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A plectic Taniyama group

We revisit Langlands' construction of the Taniyama group to define a plectic Taniyama group for any totally real field F. The construction is functorial in F and recovers Langlands' original Taniyama group when F = Q. We relate our construction to Nekovar's 'hidden symmetries' in classical CM theory, generalising the relationship between the Taniyama group and Tate's half-transfer maps.


Introduction
This article deals with new aspects of the theory of complex multiplication. In its most general form, the 'main theorem of complex multiplication' [1] can be expressed as an equality between two affine group schemes over Q (together with some additional data) -Deligne's motivic Galois group for the Tannakian category of potentially abelian motives over Q on the one hand and the Taniyama group constructed by Langlands [3] on the other. Recently Nekovář and Scholl [7] have initiated the study of extra 'plectic' symmetries which they conjecture should act on the cohomology of certain Shimura varieties. The main purpose of this article is to generalise Langlands' construction of the Taniyama group to the plectic setting.
Throughout we let Q denote the algebraic closure of Q in C and c ∈ Γ Q = Gal(Q/Q) denote the image of complex conjugation. The Taniyama group consists of an extension 1 → S → T → Γ Q → 1 of the absolute Galois group Γ Q by the Serre group S, together with a continuous finite-adelic splitting s : Γ Q → T (A Q,f ). Here S is a certain explicit pro-algebraic torus over Q and Γ Q is viewed as a profinite constant group scheme over Q. In what follows let F ⊂ Q be a fixed totally real field and Σ F = Hom Q (F, Q) denote the set of all embeddings of F into Q. In this article we ape Langlands' construction and define an extension of the form 1 → S F → T F → Aut F (F ⊗ Q Q) → 1 together with a continuous finite-adelic splitting s F : Aut F (F ⊗ Q Q) → T F (A Q,f ). Here: (1) The profinite group Aut F (F ⊗ Q Q) is the F -plectic Galois group introduced by Nekovář and Scholl. There is a canonical isomorphism between Aut F (F ⊗ Q Q) and the 'plectic group' consisting of all the automorphisms of Γ Q as a right Γ F -set. The natural inclusion Γ Q ֒→ Aut F (F ⊗ Q Q) corresponds under this isomorphism to the inclusion Γ Q ֒→ Γ Q #Γ F given by the left regular action. (2) The pro-algebraic torus S F is just the product group S ΣF . This plays the role of S in the theory of F -plectic Hodge structures 1 .
Thanks to (1) and (2), we call our extension the F -plectic Taniyama group. The construction is functorial in F and when F = Q we recover Langlands' original Taniyama group.
The classical theory. In order to explain our results in more detail we need to recall some more of the classical picture. This theory is explained very clearly, and in more detail, in the survey articles [2], [4] and [8]. For any CM number field K ⊂ Q and CM type Φ ⊂ Σ K = Hom Q (K, Q), Tate wrote down half-transfer map F Φ : Γ Q → Γ ab K which is now known to describe the action of Γ Q on the torsion points of a CM abelian variety of type Φ. This is defined by first taking a set of coset representatives w ρ for Γ K ⊂ Γ Q such that w cρ = cw ρ for all ρ ∈ Σ K and mapping g ∈ Γ Q to the element gρ gw ρ mod Γ K ab of Γ ab K , which is independent of the choice of coset representatives. There is a natural way to lift F Φ through the the Artin reciprocity map to a map f Φ : Γ Q → A × K,f /K × called the associated Taniyama element. This is determined by requiring that for any g ∈ Γ Q we have f Φ (g) 1+c = χ(g)K × where χ : Γ Q → Z × denotes the cyclotomic character.
