THE PERIOD-INDEX OBSTRUCTION FOR ELLIPTIC CURVES

CATHERINE O’NEIL

1. Introduction

Let K be a field and let E be an elliptic curve over K. Let G be the absolute Galois group of K. The elements of the group H^{1}(G, E) are in one-to-one correspondence with isomorphism classes of principal homogeneous spaces C of E. The period of such a C is its order in H^{1}(G, E), a torsion group. The index of such a C is the smallest positive integer d such that there is a K-rational line bundle of degree d on C. The period-index problem is to determine when these invariants are equal. We denote the period of C by n and the index by d; then Lang and Tate [4] that n divides d and that d divides the square of n. If K is a local field, Lichtenbaum [5] showed that d = n. If the analytic rank of an elliptic curve E is 0, Stein [11] showed that for infinitely many integers n that there exist homogeneous spaces C with n(C) = d(C) = n. If the elliptic curve E has a rational n torsion point, Lang and Tate [4] showed that there are infinitely many homogeneous spaces C with n(C) = d(C) = n. There are also examples where n ≠ d (see [4], [1], [5]). Finally, Cassels originally showed ([1]) that elements C of Sha have n(C) = d(C). In fact we give another easy proof of this below.

In this paper we will first examine the special case when E has full n torsion and relate the period-index obstruction map to the Hilbert symbol. We will then derive Hilbert symbol-like properties of this obstruction map in the general case. In particular, we show that the obstruction map is quadratic.

The author wishes to thank Jordan Ellenberg for the correct statement of Claim 4.1.

2. The Period-Index Obstruction

Let the notation be as above. If the index of a curve C is d, there is a map of C to ℙ_{K}^{d−1}, namely by taking a full set of sections of the line bundle of degree d. Assume the period n of C is prime to the characteristic of K. Define L_{E} to be the line bundle on E associated to the degree n divisor n · O_{E}. Then L_{E} gives rise to a map (again by a full set of sections) to ℙ_{K}^{n−1} which will be an embedding when n is at least 3.

Claim 2.1. The automorphism group of the diagram E → ℙ_{K}^{n−1} which acts fixedpoint free on E is E[n].

Date: April 20, 2001.

Proof. Left to the reader. □

Remark. The above claim induces a map of group schemes

r : E[n] → PGL_{n} = Aut(ℙ^{n−1}).

By general theory (see [4]) the group H^{1}(G, E[n]) parameterizes twists of the above diagram, i.e. diagrams C → S, where C is a principal homogeneous space of E and S is a twist of ℙ^{n−1}. The natural (surjective) map to H^{1}(G, E)[n] can thus be thought of as a forgetful map using the above, and we see that if C has period n there exists a diagram C → S. The period-index problem for C is thus translated to the following: is there a lift of C to a diagram C → S where S is the trivial twist? The obstruction to such a lift is given by another forgetful map (”Ob”), this time on the other side of the diagram: namely, the map H^{1}(G, E[n]) → H^{1}(G, PGL_{n}) induced by r above, where H^{1}(G, PGL_{n}) parameterizes Brauer-Severi varieties (twists of ℙ^{n−1}). Moreover, this cohomology group injects via the Δ map of nonabelian cohomology into the n torsion of the Brauer group of K. Altogether we see that the obstruction to the period-index problem lies in Br(K). Remark. When K is a global field, the above correspondence gives an easy proof that the elements of the Tate-Shafarevich group of E have trivial period-index obstruction; namely, base changing the diagram C → S to a local field gives by assumption a point on C and thus a point on S. Then S is actually projective space over that local field. Finally, Brauer-Severi varieties satisfy the Hasse Principle, so S is actually projective space over the global field. Note that this proof does not depend on the lift of an element of Sha to H^{1}(G, E[n]); therefore all elements of the Selmer group have trivial obstruction. This means we have a linear space (a subgroup) inside the kernel of a quadratic map.

3. When E[n] is rational

Now assume E[n] is totally rational. Fix a primitive nth root of unity ζ_{n} ∈ K and let e denote the level n Weil pairing on E. Choose a basis S, T for E[n] such that e(S, T) = ζ_{n}. In this case we have

H^{1}(G, E[n]) ≅ K^{∗}/K^{∗n} × K^{∗}/K^{∗n}

first by mapping to H^{1}(G, μ_{n}) × H^{1}(G, μ_{n}) by sending a cocycle ξ to the pair (e(ξ, T), e(ξ, S)), and then by identifying H^{1}(G, μ_{n}) with K^{∗}/K^{∗n} via Hilbert’s Theorem 90 (namely, a cocycle φ ∈ H^{1}(G, μ_{n}) maps to r if φ(σ) = σ(r)/r). Denote the commutator of two elements x, y as [x, y] = xyx^{−1}y^{−1}.

Claim 3.1. The Weil pairing e can be computed using the map r as follows:

e(P_{1}, P_{2}) = [M_{P_{1}}, M_{P_{2}}],

where M_{P_{1}} and M_{P_{2}} are lifts to GL_{n} of r(P_{1}) and r(P_{2}) respectively.

Proof. See Theorem 2.5 on page 6 of [6]. □

We conclude that a cocycle ξ corresponding to the pair (a, b) has

[ \overline{ξ}(σ), M_{S}] = σ(α)/α & [ \overline{ξ}(σ), M_{T}] = σ(β)/β

for all σ ∈ G, where \overline{ξ}(σ) is a lift of ξ(σ) to GL_{n}(K), and where α^{n} = a and β^{n} = b. In such a situation we have the following:

Theorem 3.1. The period-index obstruction for the diagram C → S corresponding to the the pair (a, b) ∈ K^{∗}/K^{∗n} × K^{∗}/K^{∗n} is given by the Hilbert symbol (a, b)_{Hilb} defined in [9]; symbolically, Ob (a, b) = (a, b)_{Hilb}.

Proof. Define the K-algebra A_{k,l} to be generated by elements x_{k,l}, y_{k,l} such that x_{k,l}^{n} = k, y_{k,l}^{n} = l, and x_{k,l}y_{k,l}x_{k,l}^{−1} = ζ_{n}y_{k,l}. By exercise 28 on page 139 of [2], A_{k,l} is the algebra in Br(K) corresponding to (k, l)_{Hilb}. An identification of such algebras with elements of H^{1}(G, PGL_{n}) is as follows (see [9]): the set of isomorphisms of A_{1,1} with A_{k,l} (actually, the scheme \underline{Isom}(A_{1,1}, A_{k,l})) is a principal homogeneous space of PGL_{n}(K). In particular, PGL_{n} is identified with \underline{Aut}(A_{1,1}), by sending x ∈ PGL_{n} to the automorphism y ↦ inn_{x}(y) = xyx^{−1} (here we identify A_{1,1} with the set of matrices M_{n}(K) and take any lift of x to GL_{n}). A given isomorphism λ (defined over \overline{K}) of A_{1,1} with A_{k,l} acts as a “basepoint” of \underline{Isom}(A_{1,1}, A_{k,l}) in the sense that any other isomorphism λ′ is the composition of λ with an element of PGL_{n}. Then the cocycle representation is as follows: σ ↦ h(σ), where λ^{σ} = λ ∘ inn_{h(σ)}. We will fix k = a^{−1}, l = b^{−1}, and a given basepoint λ. Then we will show the cocycle h which arises from A_{a^{−1},b^{−1}} is in the same class as r^{∗}(ξ). In other words, we will be showing that the period-index obstruction is given by (a^{−1}, b^{−1})_{Hilb}. Since the Hilbert symbol is bilinear this is enough. Take λ(x_{1,1}) = α x_{a^{−1},b^{−1}} and λ(y_{1,1}) = β y_{a^{−1},b^{−1}}, where α^{n} = a and β^{n} = b. Then λ^{−1}λ^{σ}(u) = inn_{h(σ)}(u) for all u ∈ A_{1,1}; when u = x_{1,1} we get λ^{−1}λ^{σ}(x_{1,1}) = σ(α)/α x_{1,1} = inn_{h(σ)}(x_{1,1}) = h(σ) x_{1,1} h(σ)^{−1}. Since x_{1,1} is invertible, we work in GL_{n}(K) : [ h(σ), x_{1,1}] = σ(α)/α. Similarly for u = y_{1,1} we get [ h(σ), y_{1,1}] = σ(β)/β.

