Extensions of Tori in SL(2)

Jeffrey Adams, University of Maryland

January 26, 2001

Introduction

Let { \atop SL(2, 𝔽)} be the metaplectic two-fold cover of SL(2, 𝔽), the special linear group in two variables over a local field 𝔽 of characteristic 0. The inverse image T̃ of a maximal torus T in { \atop SL(2, 𝔽)} is an abelian extension of T by ±1. We consider the question of whether this extension is trivial. We exclude the case 𝔽 = ℂ, which is trivial.

More generally suppose A is a subgroup of the circle 𝕋 containing ±1. The inclusion of ±1 in A induces a map on cohomology, and defines an extension

1 → A → T_{A} → T → 1

We say A is a splitting group for T̃ if the extension T_{A} → T splits. It is well known that 𝕋 is a splitting group. We say a splitting group A is a minimal splitting group if no proper subgroup of A is a splitting group. It is easy to see the order of a minimal splitting group is a power of 2, and hence unique, if it is finite.

Let ( , )_{𝔽} be the Hilbert symbol of 𝔽, and let μ_{n} be the n^{th} roots of unity in ℂ.

Theorem 1 The minimal splitting group A_{min} for T is given by:

(a) Suppose T ≃ 𝔽^{∗}. Then

A_{min} = \begin{cases}μ_{2} & (−1, −1)_{𝔽} = 1 \\ μ_{4} & (−1, −1)_{𝔽} = −1 \end{cases}

(b) Suppose T ≃ 𝔼^{1} for 𝔼 a quadratic extension of 𝔽. Then

A_{min} = \begin{cases}μ_{2} & (−1, −1)_{𝔽} = 1 \\ μ_{4} & (−1, −1)_{𝔽} = −1, 𝔽 non − archimedean, −1 ∉ 𝔽^{∗2} \\ μ_{8} & (−1, −1)_{𝔽} = −1, 𝔽 non − archimedean, −1 ∈ 𝔼^{∗2} \\ 𝕋 & 𝔽 = ℝ \end{cases}

Remark 2 It is well-known that (−1, −1)_{𝔽} = 1 unless 𝔽 = ℝ, ℚ_{2}, or an extension of ℚ_{2} of odd degree.

Theorem 1 is proved in sections 3,4 and 5. Here is an alternative realization of T̃. A character of T̃ is said to be genuine if it does not factor to T.

Theorem 3 Let τ(z) = z^{2} (z ∈ ℂ^{∗}). Let α̃ be a genuine character of T̃. Then α̃^{2} factors to a character α of T, and T̃ is isomorphic to the pullback of τ via α. In other words T̃ is isomorphic to the \sqrt{α}-extension of G.

From this we obtain an interpretation of the minimal splitting group of Theorem 1. Let n(T̃) be the minimal order of a genuine character of T̃. Set μ_{∞} = 𝕋.

Corollary 4 The minimal splitting group for T̃ is μ_{n(T̃)}.

For the proofs of Theorem 3 and Corollary 4 see Lemma 1.4.

We also give an explicit splitting of this extension, i.e. a function ζ : T → A_{min} whose coboundary is the cocycle defining T̃ (see §3 and Theorem 5.7).

These questions arise from the theory of the oscillator representation and dual pairs. The splitting plays a role in this context, for example see [11]. The case of 𝔽^{∗} is well known ([4], pg. 42, attributed to J. Klose), as is the existence of a 𝕋-splitting in general [2]. General results about the splitting of the metaplectic cover over subgroups are due to Kudla [7], and a splitting of the extension of an elliptic torus is found in [7], Proposition 4.8 (in the non-archimedean case it is easy to see this can be taken to be a μ_{8}-splitting). This paper grew out of an effort to simplify Kudla’s formula. In the case of a p-adic field of odd residual characteristic a formula for a μ_{2}-splitting in some cases may be deduced from [6], cf. ([4], pg. 43).

Many of the arguments, especially those of Section 1 apply to other abelian extensions of abelian groups, for example a maximal torus in the two-fold cover of Sp(2n, 𝔽). If G̃ is a non-linear n-fold cover of the 𝔽 points of an algebraic group G, then the inverse image T̃ of a maximal torus in G is typically not abelian. However similar arguments apply to the center of T̃.

Throughout 𝔽 denotes a local field of characteristic zero, and (x, y)_{𝔽} ∈ μ_{2} is the Hilbert symbol. For x ∈ 𝔽^{∗} and ψ_{𝔽} a non-trivial additive character of 𝔽, γ_{𝔽}(x, ψ_{𝔽}) ∈ μ_{4} is the Weil index. We use basic properties of the Hilbert symbol and the Weil index without further comment, see ([10], Appendix) for details. We make repeated use of the identities

(1)  γ_{𝔽}(x, ψ_{𝔽})γ_{𝔽}(y, ψ_{𝔽}) = (x, y)_{𝔽}γ_{𝔽}(xy, ψ_{𝔽})
(2)  γ_{𝔽}(x, ψ_{𝔽})^{2} = (−1, x)_{𝔽}

If 𝔼 is a quadratic extension of 𝔽 then

(3)  (x, z)_{𝔼} = (x, Nz)_{𝔽} (x ∈ 𝔽^{∗}, z ∈ 𝔼^{∗})
(4)  (x, y)_{𝔼} = 1 (x, y ∈ 𝔽^{∗})
(5)  γ_{𝔼}(x, ψ_{𝔼})γ_{𝔼}(y, ψ_{𝔼}) = γ_{𝔼}(xy, ψ_{𝔼}) (x, y ∈ 𝔽^{∗})

I would like to thank Steve Kudla and Jonathan Rosenberg for many useful discussions, and Jim Schafer for assistance with the arguments in section 1.

1 Abstract Groups

In this section we ignore the topology on T and consider it as an abstract group. We recall some standard facts from group cohomology and establish some notation. For example see [1].

Suppose G is a group, A is an abelian group, and G acts trivially on A. The equivalence classes of central extensions of G by A are parametrized by the group cohomology H^{2}(G, A). Given an extension p : H → G let s : G → H be a section, i.e. p ∘ s = 1. The cohomology class of the extension is represented by the 2-cocycle c_{s}(g, h) = s(gh)s(h)^{−1}s(g)^{−1}. When there is no danger of confusion we do not distinguish between c_{s} and its image \overline{c}_{s} in H^{2}(G, A). Any other such splitting s′ is given by s′(g) = s(g)ζ(g) for some map ζ : G → A, and then c_{s′}(g, h) = c(g, h)ζ(gh)ζ(h)^{−1}ζ(g)^{−1}. Thus c_{s′} = c_{s}dζ, and \overline{c}_{s} = \overline{c}_{s′}.

Conversely given a cocycle c we define H to be equal to G × A as a set, with multiplication (g, a)(g′, a′) = (gg, aa′c(g, g′)). The cocyle c is trivial in cohomology if and only if

(6)  c(g, h) = ζ(g)ζ(h)ζ(gh)^{−1}

for some ζ, i.e. dζ = c. We say ζ is a splitting of the cocycle. Equivalently the splitting map s(g) = (g, ζ^{−1}(g)) is a homomorphism. Any other splitting is then of the form ζ′ = ζα with α : G → A a homomorphism.

Suppose A = μ_{2}, with cocycle c, and A ⊂ μ_{ab} with b odd. If ζ : G → μ_{ab} is a splitting of c, then ζ^{b} is a μ_{a} splitting. Therefore we will restrict consideration to μ_{n} with n a power of 2.

Now suppose G is abelian. The universal coefficient theorem for group cohomology gives an exact sequence:

1 → Ext(G, A) → H^{2}(G, A) →^{ϕ} Hom(Λ^{2}G, A) → 1.

Here G and A are considered as ℤ-modules, Hom = Hom_{ℤ}, Ext = Ext_{ℤ}, and Hom(Λ^{2}G, A) consists of alternating, bilinear maps G × G → A.

If c is a 2-cocycle, representing the class \overline{c} ∈ H^{2}(G, A), then ϕ(\overline{c})(g, h) = c(g, h)c(h, g)^{−1}. In terms of the group, suppose p : H → G is the corresponding extension. For g, h ∈ G and any section s let {g, h} be the commutator s(g)s(h)s(g)^{−1}s(h)^{−1}. This is contained in A, is independent of the choice of s, and ϕ(\overline{c})(g, h) = \{g, h\}. In particular ϕ(\overline{c}) = 1 if and only if H is abelian, so Ext(G, A) ⊂ H^{2}(G, A) parametrizes the abelian extensions of G by A.

Let G^{n} = {g^{n}| g ∈ G} and _{n}G = {g ∈ G |g^{n} = 1}. The next result is presumably well known to the experts.

Lemma 1.1 For any positive integer n, inclusion ι : _{n}G ↪ G induces an isomorphism:

Ext(G, μ_{n}) ≃ Ext(_{n}G, μ_{n})

Proof. Consider the maps

G →^{α} G^{n} →^{β} G

where α(g) = g^{n} and β is inclusion. The induced map α^{∗}β^{∗} : Ext(G, μ_{n}) → Ext(G, μ_{n}) is induced by the n^{th} power map g → g^{n} on G. This is the same map as that induced by the n^{th} power map on μ_{n}, and therefore α^{∗}β^{∗} = 0.

Now the long exact cohomology sequence corresponding to 0 → G^{n} →^{β} G → G/G^{n} → 0 has final two terms Ext(G, A) →^{β^{∗}} Ext(G^{n}, A) → 0. Therefore β^{∗} is surjective, which implies α^{∗} = 0. On the other hand the short exact sequence

0 → _{n}G →^{ι} G →^{α} G^{n} → 0

gives rise to the long exact sequence

0 → Hom(G^{n}, A) → Hom(G, A) → Hom(_{n}G, A) → (7)

Ext(G^{n}, A) →^{α^{∗}} Ext(G, A) →^{ι^{∗}} Ext(_{n}G, A) → 0

Since α^{∗} = 0, ι^{∗} is an isomorphism.

Remark 1.2 In our setting _{2}T = ±1. For the μ_{2} extension T̃ to split it is necessary that it splits over ±1. Perhaps surprisingly the converse holds as well by the Lemma.

For later use we note an explicit formula for a splitting of α^{∗}β^{∗}c. We drop the assumption that H is abelian, so let p : H → G be an extension, with section s and corresponding cocycle c.

Lemma 1.3 Let

\begin{aligned}τ(g) & = s(g^{n})s(g)^{−n} \\ & = c(g, g)^{−1}c(g, g^{2})^{−1} … c(g, g^{n−1})^{−1} ∈ A. \end{aligned}

Then

(8)  c(g^{n}, h^{n}) = τ(g)τ(h)τ(gh)^{−1}{g, h}^{n(n−1)/2}

If H is abelian then dτ = α^{∗}β^{∗}c.

Note that {g, h}^{n(n−1)/2} = ±1, and is identically 1 if n is odd. Compare ([3], pg. 130) and ([5], §4). Proof. This follows from the identity

[s(g)s(h)]^{n} = s(g)^{n}s(h)^{n}{h, g}^{n(n−1)/2}.

