Pries: 605C Elliptic Curves 2011. Homework 3.
Assignment: Due Friday 4/8.
Galois representations
1. Let E be the elliptic curve y 2 = x3 + x.
(i) Find a basis {Q1 , Q2 } for E[4] such that 2Q1 = (0, 0) and 2Q2 = (i, 0).
(ii) Find a generating set σ, τ for G4 = Gal(K4 : Q).
(ii) Compute the Galois representation ρ4 : G4 → GL2 (Z/4).
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2. Let E be the elliptic curve y 2 = x3 + x and let β be the 4th root of (8 3 − 12)/9. The
3-torsion points of E are listed on page 191 of Silverman-Tate; they are defined over
K3 = Q(β, i).
(i) Show that the Galois group of K3 over Q(i) is an abelian group of order 8.
(ii) Show that the Galois group G3 of K3 over Q is a non-abelian group of order 16.
(iii) Find a basis {Q1 , Q2 } for E[3].
(iv) Compute the representation ρ3 : G3 → GL2 (Z/3).
3. Let E be√the elliptic curve y 2 = x3 + 1 (which has complex multiplication). Let
Kn = Q( −3)(E[n]).
(i) Prove that Kn is a Galois extension of Q.
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(ii) Prove that the Galois group of Kn over Q( −3) is abelian.
4. Let E be an elliptic curve defined over K. Let m and n be relatively prime integers.
Let Kn = K(E[n]) be the field obtained by adjoining the coordinates of all n-torsion
points.
(i) Prove that the compositum Km Kn of Km and Kn is contained in Knm .
(ii) Prove that Kn Km = Knm . Hint: show that every mn-torsion point is a sum of an
n-torsion point and an m-torsion point. If σ is an automorphism of Knm which
fixes Kn Km show that σ is the identity.
5. Let E be the elliptic curve y 2 = x3 + x and let Kn0 = Q(i)(E[n]). We proved that Kn0
is Galois over Q, let G0n be its Galois group. We proved also that the Galois group Hn0
of Kn0 over Q(i) is abelian.
(i) Let τ denote complex conjugation. Prove that every σ ∈ G0n can be written
uniquely as σ = st for some s ∈ Hn0 and t ∈ {id, τ }.
(ii) Prove that sτ = τ s−1 for all s ∈ Hn0 .
(iii) Prove that G0n is abelian if and only if every s ∈ Hn0 has order dividing 2.