Lemma 3.2. There exists a G ∈ GL_{n}(K) such that x_{1,1} = G^{−1}M_{S}G and y_{1,1} = G^{−1}M_{T} G where M_{S} and M_{T} are lifts of r(S) and r(T).

Proof of Lemma. This follows from the results of [6] or an application of the Skolem-Noether Theorem. □

In light of the above lemma, we can rewrite the above formulas as:

[ (Gh(σ)G^{−1}), M_{S}] = σ(α)/α & [ (Gh(σ)G^{−1}), M_{T}] = σ(β)/β.

We see that in PGL_{n}, Gh(σ)G^{−1} commutes with r(E[n]); this is enough to show it in inside r(E[n]) (a computation left to the reader). Moreover, by Claim 3.1, ξ(σ) and Gh(σ)G^{−1} pair the same way with the basis elements S and T. By the nondegeneracy of e, we conclude that ξ(σ) = Gh(σ)G^{−1}. Since G is K-rational, this is exactly saying that ξ and h represent the same cocycle class. □

Remark. The Hilbert symbol depends on the choice of nth root of unity ζ which is the Weil pairing of S and T; a different choice of basis for E[n] would also change the Hilbert symbol.

Corollary 3.3. If we restrict to the curves of the form (1, a), we get a subgroup of H^{1}(G, E[n]) all of whose elements have trivial period-index obstruction (hence there exist “addition formulas” for such curves).

Proof. trivial. □

Remarks. 1) The above theorem demonstrates the existence of an infinite number of curves C with trivial and nontrivial period-index obstruction. 2) We can start with different line bundles on E to produce different correspondences between H^{1}(G, E[n]) and {C → S}; two such line bundles will differ by a degree 0 line bundle on E which can be identified with a point of E(K). The correspondences then will be translates of each other by the image of that point under the natural map E(K) → H^{1}(G, E[n]). This “shift” will be seen below (section 5) to define the Tate pairing in the case where K is a local field.

A non-trivial characteristic of the Hilbert symbol is that (t, 1−t)_{Hilb} = 1 whenever t (t − 1) ≠ 0. For any E with full n-torsion as above then we have a ℙ_{K}^{1} of examples of homogeneous spaces of E embeddable in ℙ^{n−1}. In other words we have a sampling surface, isomorphic to ℙ^{1} × X(n), of such curves.

Claim 3.2. For n = 3 the family C_{(t,1−t)} is given by

(3 t (1 − ζ_{3}) + λ (1 − t)) (t^{2}X^{3} + tY^{3} + Z^{3}) + 9 t (2ζ_{3} + 1) (tX^{2}Z + tY^{2}X + Z^{2}Y) +

9 t (ζ^{2} + 1) (tX^{2}Y + Y^{2}Z + Z^{2}X) + 3 t (6 t (1 − ζ_{3}) − λ (1 − t)) XY Z = 0,

where the Jacobian X(3) is given by E : X^{3} + Y^{3} + Z^{3} + λXY Z = 0.

Proof. This is a calculation using Theorem 3.1 of [7].

