Lecture 12: Kummer Theory

William Stein

Feb 8, 2010

1 Kummer Theory of Fields

Kummer theory is concerned with classifying the abelian extensions of exponent n of a field K, assuming that K contains the nth roots of unity. It’s a generalization of the correspondence between quadratic extensions of ℚ and non-square squarefree integers.

Let n be a positive integer, and let K be a field of characteristic prime to n. Let L be a separable closure of K. Let μ_{n}(L) denote the set of elements of order dividing n in L.

Lemma 1.1. μ_{n}(L) is a cyclic group of order n.

Proof. The elements of μ_{n}(L) are exactly the roots in L of the polynomial x^{n} −1. Since n is coprime to the characteristic, all roots of x^{n} − 1 are in L, so μ_{n}(L) has order at least n. But K is a field, so x^{n} − 1 can have at most n roots, so μ_{n}(L) has order n. Any finite subgroup of the multiplicative group of a field is cyclic, so μ_{n}(L) is cyclic.

Consider the exact sequence

1 → μ_{n}(L) → L^{∗} ⟶^{x ↦ x^{n}} L^{∗} → 1

of G_{K} = Gal(L/K)-modules. The associated long exact sequence of Galois cohomology yields

1 → K^{∗}/(K^{∗})^{n} → H^{1}(K, μ_{n}(L)) → H^{1}(K, L^{∗}) → · · · .

We proved that H^{1}(K, L^{∗}) = 0, so we conclude that

K^{∗}/(K^{∗})^{n} ≅ H^{1}(K, μ_{n}(L)),

where the isomorphism is via the δ connecting homomorphism. If α ∈ L^{∗}, we obtain the corresponding element δ(α) ∈ H^{1}(K, μ_{n}(L)) by finding some β ∈ L^{∗} such that β^{n} = α; then the corresponding cocycle is σ ↦ σ(β)/β ∈ μ_{n}(L).

As a special case, consider n = 2 and K = ℚ. Then we have μ_{2}(\overline{ℚ}) = \{±1\}, on which G_{ℚ} acts trivially. Recall that H^{1}(G, A) = Hom(G, A) when G acts trivially on A. Thus

ℚ^{∗}/(ℚ^{∗})^{2} ≅ Hom(G_{ℚ}, {±1}),

where the homomorphisms are continuous. The set of squarefree integers are representative elements for the left hand side of the above isomorphism. The right hand side is the set of continuous homomorphisms φ : G_{ℚ} → {±1}. To give such a nontrivial homorphism φ is exactly the same as giving a quadratic extension of ℚ. We thus recover—in a conceptual way—the standard bijection between quadratic fields and squarefree integers ≠ 1, which is one of the basic facts one learns in a first algebraic number theory course.

We generalize the above construction as follows. Suppose μ_{n} ⊂ K, i.e., all the nth roots of unity are already in K. Then we have

K^{∗}/(K^{∗})^{n} ≅ Hom(G_{K}, ℤ/nℤ),  (1.1)

where as usual the homomorphisms are continuous. We associate to a homomorphism φ : G_{K} → ℤ/nℤ an extension L^{H} of K, where H = ker(φ), and by Galois theory, Gal(L^{H}/K) ≅ image(φ) ⊂ ℤ/nℤ. Conversely, given any Galois extension M/K with Galois group contained in ℤ/nℤ, there is an associated homorphism φ : G_{K} → Gal(M/K) ⊂ ℤ/nℤ. Define an equivalence relation ∼ on Hom(G_{K}, ℤ/nℤ) by φ ∼ ψ if ker(φ) = ker(ψ) (equivalently, φ = mψ for some integer m coprime to n). Then we have a bijection

Hom(G_{K}, ℤ/nℤ)_{/∼} ⟶^{≅} { Galois extensions M/K with Gal(M/K) ⊂ ℤ/nℤ }.

Using Equation 1.1 along with the explicit description of δ mentioned above, we thus see that the Galois extensions of K with Gal(M/K) ⊂ ℤ/nℤ are the extensions of the form K( \sqrt[n]{α}) for some α ∈ K^{∗}. An element σ ∈ Gal(M/K) acts by \sqrt[n]{α} ↦ \sqrt[n]{α}^{b} for some b, and the map Gal(M/K) ⊂ ℤ/nℤ is σ ↦ b.

The above observation is Kummer theory: There is a conceptually simple description of the exponent n abelian extensions of K, assuming that all nth roots of unity are in K. Of course, understanding K^{∗}/(K^{∗})^{n} well involves understanding the failure of unique factorization into primes, hence understanding the unit group and class group of the ring of integers of K well.

When the nth roots of unity are not in K, the situation is much more complicated, and is answered by Class Field Theory.

