REVIEW SHEET FOR LECTURE 11/17
MING-LUN HSIEH
1. The method of 2-descent
For every abelian group A, we let [2] : A → A, [2](a) := 2a be the homomorphism of multiplication by 2. The following observation is the basic idea of 2-descent.
Lemma 1.1. Let ϕ : A→ B and ψ : B → A two homomorphisms between abelian groups A and B such that
ψ◦ ϕ = [2], ϕ ◦ ψ = [2].
Suppose that
B/ϕ(A) and B/ψ(A) are finite sets.
Then A/2A is a finite abelian group and
#(A/2A)≤ #(A/ψ(B)) · #(B/ϕ(A)).
Let f (x) = x3+ ax2+ bx and E : y2 = f (x) and let f′(x) := x3+ a′x2+ b′x with a′ =−2a and b′ = a2− 4b and E′ : y2 = f′(x). Then E and E′ have an obvious 2-torsion point (0, 0).
Define ϕ : E→ E′ by
ϕ(x, y) = (y2
x2,y(x2− b) x2 ).
and ψ : E′ → E by
ψ(x, y) = ( y2
4x2,y(x2− b′) 8x2 ).
Proposition 1.2. The maps ϕ and ψ are group homomorphisms. In addition, ϕ◦ ψ = [2]
and ψ◦ ϕ = [2].
Definition 1.3 (Kummer map). We define the Kummer map:
α : E(Q)/ψ(E′(Q))−→ Q×/(Q×)2
P 7→ x (mod (Q×)2) if P = [x, y, 1]̸= O, (0, 0).
(0, 0) = [0, 0, 1]7→ b, O = [0, 1, 0] 7→ 1.
We can define α′ : E′(Q)/ϕ(E(Q))→ Q×/(Q×)2 similarly.
Proposition 1.4.
(1) The Kummer maps α and α′ are injective group homomorphisms,
(2) the image α(E(Q)) is contained in the subgroup of Q×/(Q×)2 generated by prime divisors of b.
(3) E(Q)/ψ(E′(Q)) is finite.
Date: December 20, 2011.
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2 M.L. HSIEH
The following corollary is an immediate consequence of the above proposition combined with Lemma 1.1.
Corollary 1.5. E(Q)/2E(Q) is finite.
2. Computation of E(Q) via 2-descent
Let E : y2 = x3+ ax2+ bx be an elliptic curve with a rational 2-torsion point (0, 0). We have proved the Mordell-Weil Theorem for such E. As a consequence of the fundamental theorem of finitely generated abelian groups, we have the following structure theorem of E(Q).
Theorem 2.1.
E(Q)≃ ZrE ⊕ E(Q)tor for some integer rE ∈ Z≥0. The integer rE is called the Mordell-Weil rank of E.
In general, it is very difficult to compute the Mordell-Weil rank rE. We give a few example to compute rE by the method of 2-descent in some favorable cases. First, the following lemma reduces the problem to the computation of images of α and α′.
Lemma 2.2.
4· 2rE = #(α(E(Q)))· #(α′(E′(Q)).
We describe a method to calculate the image of α. For a rational point P = (x, y)∈ E(Q), we write x = m/e2 and y = n/e3 with (m, e) = (n, e) = 1. Then we have
n2 = m3+ am2e2+ bme4. Write m = dm1 and b = db1 with d = (m, b). Then
n2 = d2m1(m1m + am1e2+ b1e4).
This implies that m1 = M2 and n = dM N for some M, N ∈ Z. We conclude that (2.1) N2 = dM4+ aM2e2+ b1e4,
and α(P ) = x ≡ d (mod (Q×)2). Conversely, for a divisor d|b, if the equation (2.1) has an integral solution (M, N, e), then
(dM2/e2, dM N/e3)∈ E(Q).
In summary, we have
Proposition 2.3. Let Sa,b be the set of divisors d of b such that d̸= 1, b and
N2 = dM4+ aM2e2+ (b/d)e4 has a solution (M, N, e)∈ Z3 with M, e̸= 0.
Then α(E(Q)) and α′(E′(Q)) are the subgroups of Q×/(Q×)2 generated by the the image of {1, b}⊔
Sa,b and {1, b′}⊔
Sa′,b′ respectively.
For each d|b, define the curve Cd(Q) by Cd(Q) :={
(w, z)∈ Q2 | dw2 = d2 + adz2 + bz4} . The above proposition can be rephrased as follow:
Proposition 2.4. The groups α(E(Q)) is generated by the set of divisors d of b such that Cd(Q)̸= ∅.
REVIEW SHEET FOR LECTURE 11/17 3
Example 2.5. Determine the Mordell-Weil group E(Q) for the following elliptic curves (1) E : y2 = x3+ 2x2− 3x (E(Q) ≃ Z/4Z×Z/2Z).
(2) E : y2 = x3− 6x2+ 17x (E(Q)≃ Z×Z/2Z).
3. Exercises
Exercise 3.1. Compute the Mordell-Weil group E(Q) for the following elliptic curves E:
(1) E : y2 = x3− 48x2+ 432x.
(2) E : y2 = x3+ 9x2− x.
(3) E : y2 = x3+ 17x. (Optional)
Exercise 3.2. Let p be an odd prime such that p≡ 3 (mod 8). Let E(p) : y2 = x3− p2x.
(1) Determine the torsion subgroup E(p)(Q)tor.
(2) Determine the Mordell-Weil group E(p)(Q) by the method of 2-descent.
Department of Mathematics, National Taiwan University, Taipei, Taiwan E-mail address: [email protected]