4. The General Case

Let G be a group. Given a central extension of G modules:

0 ⟶ A ⟶ B ⟶ C ⟶ 0,

where C is abelian, we have two maps from H^{1}(G, C) → H^{2}(G, A). The first is the Δ map of nonabelian cohomology (see page 123 of [9]) and the second, Γ, is induced from the natural pairing of elements of C given by < c_{1}, c_{2} >= [\overline{c_{1}}, \overline{c_{2}}], where \overline{c_{i}} is a lift of c_{i} to B. Namely, Γ is given by:

Γ : H^{1}(G, C) ⟶^{diag} H^{1}(G, C) × H^{1}(G, C) ⟶^{∪} H^{2}(G, C ⊗ C) ⟶^{<,>^{∗}} H^{2}(G, A).

Let B_{Δ} be the bilinear form associated to Δ, namely, for ξ, η ∈ H^{1}(G, C),

B_{Δ}(ξ, η) = Δ(ξ + η) − Δ(ξ) − Δ(η).

Claim 4.1. B_{Δ} =<, >^{∗} ∘ ∪ .

Proof. We work on the level of cocycles, where we may assume that for all σ ∈ G, \overline{ξ + η} (σ) = \overline{ξ} (σ) \overline{η} (σ):

B_{Δ}(ξ, η)(σ, τ) = Δ(ξ + η)(σ, τ) − Δ(ξ)(σ, τ) − Δ(η)(σ, τ) =

[\overline{ξ + η} (σ) \overline{ξ + η} (τ)^{σ} \overline{ξ + η} (στ)^{−1}][\overline{ξ} (σ) \overline{ξ} (τ)^{σ} \overline{ξ} (στ)^{−1}]^{−1}[\overline{η} (σ) \overline{η} (τ)^{σ} \overline{η} (στ)^{−1}]^{−1}. This computation takes place in B, but the elements in braces are in its center A, so we may move them at will to get:

\overline{ξ + η} (σ) \overline{ξ} (τ)^{σ} \overline{η} (τ)^{σ} \overline{η} (στ)^{−1} \overline{ξ} (στ)^{−1}\overline{ξ} (στ) \overline{η} (στ) \overline{η} (τ)^{σ −1} \overline{η} (σ)^{−1} \overline{ξ} (τ)^{σ −1} \overline{ξ} (σ)^{−1},

which simplifies to

\overline{ξ} (σ) \overline{η} (σ) \overline{ξ} (τ)^{σ} \overline{η} (σ)^{−1}\overline{ξ} (τ)^{σ −1} \overline{ξ} (σ)^{−1};

we can cancel the two ends because the whole thing is a central element in B to get

B_{Δ}(ξ, η)(σ, τ) = [ \overline{η} (σ), \overline{ξ} (τ)^{σ}] =<, >^{∗} ∘ ∪ (η, ξ)(σ, τ).

Since B_{Δ}(ξ, η) = B_{Δ}(η, ξ) we are done.□

In our situation we have

0 ⟶ μ_{n}(\overline{K}) ⟶ H ⟶ E[n](\overline{K}) ⟶ 0,

where H is the Heisenberg group of dimension n (for more about the Heisenberg group, see page 11 of [3]). Here <, > is the Weil pairing e and Δ is none other than the period-index obstruction map Ob.

Claim 4.2. Denote by ξ the cocycle (class) representing the diagram C → S in H^{1}(G, E[n]). Let a be an integer. Then Ob ( a ξ) = a^{2} Ob (ξ).

Proof. We will use an alternative definition of the Ob map; see the definition of ψ_{1} on page 1212 of [5]. Namely, a diagram C → S corresponds to a pair (C, D) where C is the same genus one curve and where D is a degree n \overline{K}-rational divisor on C whose class is G-invariant, in other words, for all σ ∈ G, D ∼ D^{σ}.^{1} For every σ, fix a function f_{σ} ∈ K(C) such that div(f_{σ}) = D − D^{σ}. Then the cocycle in Br(K) representing (C, D) is

1Over \overline{K}, the class of D gives rise to the line bundle L(D) which in turn induces the diagram C\overline{K} → ℙn−1 ; \overline{K}. Since [D] is G-invariant, this diagram descends to K as C → S; S will be isomorphic to ℙ^{n−1} over K exactly when [D] is representable by a K-rational divisor.