As we now explain, Langlands' Taniyama group is built from a universal version of these Taniyama elements. To understand this, we need to review a few more of the details of Langlands' construction. First, for any CM number field E ⊂ Q the level E Serre group is the quotient of the Q-rational torus T E = Res E/Q G m with character group X * (S E ) = { σ n σ [σ] : there is some w ∈ Z such that n σ + n cσ = w for all σ ∈ Hom Q (E, Q) } The identity embedding E ⊂ Q gives rise to a canonical cocharacter µ E of the torus T E and its quotient S E . The full Serre group S featuring in Langlands' Taniyama group is then the projective limit S = lim ← −E S E over all CM fields E ⊂ Q with respect to the algebraic norm maps. The Taniyama group is itself built from extensions of the form 1 → S E → T E → Γ E ab /Q → 1 together with continuous finite-adelic splittings s E : Γ E ab /Q → T E (A Q,f ), where here E ⊂ Q is a CM number field which is Galois over Q. At this finite level, such extensions are easily classified. Any such extension gives rise to and is completely determined by a map and it is known precisely which such maps arise in this way (see [5] Proposition 2.7). The data of Langlands' level E Taniyama extension is equivalent to the data of a certain explicit map f E : Γ E ab /Q → S E (A E,f )/S E (E) called the universal Taniyama element. Unsurprisingly, to define this map we need to make similar choices as we did for Tate's half-transfer maps. Let W E/Q,f denote the finite-adelic 1 For a totally real field F , an F -plectic R-Hodge structure (see [7], §16) is a finite-dimensional real vector space V together with a decomposition of the C-vector space V C = V ⊗ R C of the form version of the Weil group of the extension E/Q, which sits in a diagram and pick a liftc ∈ W E/Q,f of complex conjugation and a set {w σ } of coset representatives for is nothing more than the image of the element is well-defined independent of the choice of lift g or the choices ofc and w σ . With our choices of normalisations (we normalise the Artin reciprocity map by insisting that local uniformisers get mapped to geometric Frobenius elements), the level E Taniyama extension is defined by the map b The precise link between the universal Taniyama element and Tate's half-transfer maps is given by the Hodge theory of CM abelian varieties. Take K ⊂ Q a CM field and Φ a CM type for K. Let T K = Res K/Q G m and µ Φ denote the cocharacter which we can think of as defining the Hodge filtration on a CM abelian variety of type (K, Φ). Let E ⊂ Q be a CM number field which is Galois over Q and contains the reflex field of (K, Φ). Then since µ Φ is defined over E and has weight µ −(1+c) Φ defined over Q, there is a unique representation taking the canonical cocharacter µ E of S E to µ Φ . This representation is nothing more than the reflex norm of classical CM theory. The map f E really is then a 'universal Taniyama element' in the sense that whenever (K, Φ) is a CM type with reflex field contained in E we have for all g in Γ Q . In the case when E actually contains K we can identify Σ K with Hom Q (K, E) and decompose The above assertion is equivalent to saying that for any ρ : This article. From now on we fix a totally real number field F ⊂ Q and let Σ F = Hom Q (F, Q). In what follows we recover the classical theory when we take F to be Q. Our construction of the F -plectic Taniyama group is inspired by Nekovář's paper [6] 2 , where it is shown that if K ⊂ Q is any CM field with totally real subfield F then Tate's half-transfer map attached to a CM type Φ of K canonically extends to a map This can now be viewed as evidence for the full plectic conjecture of Nekovář and Scholl. The first main contribution of this article is to reinterpret Nekovář's half-transfer maps along the lines of Tate's original definition.
Theorem 1.1. Let w ρ be a set of coset representatives for Γ K ⊂ Γ Q such that w cρ = cw ρ for all ρ ∈ Σ K . Then Nekovář's half-transfer map F Φ sends any g ∈ Aut F (F ⊗ Q Q) to the element In the remainder of this paper we take Theorem 1.1 as our starting point and revisit Langlands' original definition to construct an F -plectic Taniyama group for any totally real number field F . This takes the form of an extension 1 → S F → T F → Aut(F ⊗ Q Q) → 1 of the F -plectic absolute Galois group Aut(F ⊗ Q Q) by the F -plectic Serre group S F , together with a continuous finite-adelic splitting s F : Aut(F ⊗ Q Q) → T F (A Q,f ). Here S F denotes the F -plectic Serre group S ΣF , which plays the role of S in the theory of F -plectic Hodge structures. The group T F is built from extensions of the form Here E ