defs

Question 1

State Mordell’s theorem.

Answer 1

If E is an elliptic curve defined over Q. Then the group E(Q) is a finitely generated abelian group.

Question 2

Let E be an elliptic curve over R. State (without proof) a criterion that links the discriminant ∆ to the number of 2-torsion points in E(R).

Answer 2

If ∆ > 0 then E(R)[2] has 4 elements and if ∆ < 0 it has 2 elements

Question 3

Let p be a prime number. Give the precise definition of the reduction map from P²(Q) to P²(F_p).

Answer 3

Let (X : Y : Z) be a point in P²(Q).

First we multiply with an integer scalar which is a multiple of the denominators of X, Y and Z. We obtain (X’:Y’:Z’) = (X : Y : Z) with X’, Y’, Z’ in Z.

Then we divide through by the greatest common divisor of X’, Y’ and Z’. This yields (X’’ : Y’’ : Z’’) = (X : Y : Z) with X’’, Y’’, Z’’ in Z and gcd(X’’, Y’’, Z’’) = 1.

The reduction of (X : Y : Z) is defined to be (X’’ + pZ : Y’’ + pZ : Z’’ + pZ) ∈ P²(F_p).

Question 4

Give the definition of a degenerate conic.

Answer 4

A conic is a homogeneous equation of degree 2. If this equation factors into two linear terms over some field K containing k then it is degenerate.

Question 5

Give the definition of a line defined over a field k in the projective plane P².

Answer 5

A line defined over k is given by an equation of the form L : aX+bY +cZ = 0 with constants a, b, c ∈ k of which at least one is non-zero.

Question 6

State the definition of the projective plane ℙ²_𝑘 .

Answer 6

P_²k = k³{(0, 0, 0)}/ ∼, where the equivalence relation ∼ is given by (a, b, c) ∼ (λa, λb, λc) for all λ ∈ k^×. The equivalence classes of P²_k are denoted (a : b : c).

Question 7

Let 𝐶 ∶ 𝑓 (𝑥, 𝑦) = 0 be the curve in 𝑘² given by 𝑓. State the definition of the projective closure 𝐶 of 𝐶 in ℙ²_𝑘 .

Answer 7

Let F ∈ k[x, y, z] be the homogenisation of f. That is, F is the unique homogeneous polynomial of degree equal to the degree of f, such that F(x, y, 1) = f(x, y) for all x, y. Then the projective closure C of C is the projective curve C : F = 0 in P²_k .

Question 8

State the definition of the tangent to 𝐶 at a point 𝑃 = (𝑎, 𝑏) ∈ 𝐶. State also the definition of a singular point 𝑃 on 𝐶.

Answer 8

The tangent at P = (a, b) ∈ C is given by xf_x(a, b) + yf_y(a, b) = e, where e = af_x(a, b) + bf_y(a, b). 2 We say that P ∈ C is a singular point of C if f_x(a, b) = f_y(a, b) = 0.

Question 9

State carefully the definition of an elliptic curve.

Answer 9

Let k be a field and let f ∈ k[x, y] be a cubic polynomial. The planar curve E : f = 0 is an elliptic curve if E^_ is a non-singular projective curve containing at least one point O with coefficients in k.

Question 10

Carefully state Bézout’s Theorem in the case of a curve and a line.

Answer 10

Let k be an algebraically closed field, and suppose that C and L are, respectively, a curve and a line in P²_k such that C ∩L is finite. Then the global intersection multiplicity I(C,L) is equal to the degree of the homogeneous polynomial defining C.