Using s(g)s(h) = s(gh)c(g, h) and s(g)^{n} = s(g^{n})τ(g)^{−1}, the left hand side is equal to

s(gh)^{n} = s(g^{n}h^{n})τ(gh)^{−1}

The right hand side is

s(g^{n})s(h^{n})τ(g)^{−1}τ(h)^{−1}{h, g}^{n(n−1)/2} =
s(g^{n}h^{n})c(g^{n}, h^{n})τ(g)^{−1}τ(h)^{−1}{h, g}^{n(n−1)/2}

and the first assertion follows. Since α^{∗}β^{∗}c(g, h) = c(g^{n}, h^{n}), the second assertion is equivalent to

(9)  c(g^{n}, h^{n}) = τ(g)τ(h)τ(gh)^{−1}

which is (8) for H abelian.

For H an abelian extension (9) implies τ is a character when restricted to _{n}G. In terms of the exact sequence (7), c ∈ Ext(G, A), β^{∗}c ∈ Ext(G^{n}, A), α^{∗}β^{∗}c = 0, and β^{∗}c is the image of τ ∈ Hom(_{n}G, A). Thus β^{∗}c = 0 if τ extends to an element of Hom(G, A).

More generally, we try to find a splitting subgroup A of β^{∗}c, i.e. c restricted to G^{n}, together with an explicit formula. Note that (9) does not necessarily define such a splitting since the function g^{n} → τ(g) is not necessarily well defined. Let α be a character of G whose restriction to _{n}G is equal to τ^{−1}. Then ζ_{α}(g^{n}) := τ(g)α(g) is well-defined, and dζ_{α} = c. The minimal splitting subgroup for β^{∗}c is thus the minimal subgroup A of 𝕋, containing μ_{n}, such that τ restricted to _{n}G can be extended to a character of G with values in A.

Characters and the \sqrt{α} extension

For the remainder of this section let p : G̃ → G be a μ_{2} extension of a group G. We do not assume that G or G̃ is abelian. If α is a character of G, and τ(z) = z^{2} (z ∈ ℂ^{∗}) then the pullback of τ via α is a μ_{2} extension of G, and may be realized as the subgroup of G × ℂ^{∗} given by {(g, z) | α(g) = τ(z)}. Projection on the second factor is a genuine character α̃ of G̃ satisfying α̃^{2} = α ∘ p. This is sometimes denoted the \sqrt{α}-extension of G. It may or may not be the trivial extension.

We see that G̃ has a 𝕋-splitting if and only if there is a genuine character of G̃. More precisely:

Lemma 1.4 Suppose there is a genuine character α̃ of G̃. Then α̃^{2} factors to a character α of G, and G̃ is isomorphic to the \sqrt{α} extension of G. If Image(α̃) ⊂ A ⊂ 𝕋 then G_{A} ≃ G × A. The minimal splitting group for G is μ_{n(G̃)} where n(G̃) is the minimal order of a genuine character of G̃.

Conversely if G̃_{A} ≃ G × A then there is a genuine character of G̃ with values in A.

Note that there exists a genuine character α̃ of G̃ if and only if z ∉ [G̃, G̃] where z is the non-trivial element in the inverse image of 1. In particular this holds if G̃ is abelian, which proves the existence of a 𝕋-splitting (cf. [2]).

Proof. The map ϕ : g → (p(g), α̃(g)) ⊂ G × 𝕋 is an isomorphism of G̃ with the pullback of τ via α. This is a subgroup of G × A, and ϕ extends to an isomorphism of G̃_{A} with G × A. The final two assertions are immediate.

Remark 1.5 In the setting of the Lemma, suppose ζ is a 𝕋-splitting of the cocycle defining G̃ (with respect to a section s). Then α := ζ^{2} is a character of G, and G̃ is isomorphic to the \sqrt{α} cover of G.

Theorem 3 and Corollary 4 are immediate consequences of the Lemma. Theorem 1 also follows from the lemma, from a computation of n = n(T̃): n is the minimal power of 2 such that z ∉ T̃^{n}. We follow a different approach, by giving explicit formulas for the minimal splitting ζ in sections 3–5.

2 Moore cohomology and { \atop SL(2, 𝔽)}

Now suppose G and A are locally compact topological groups, A is abelian, and G acts continuously on A. In our applications A will either be μ_{n} or 𝕋, with trivial G action. C. Moore has defined cohomology groups H_{top}^{n}(G, A) using measurable cochains [8]. In the case of a totally disconnected group it is equivalent to use continuous cochains. Viewing G and A as abstract groups, there is a natural homomorphism H_{top}^{2}(G, A) → H^{2}(G, A). In general it is neither surjective nor injective.

Now let G = GL(2, 𝔽). We recall the definition of the standard cocycle

on G [6], cf. ([4], pg. 41). For g = (\begin{matrix}a & b \\ c & d \end{matrix}) let x(g) = c (resp. d) if c ≠ 0

(resp. c = 0). Then

(10)  c(g, h) = (x(g)x(gh), x(h)x(gh))_{𝔽}(det(g), x(g)x(gh))_{𝔽}.

This defines a μ_{2}–extension { \atop GL(2, 𝔽)} of GL(2, 𝔽), with distinguished section s. We write { \atop GL(2, 𝔽)} in cocycle notation as usual. The restriction to SL(2, 𝔽) is isomorphic to { \atop SL(2, 𝔽)}.