Ob(ξ)(σ, τ) = \frac{f_{σ}f_{τ}^{σ}}{f_{στ}} .

There is a natural map φ_{a} from C to a C. One way to visualize φ_{a} is to identify C with \underline{P ic}^{1}(C) and a C with \underline{P ic}^{a}(C) (via the Riemann-Roch Theorem) and then φ_{a} is given by sending a geometric point x to the point ax. Then φ_{a} is of degree a^{2}, since over \overline{K} these schemes are elliptic curves and the map is essentially multiplication by a. Moreover, φ_{a} induces “multiplication by a” on the level of cocycles, since (a ξ) (σ) = a(x_{0}^{σ} − x_{0}) = ax_{0}^{σ} − ax_{0} = φ(x_{0})^{σ} − φ(x_{0}) = φ^{∗}(ξ)(σ). Finally, φ_{a} induces an injection of function fields K(aC) ⊂ K(C), of relative degree a^{2}. The norm map N goes the other way, and brings the functions f_{σ} to functions on aC which have the property that div(N(f_{σ})) = a(D^{σ} − D). Then we have Ob(aξ) is represented by the cocycle

Ob(aξ)(σ, τ) = \frac{N(f_{σ})N(f_{τ}^{σ})}{N(f_{στ})} = N (\frac{f_{σ}f_{τ}^{σ}}{f_{στ}} ) .

Since \frac{f_{σ}f_{τ}^{σ}}{f_{στ}} ∈ K, we have

Ob(aξ)(σ, τ) = (\frac{f_{σ}f_{τ}^{σ}}{f_{στ}} )^{deg(K(aC)/K(C))} = a^{2}Ob(ξ)(σ, τ).□

Corollary 4.1. 2 Ob = Γ.

Proof. By Claim 4.2, B_{Ob}(ξ, ξ) = Ob(2ξ) − 2Ob(ξ) = 2Ob(ξ). By Claim 4.1, B_{Ob}(ξ, ξ) = e^{∗}(ξ ∪ ξ) = Γ(ξ). □

Remark. In light of the representation we now have for Ob in terms of the cup product, the fact that it specializes to the Hilbert symbol is not surprising; see [9], Proposition 5, p. 207 and [12] for the known relationship between the Hilbert symbol and the cup product.

Corollary 4.2. Ob is quadratic as a function on the ℤ-module H^{1}(G, E[n]).

Proof. By Definition 1 on page 27 of [8], Ob is quadratic if Ob ( a ξ) = a^{2} Ob (ξ) for a ∈ ℤ and for ξ ∈ H^{1}(G, E[n]) and if the associated function B_{Ob} defined as above is bilinear. These two facts follow from Claims 4.1 and 4.2. □

5. Relation to the Tate Pairing

Let K be local. Then we have a perfect pairing, called the Tate pairing, as follows:

<, >_{T}: E(K)/nE(K) × H^{1}(G, E)[n] → Br(K).

In [5], it was shown that we can represent the Tate pairing in terms of Ob as follows: to compute < P, C >_{T} , choose any lift C̃ of C to the group H^{1}(G, E[n]) and identify P with its image in that group via the δ map of cohomology. Then < P, C >_{T} = Ob(C̃ + P) − Ob(C̃). Moreover, since Ob vanishes on the Selmer group, and in particular Ob(P) = 1, we can rewrite the above as < P, C >_{T} = B_{Ob}(P, C̃), where B_{Ob} is the bilinear form associated to Ob. In this notation the proof of the equality of the period and index over a local field can reproduced from [5] as follows: The period of C is d ⇔ Ob(C̃)^{d} = 1 for all lifts C̃ of C ⇒ B_{Ob}(P, C̃)^{d} = 1 ∀P ⇔ B_{Ob}(P, dC̃) = 1 ∀P ⇔< P, dC >_{T} = 1 ∀P ⇔ dC = 0 ⇔ n | d ⇒ d = n.

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