Remark 1.2. A concise general reference about Kummer theory of fields is Birch’s article Cyclotomic Fields and Kummer Extensions in Cassels-Frohlich. For a Galoiscohomological approach to Class Field Theory, see the whole Cassels-Frohlich book.

2 Kummer Theory for an Elliptic Curve

Let n be a positive integer, and let E be an elliptic curve over a field K of characteristic coprime to n, and let L = K^{sep}. We mimic the previous section, but for the G_{K}-module E(L) instead of L^{∗}. Consider the exact sequence

0 → E[n] → E ⟶^{[n]} E → 0.

Taking cohomology we obtain an exact sequence

0 → E(K)/nE(K) → H^{1}(K, E[n]) → H^{1}(K, E)[n] → 0.

Unlike the above situation where H^{1}(K, L^{∗}) = 0, the group H^{1}(K, E)[n] is often very large, e.g., when K is a number field, this group is always infinite.

In Kummer theory, we obtained a nice result under the hypothesis that μ_{n} ⊂ K. The analogous hypothesis in the context of elliptic curves is that every element of E[n] is defined over K, in which case

H^{1}(K, E[n]) ≈ Hom(G_{K}, (ℤ/nℤ)^{2}),

where we have used that E[n](L) ≈ (ℤ/nℤ)^{2}, which is a standard fact about elliptic curves, and as usual all homomorphisms are continuous. Another consequence of our hypothesis that E[n](K) = E[n] is that μ_{n} ⊂ K; this later fact can be proved using the Weil pairing, which is a nondegenerate G_{K}-invariant map

E[n] ⊗ E[n] → μ_{n}.

As above, we can interpret the elements φ ∈ Hom(G_{K}, (ℤ/nℤ)^{2}) (modulo an equivalence relation) as corresponding to abelian extensions M of K such that Gal(M/K) ⊂ (ℤ/nℤ)^{2}. Moreover, we have upon fixing a choice of basis for E[n], an exact sequence

0 → E(K)/nE(K) → Hom(G_{K}, (ℤ/nℤ)^{2}) → H^{1}(K, E)[n] → 0,

or, using Kummer theory from the previous section,

0 → E(K)/nE(K) → (K^{∗}/(K^{∗})^{n})^{2} → H^{1}(K, E)[n] → 0.

Another standard fact about elliptic curves—the (weak) Mordell-Weil theorem—is that when K is a number field, then E(K)/nE(K) is finite. Thus when E[n](K) = E[n] , we have a fairly explicit description of H^{1}(K, E)[n] in terms of K^{∗} and E(K). This idea is one of the foundations for using descent to compute Mordell-Weil groups of elliptic curves.

If we restrict to classes whose restriction everywhere locally is 0 we obtain the sequence

0 → E(K)/nE(K) → Sel^{(n)}(E/K) → � (E/K)[n] → 0.

Here

Sel^{(n)}(E/K) = ker (H^{1}(K, E[n]) → ⨁_{all v} H^{1}(K_{v}, E)) ,

and

� (E/K) = ker (H^{1}(K, E) → ⨁_{all v} H^{1}(K_{v}, E)) .

When K is a number field, it is possible to describe Sel^{(n)}(E/K) so explicitly as a subgroup of (K^{∗}/(K^{∗})^{n})^{2} that one can prove that Sel^{(n)}(E/K) is computable.

Theorem 2.1. Given any elliptic curve E over any number field K, and any integer n, the group Sel^{(n)}(E/K) defined above is computable.

It is a major open problem to show that E(K) is computable. A positive solution would follow from the following conjecture:

Conjecture 2.2 (Shafarevich-Tate). The group � (E/K) is finite.

Conjecture 2.2 is extremely deep; for example, it is a very deep (hundreds of pages!) theorem when E/ℚ has “analytic rank” 0 or 1, and is not known for even a single elliptic curve defined over ℚ with analytic rank ≥ 2.

Example 2.3. Consider an elliptic curve E over ℚ of the form y^{2} = x(x − a)(x + b), so that all the 2-torsion of E is ℚ-rational. As above, we obtain an exact sequence

0 → E(ℚ)/2E(ℚ) → ((ℚ^{∗})/(ℚ^{∗})^{2})^{2} → H^{1}(ℚ, E)[2] → 0.

From this diagram and the fact that E(ℚ)/2E(ℚ) is finite, we see that H^{1}(ℚ, E)[2] is infinite. Moreover, given any pair (α, β) of nonzero rational numbers, we can write down an explicit Galois cohomology class in H^{1}(ℚ, E)[2], and given any rational point P ∈ E(ℚ) we obtain a pair of rationals in ((ℚ^{∗})/(ℚ^{∗})^{2})^{2}.