Up to conjugation GL(2, 𝔽) contains one hyperbolic torus isomorphic to 𝔽^{∗} ×𝔽^{∗}, and for each quadratic extension 𝔼 of 𝔽 one elliptic torus isomorphic to 𝔼^{∗}. The commutator of two elements z, w of 𝔼^{∗} is given by ([3], pg. 128)

(11)  \begin{aligned}\{z, w\} & = (z, \overline{w})_{𝔼} \\ & = (z, w)_{𝔼}(Nz, Nw)_{𝔽}. \end{aligned}

Here and elsewhere we suppress the map ι : 𝔼^{∗} ↪ GL(2, 𝔽) from the notation, and write {z, w} := {ι(z), ι(w)}. The commutator is trivial when restricted to 𝔼^{1} = 𝔼^{∗} ∩ SL(2, 𝔽).

3 The hyperbolic torus

Let ι : 𝔽^{∗} ↪ SL(2, 𝔽), so T = ι(𝔽^{∗}) is a hyperbolic torus. After conjugation we may assume ι(x) = diag(x, x^{−1}). We drop ι from the notation and identify x with ι(x). We prove Theorem 1 in this case, together with a formula for the minimal splitting. While these results are well known they are not easy to find in the literature, and the calculation illustrates some of the ideas in the next section.

The restriction of { \atop SL(2, 𝔽)} to T defines a μ_{2} extension T̃ of T with cocycle

c(x, y) = (x, y)_{𝔽}.

In particular (−I, ϵ)^{2} = (I, (−1, −1)_{𝔽}), so the restriction of the extension to ±I is isomorphic to ℤ/2ℤ × ℤ/2ℤ if (−1, −1)_{𝔽} = 1, or ℤ/4ℤ if (−1, −1)_{𝔽} = −1. By Lemma 1.1 the minimal splitting group for T̃, considered as an abstract group, is μ_{2} (resp. μ_{4}) if (−1, −1)_{𝔽} = 1 (resp. −1). It remains to show this splitting is measurable. We do this by computing it explicitly.

Fix ψ and write γ(x) = γ_{𝔽}(x, ψ). The key point is that properties (1) and (2) of the Weil index shows that dγ = c and γ^{4} = 1, so γ is a measurable μ_{4} splitting of c. This completes the proof in case (−1, −1)_{𝔽} = −1, so assume (−1, −1)_{𝔽} = 1. Let α be a character of 𝔽^{∗} satisfying α(x^{2}) = (−1, x). To see that such a character exists, define α restricted to 𝔽^{∗2} by this formula; it is well-defined since α((−x)^{2}) = (−1, −x)_{𝔽} = (−1, −1)_{𝔽}α(x^{2}) = α(x^{2}). Extend arbitrarily from 𝔽^{∗2} to 𝔽^{∗}. By (2) α(x)^{2}γ(x)^{2} = (−1, x)^{2} = 1. Let ζ_{α} = γα, i.e.

ζ_{α}(x) = γ(x, ψ_{𝔽})α(x).

Then dζ_{α} = dγ = c and ζ_{α} is a μ_{2}-splitting of c.

Choose representatives a_{1}, a_{2}, …, a_{m} ∈ 𝔽^{∗} of generators of 𝔽^{∗}/𝔽^{∗2} ≃ (ℤ/2ℤ)^{m}. (By [4], Lemma 0.3.2, 2^{m} = 4/|2|_{𝔽}.) Given any choice of signs ϵ_{i} we may choose α so that ζ(a_{i}) = ϵ_{i}, and α(x^{2}) = (−1, x)_{𝔽} for all x ∈ 𝔽. Then ζ extends uniquely to a splitting.

For example if −1 ∈ 𝔽^{∗2} then we may take α = 1 and ζ(x) = γ_{𝔽}(x, ψ_{𝔽}) = ±1. On the other hand suppose −1 ∉ 𝔽^{∗2} and the residual characteristic of 𝔽 is odd. We may take representatives ±1, ±ϖ for 𝔽^{∗}/𝔽^{∗2} (ϖ is a uniformizing parameter) and then choose ζ satisfying:

\begin{aligned}ζ(±x^{2}) & = (−1, x)_{𝔽} \\ ζ(±ϖx^{2}) & = ±(−1, x)_{𝔽}. \end{aligned}

4 Elliptic Tori

Let T be an elliptic torus of SL(2, 𝔽) as in §2. Thus 𝔼 is a quadratic extension of 𝔽, ι : 𝔼^{1} ↪ SL(2, 𝔽) is an embedding, and T = ι(𝔼^{1}). As in §3 we fix ι and drop it from the notation.

As in the case of the hyperbolic torus, (−I, ϵ) has order 2 or 4 depending on whether (−1, −1) = +1 or −1. By Lemma 1.1 this proves μ_{2} is a splitting group if and only if (−1, −1) = 1, so assume (−1, −1) = −1.

Let 𝔽 = ℝ, 𝔼 = ℂ. Since T = T^{2}, it is enough to find a splitting of β^{∗}c as in §2. Choose a character α of 𝔼^{1} such that α(−1) = −1, i.e. α(z) = z^{n} for n odd. It is easy to see c(−z, −z) = −c(z, z), and the discussion in §2 shows that

(12)  ζ_{α}(z^{2}) := c(z, z)α(z)

is a well defined (measurable) 𝕋-splitting. Since ζ_{α} is surjective onto 𝕋 for any α, this shows that there is no A-splitting for any proper subgroup A of 𝕋.

We now assume 𝔽 is non-archimedean. Since (−1, −1)_{𝔽} = −1, 𝔽 is an extension of ℚ_{2} of odd degree, and −1 ∉ 𝔽^{∗2}. If −1 ∉ 𝔼^{∗2} then _{4}T = {±1}. Since (−I, ζ)^{4} = (I, ±ζ^{2})^{2} = I for all ζ ∈ μ_{4}, the μ_{4}-extension splits over _{4}T.

