Pries: 605C Elliptic Curves 2011. Homework 1.
Assignment: Do 4-5 of the problems below by Friday 2/18.
Group law and endomorphisms - less theory
1. The point P = (3, 8) is on the elliptic curve y 2 = x3 − 43x + 166. Compute P , 2P , 4P ,
and 8P . Comparing 8P with P , what can you conclude about the order of P ?
2. If E : y 2 = x3 + 1, find the eight points P ∈ E(C) such that 3P = OE .
3. If E : y 2 = x3 + 1, find an element of order 6 in Aut(E) and a quadratic imaginary
ring R ⊂ End(E). Find a lattice L ⊂ C which has an automorphism of order 6.
4. Automorphisms of C∪{∞} are of the form f (z) = (az +b)/(cz +d) where a, b, c, d ∈ R.
Prove that f (z) is invertible iff |ad − bc| = 1 and that f (z) stabilizes the upper half
plane iff ad − bc > 0.
5. (Silverman III, 3.3) Suppose char(K) 6= 3 and A ∈ K ∗ . Let E ⊂ P2 be the variety of
F (X, Y, Z) = X 3 + Y 3 − AZ 3 .
(a) Show that E is smooth and has genus 1 (hint: Plücker formula).
(b) Find the point P∞ on E.
(c) Under the rubric that three points add together if and only if they are collinear:
if P = [X : Y : Z] ∈ E, show that −P = [Y : X : Z] and
[2](P ) = [−Y (X 3 + AZ 3 ), X(Y 3 + AZ 3 ), X 3 Z − Y 3 Z].
(d) Prove that E has j-invariant 0.
Group law and endomorphisms - more theory
1. A hyperelliptic cover φ : X → P1 has an equation y 2 = f (x) for some f (x) ∈ C[x]
with degree d. Where is φ branched? How does the genus of X depend on d? (Hint:
Riemann-Hurwitz).
2. Suppose X is a smooth projective curve of genus 1 and P ∈ X. This problem is
important for showing that there is a cubic equation in Weierstrauss form for X.
(a) For r ∈ N, what is dim(L(rP )) (the vector space of functions on X with no poles
except at P , where the order of the pole is at most r)? (Hint: Riemann-Roch).
(b) Prove there exist functions x ∈ K(X) and y ∈ K(X) such that x and y have poles
only at P where x (resp. y) has a pole of exact order 2 (resp. 3).
(c) Show that there is a linear relation between {1, x, y, x2 , xy, y 2 , x3 }.
3. (Silverman II, 2.6) Let C be a smooth curve of genus 1 and let P0 ∈ C. Prove that:
(a) For all P, Q ∈ C, there exists a unique R ∈ C such that P + Q ≡ R + P0 in
Pic0 (C). Let σ(P, Q) = R.
(b) The map σ : C × C → C makes C into an abelian group with identity P0 .
(c) Define κ : C → Pic0 (C) by sending P to the divisor class of P − P0 . Prove that
κ is a bijection of sets.
(d) Define a group law on C by the rule P + Q = κ−1 (κ(P ) + κ(Q)). Prove that this
is the same as the group law in part (b).