Suppose −1 = δ^{2} (δ ∈ 𝔼^{∗}). We claim _{4}T = _{8}T = {±1, ±δ}. It is enough to show 𝔼^{∗} does not contain a primitive eighth root of unity, or equivalently δ ∉ 𝔼^{∗2}. Since 𝔽 is an extension of ℚ_{2} of odd degree, 2 ∉ 𝔽^{∗2}. But then (a + bδ)^{2} = δ implies a^{2} = ±\frac{1}{2}, which is a contradiction.

For any ζ ∈ μ_{8} we compute (δ, ζ)^{8} = (−I, ±ζ^{2})^{4} = (I, −ζ^{4})^{2} = I, which implies the extension splits over _{8}T.

Therefore μ_{4} (resp. μ_{8}) is a minimal splitting group for T if 𝔽 is nonarchimedean, (−1, −1) = −1, and −1 ∉ 𝔼^{∗2} (resp. −1 ∈ 𝔼^{∗2}). It remains to show these splittings can be chosen to be measurable. In the next section we give explicit such splittings.

Remark 4.1 We have shown the cohomology class \overline{c} ∈ H_{top}^{2}(G, A) has image 0 in H^{2}(G, A) in the given cases. An argument due to Jonathan Rosenberg shows that the map H_{top}^{2}(G, A) → H^{2}(G, A) is injective in this situation. Since we are interested in explicit formulas for the splittings in any case, we do not pursue this approach. For G perfect (not at all the case here!) the injectivity of ϕ is known ([9], Theorem 2.3).

5 Explicit splittings for elliptic tori

We continue with the notation of the previous section. The embedding ι : 𝔼^{1} ↪ SL(2, 𝔽) extends to an embedding ι : 𝔼^{∗} ↪ GL(2, 𝔽). We will make use of the non-abelian μ_{2} extension of 𝔼^{∗} obtained by restricting the extension

{ \atop GL(2, 𝔽)} of GL(2, 𝔽).

We proceed as follows. Since \{z, w\} = (z, \overline{w})_{𝔼}, the commutator is trivial on 𝔼^{∗2}. By the method of §1 we find a splitting of this extension. As in §3 we also find a splitting of the extension of 𝔽^{∗} ⊂ 𝔼^{∗}. The extension of 𝔼^{∗2}𝔽^{∗} is abelian, and by an explicit version of the Mayer-Vietoris sequence we obtain a splitting of this extension. Finally 𝔼^{1} is contained in 𝔼^{∗2}𝔽^{∗}, and we restrict to obtain a splitting of the extension of 𝔼^{1}.

Fix non-trivial additive characters ψ_{𝔽} of 𝔽 and ψ_{𝔼} of 𝔼. Recall (5) the restriction of γ_{𝔼}(·, ψ_{𝔼}) to 𝔽^{∗} is a quadratic character.

Lemma 5.1 Suppose λ (respectively μ) is a splitting of the cocycle restricted to 𝔼^{∗2} (respectively 𝔽^{∗}). Assume λ(x) = μ(x) for x ∈ 𝔼^{∗2} ∩ 𝔽^{∗}. For z ∈ 𝔼^{∗}, x ∈ 𝔽^{∗} let ζ_{λ,μ}(z^{2}x) := λ(z^{2})μ(x)c(x, z^{2}). Then ζ_{λ,μ} is a well defined splitting of the cocycle restricted to 𝔼^{∗2}𝔽^{∗}.

Conversely if ζ is any splitting of the cocycle restricted to 𝔼^{∗2}𝔽^{∗} then ζ = ζ_{λ,μ} with λ = ζ|_{𝔼^{∗2}} and μ = ζ|_{𝔽^{∗}}.

Proof. Let ζ be any splitting of the cocycle. Then ζ(z^{2}x) = ζ(z^{2})ζ(x)c(x, z^{2}) by (6) and the second assertion is immediate.

Given λ and μ, choose any splitting ζ. Then λ and ζ|_{𝔼^{∗2}} both define splittings, so λ(z^{2}) = ζ(z^{2})α(z^{2}) for some character α of 𝔼^{∗2}. Similarly μ(x) = ζ(x)β(x) for some character β of 𝔽^{∗}. Let τ be a character of 𝔼^{∗} extending α and β; this exists since, for x ∈ 𝔼^{∗2} ∩ 𝔽^{∗}, λ(x) = μ(x) implies α(x) = β(x). Then

λ(z^{2})μ(x)c(x, z^{2}) = ζ(z^{2})α(z^{2})ζ(x)β(x)c(x, z^{2})
= ζ(z^{2}x)α(z^{2})β(x)
= ζ(z^{2}x)τ(z^{2}x).

This shows that ζ_{λ,μ} is well defined and is a splitting of the cocycle.

Lemma 5.2 (1) Choose a character α of 𝔼^{∗} satisfying

(13)  α(−1) = (−1, −1)_{𝔽}γ_{𝔼}(−1, ψ_{𝔼})

and let

λ_{α}(z^{2}) = c(z, z)γ_{𝔼}(z, ψ_{𝔼})γ_{𝔽}(Nz, ψ_{𝔽})α(z).

Then λ_{α} is a well defined splitting of the cocycle restricted to 𝔼^{∗2}. Furthermore every splitting of the cocycle restricted to 𝔼^{∗2} is equal to λ_{α} for some α satisfying (13).

(2) Let β be a character of 𝔽^{∗} and let

μ_{β}(x) = γ_{𝔽}(x, ψ_{𝔽})β(x).

Then μ_{β} is a splitting of the cocycle restricted to 𝔽^{∗}, and every splitting of the cocycle restricted to 𝔽^{∗} is equal to μ_{β} for some β.

(3) Suppose α ∈ 𝔼̂^{∗}, β ∈ 𝔽̂^{∗} satisfy

(14)  α(z) = γ_{𝔽}(z^{2}, ψ_{𝔽})γ_{𝔽}(Nz, ψ_{𝔽})γ_{𝔼}(z, ψ_{𝔼})c(z, z)β(z^{2}) (z^{2} ∈ 𝔽^{∗}).

In particular α satisfies (13). For z ∈ 𝔼^{∗}, x ∈ 𝔽^{∗} define

(15) \begin{aligned}ζ_{α,β}(z^{2}x) & := λ_{α}(z^{2})μ_{β}(x)c(x, z^{2}) \\ & = γ_{𝔼}(z, ψ_{𝔼})γ_{𝔽}(Nz, ψ_{𝔽})γ_{𝔽}(x, ψ_{𝔽})α(z)β(x)c(z, z)c(x, z^{2}). \end{aligned}

Then ζ is a well defined splitting of the cocycle restricted to 𝔼^{∗2}𝔽^{∗}. Furthermore every splitting of the cocycle restricted to 𝔼^{∗2}𝔽^{∗} is equal to ζ_{α,β} for some α, β satisfying (14).

Proof. Part (1) is an extension of (8) to the case of a non-abelian group. Thus by (8) and (11),

\begin{aligned}c(z^{2}, w^{2}) & = c(z, z)c(w, w)c(zw, zw)\{z, w\} \\ & = c(z, z)c(w, w)c(zw, zw)(z, w)_{𝔼}(Nz, Nw)_{𝔽}. \end{aligned}

Replacing (z, w)_{𝔼} by γ_{𝔼}(z, ψ_{𝔼})γ_{𝔼}(w, ψ_{𝔼})γ_{𝔼}(wz, ψ_{𝔼})^{−1}, and similarly (Nz, Nw)_{𝔽} gives

c(z^{2}, w^{2}) = τ(z)τ(w)τ(zw)^{−1}

with τ(z) = c(z, z)γ_{𝔼}(z, ψ_{𝔼})γ_{𝔽}(Nz, ψ_{𝔽}). The same relation holds with τ(z) replaced by τ(z)α(z).

We check the condition that λ_{α}(z^{2}) := τ(z)α(z) be well defined:

\begin{aligned}τ(−z)α(−z) & = c(−z, −z)γ_{𝔼}(−z, ψ_{𝔼})γ_{𝔽}(N(−z), ψ_{𝔽})α(−z) \\ & = c(−z, −z)γ_{𝔼}(−1, ψ_{𝔼})γ_{𝔼}(z, ψ_{𝔼})(−1, z)_{𝔼}γ_{𝔽}(Nz, ψ_{𝔽})α(z)α(−1). \end{aligned}

A simple calculation using (10) gives

(16)  c(−z, −z) = (−1, −1)_{𝔽}(−1, Nz)_{𝔽}c(z, z)

and inserting this gives

\begin{aligned}τ(−z)α(−z) & = (−1, −1)_{𝔽}γ_{𝔼}(−1, ψ_{𝔼})α(−1)τ(z)α(z) \\ & = τ(z)α(z) by (13). \end{aligned}

Fix α satisfying (13). If λ is any splitting of the cocycle restricted to 𝔼^{∗2} then λ = λ_{α}δ for some character δ of 𝔼^{∗2}. Extend δ to a character δ^{∗} of 𝔼^{∗}. Then λ_{α}δ = λ_{αδ^{∗2}}, and αδ^{∗2} satisfies (13). This proves (1).

By (10), c(diag(x, x), diag(y, y)) = (x, y)_{𝔽}, and (2) follows as in Section 3.

For (3) apply Lemma 5.1. Inserting λ_{α}, μ_{β} in the condition of the Lemma gives (14). The final assertion follows as in the proof of (1). This completes the proof.

Let ζ be any splitting of the cocycle restricted to 𝔼^{∗2}𝔽^{∗}. Then ζ = ζ_{α,β} for some α, β satisfying (14). This implies

Lemma 5.3 The map α(z) := γ_{𝔽}(z^{2}, ψ_{𝔽})γ_{𝔽}(Nz, ψ_{𝔽})γ_{𝔼}(z, ψ_{𝔼})c(z, z) is a character of {z ∈ 𝔼^{∗} | z^{2} ∈ 𝔽^{∗}}.

For completeness we also prove this directly:

α(zw) = γ_{𝔽}(z^{2}w^{2}, ψ_{𝔽})γ_{𝔽}(N(zw), ψ_{𝔽})γ_{𝔼}(zw, ψ_{𝔼})c(zw, zw)

= γ_{𝔽}(z^{2}, ψ_{𝔽})γ_{𝔽}(w^{2}, ψ_{𝔽})(z^{2}, w^{2})_{𝔽}γ_{𝔽}(Nz, ψ_{𝔽})γ_{𝔽}(Nw, ψ_{𝔽})(Nz, Nw)_{𝔽}

γ_{𝔼}(z, ψ_{𝔼})γ_{𝔼}(w, ψ_{𝔼})(z, w)_{𝔼}c(zw, zw)
= α(z)α(w)c(z, z)c(w, w)c(z^{2}, w^{2}){z, w}c(zw, zw) by (11)
= α(z)α(w) by (8).

Remark 5.4 Given α ∈ 𝔼̂^{∗} there exists β ∈ 𝔽̂^{∗} satisfying (14) if and only if (13) holds.

This follows by an argument as in the proof of Lemma 5.2 (1): define β(z^{2}) = α(z)γ_{𝔽}(z^{2}, ψ_{𝔽})^{−1}γ_{𝔽}(Nz, ψ_{𝔽})^{−1}γ_{𝔼}(z, ψ_{𝔼})^{−1}c(z, z); this is well defined if (13) holds, and extends to a character of 𝔽^{∗}.

Suppose z ∈ 𝔼^{1}. By Hilbert’s Theorem 90, z = w/\overline{w} for some w ∈ 𝔼^{∗}. Then z = w^{2}/N(w) ∈ 𝔼^{∗2}𝔽^{∗}, so ζ restricts to a splitting of the extension of 𝔼^{1}. We now make explicit choices such that ζ is a 𝕋 or μ_{n} splitting as in Theorem 1.

Lemma 5.5 We may choose ψ_{𝔼} so that

(17)  γ_{𝔼}(x, ψ_{𝔼}) = 1 (x ∈ 𝔽^{∗}).

Proof. Since γ_{𝔼}(·, ψ_{𝔼}) is a quadratic character of 𝔽^{∗}, γ_{𝔼}(x, ψ_{𝔼}) = (x, y)_{𝔽} for some y ∈ 𝔽^{∗}. Suppose 𝔼 = 𝔽(\sqrt{Δ}). Then γ_{𝔼}(Δ, ψ_{𝔼}) = 1 since Δ is a square in 𝔼^{∗}. Therefore (Δ, y)_{𝔽} = 1, so y = Nw for some w ∈ 𝔼^{∗}. Replacing ψ_{𝔼} by wψ_{𝔼} gives (cf. [10], Appendix)

\begin{aligned}γ_{𝔼}(x, wψ_{𝔼}) & = (x, w)_{𝔼}γ_{𝔼}(x, ψ_{𝔼}) \\ & = (x, Nw)_{𝔽}(x, y)_{𝔽} = 1. \end{aligned}

This completes the proof.

Lemma 5.6 Fix ψ_{𝔼} satisfying Lemma 5.5. Choose α, β satisfying (14) and let ζ = ζ_{α,β}. Then

ζ(z)^{2} = α(z) (z ∈ 𝔼^{1})

Proof. Writing z = w/\overline{w} = w^{2}/N(w) and applying the definition (15) gives

(18) ζ(w/\overline{w}) = (−1, Nw)_{𝔽}γ_{𝔼}(w, ψ_{𝔼})α(w)β(Nw^{−1})c(w, w)c(Nw^{−1}, w^{2})

and (19)

ζ(w/\overline{w})^{2} = (−1, w)_{𝔼}α(w^{2})β(Nw^{−2})
= (−1, Nw)_{𝔽}α(w^{2})α(Nw^{−2})(−1, Nw)_{𝔽}γ_{𝔼}(Nw^{−1}, ψ_{𝔼}) by (14)
= α(w/\overline{w})γ_{𝔼}(Nw, ψ_{𝔼})
= α(w/\overline{w}) by (17).

We see that ζ_{α,β} is a μ_{2n} splitting if and only if α(z)^{n} = 1 for all z ∈ 𝔼^{1}. We now complete the proof of Theorem 1. Proof of Theorem 1: By (16) and (17) we have

α(−1) = (−1, −1)_{𝔽}.

Therefore we may choose α = 1 if (−1, −1)_{𝔽} = 1. Assume (−1, −1)_{𝔽} = −1. If 𝔽 = ℝ then α(z) = z^{n} for n odd as in Section 4, so assume 𝔽 is nonarchimedean. If −1 ∉ 𝔼^{∗2} we may choose α^{2} = 1. If −1 ∈ 𝔼^{∗2} then −1 ∉ 𝔼^{∗4} (cf. §4) and we may choose α^{4} = 1.

We make some explicit choices and summarize the preceding discussion. If (−1, −1)_{𝔽} = −1 and −1 ∉ 𝔼^{∗2} choose z_{1} ∈ 𝔼^{∗} with (z_{1}, −1)_{𝔼} = −1. If (−1, −1)_{𝔽} = −1 and −1 ∈ 𝔼^{∗2}, i.e. 𝔼 = 𝔽(\sqrt{−1}), then the norm

residue symbol (w, z)_{𝔼,4} is defined. In particular the map z → (w, z)_{E,4} is a character of 𝔼^{∗} of order 4. Choose z_{2} ∈ 𝔼^{∗} satisfying (z_{2}, −1)_{𝔼,4} = −1.

Theorem 5.7 Choose a non-trivial character ψ_{𝔼} of 𝔼 such that γ_{𝔼}(x, ψ_{𝔼}) = 1 for all x ∈ 𝔽^{∗} (Lemma 5.5). For z ∈ 𝔼^{∗} let

α(z) := \begin{cases}z & 𝔽 = ℝ \\ 1 & (−1, −1)_{𝔽} = 1 \\ (z_{1}, z)_{𝔼} & (−1, −1)_{𝔽} = −1, −1 ∉ 𝔼^{∗2} \\ (z_{2}, z)_{𝔼,4} & (−1, −1)_{𝔽} = −1, −1 ∈ 𝔼^{∗2} \end{cases}

Then α is a character of 𝔼^{∗} of order ∞, 1, 2 or 4 respectively, satisfying (13). Choose a character β of 𝔽^{∗} satisfying

(20) β(z^{2}) = γ_{𝔽}(z^{2}, ψ_{F})^{−1}γ_{𝔽}(Nz, ψ_{𝔽})^{−1}γ_{𝔼}(z, ψ_{𝔼})^{−1}c(z, z)α(z) (z^{2} ∈ 𝔽^{∗})

(cf. Remark 5.4). In particular

β(x^{2}) = (−1, x)α(x) (x ∈ 𝔽^{∗}).

Let

(21)  ζ(w/\overline{w}) = γ_{𝔼}(w, ψ_{𝔼})α(w)β(Nw^{−1})c(w, w)c(Nw^{−1}, w^{2}).

Then ζ is a splitting of the cocycle restricted to 𝔼^{1}. Furthermore for z ∈ 𝔼^{1} ζ(x)^{2} = α(z), and for 𝔽 non-archimedean this gives ζ(z)^{n} = α(z)^{n/2} = 1 with

n = \begin{cases}2 & (−1, −1)_{𝔽} = 1 \\ 4 & (−1, −1)_{𝔽} = −1, −1 ∉ 𝔼^{∗2} \\ 8 & (−1, −1)_{𝔽} = −1, −1 ∈ 𝔼^{∗2} \end{cases}

Remark 5.8 With α, β as in the Theorem,

ζ(w/\overline{w}) = ζ_{α,β}(w/\overline{w})(−1, Nw)_{𝔽}.

We have dropped the term (−1, Nw), which is allowed since w/\overline{w} → (−1, Nw) is a quadratic character of 𝔼^{1}.

Henceforth write 𝔼 = 𝔽(δ) with Δ := δ^{2} ∈ 𝔽.

Remark 5.9 Condition (20) is equivalent to

\begin{aligned}β(x^{2}) & = (−1, x)α(x) (x ∈ 𝔽^{∗}) \\ β(Δ) & = γ_{𝔽}(−1, ψ_{𝔽})^{−1}γ_{𝔼}(δ, ψ_{𝔼})^{−1}c(δ, δ)α(δ). \end{aligned}

The splitting has a simple formula on T^{2}:

ζ(z^{2}) = c(z, z)α(z) (z ∈ 𝔼^{1}),

which is independent of β, ψ_{𝔽} and ψ_{𝔼}. Note that any two μ_{2} splittings of T have the same restriction to T^{2} since they differ by a quadratic character.

The map w/\overline{w} → Nw induces an isomorphism T/T^{2} ≃ N𝔼^{∗}/𝔽^{∗2}. Choose representatives a_{1}, …, a_{n} of generators of N𝔼^{∗}/𝔽^{∗2}, with corresponding elements z_{1}, …, z_{n} ∈ T. Given α there are two choices of each β(a_{i}) differing by sign, and these signs may may be chosen arbitrarily. The following result follows easily.

Corollary 5.10 Choose representatives z_{1} …, z_{m} of generators of T/T^{2} ≃ N𝔼^{∗}/𝔽^{∗2}. Choose α as in Theorem 5.7. Define

ζ(z^{2}) = c(z, z)α(z) (z ∈ 𝔼^{1})

and for 1 ≤ i ≤ m let ζ(z_{i}) be either square root of α(z_{i}). Then ζ extends uniquely to a splitting of the cocycle as in Theorem 5.7.

In the non-archimedean case by ([4], Lemma 0.3.2) |T/T^{2}| = 2/|2|_{𝔽}, which equals 2 if the residual characteristic of 𝔽 is odd.

We conclude with a few remarks about the definition of ζ.

From the definition we have for w ∈ 𝔼^{∗}:

c(w, w) = (−Nw, x(w)x(w^{2}))_{𝔽}

and

c(Nw^{−1}, w^{2}) = (Nw, x(w^{2}))_{𝔽}.

Note that for λ ∈ 𝔽^{∗}, w ∈ 𝔼^{∗} we have

c(λ, w) = (λ, x(w))_{𝔽}.

Fix u ∈ 𝔼^{∗} with trace(u) = 0, and define Tr_{u} : 𝔼^{∗} → 𝔽^{∗} by

Tr_{u}(z) = \begin{cases}trace(z) & trace(z) ≠ 0 \\ uz & trace(z) = 0 \end{cases}

Up to conjugation by SL(2, 𝔽) we may assume ι(x + yδ) = (\begin{matrix}x & yΔ/a \\ ya & x \end{matrix})

for some a ∈ 𝔽^{∗}. If w = x + yδ with xy ≠ 0 we have x(w)x(w^{2}) = 2xy^{2}a^{2}. Considering the cases with xy = 0 separately gives

c(w, w) = (−Nw, Tr_{aδ}(w))_{𝔽} (z ∈ 𝔼^{∗})

and

ζ(z^{2}) = (−1, Tr_{aδ}(z))_{𝔽}α(z) (z ∈ 𝔼^{1}).

For example if p is odd and 𝔼 ≠ 𝔽(\sqrt{−1}) then m = 1 (cf. Corollary 5.10), we may take z_{1} = −1, α = 1 and ζ(−1) = 1 which gives

ζ(ϵz^{2}) = c(z, z)c(ϵ, z^{2}) (ϵ = ±1, z ∈ 𝔼^{1}).

For example let 𝔽 = ℝ and define ι(x+iy) = (\begin{matrix}x & y \\ −y & x \end{matrix}). We take α(z) = z

and β(x) = ±\sqrt{|x|}. Then for z ∈ 𝔼^{1},

ζ(z^{2}) = (−1, Tr_{−i}(z))_{ℝ}z = sgn(Tr_{−i}(z))z

(independent of β). Note that a = −1 and Tr_{−i}(iy) = y (y ∈ ℝ^{∗}).

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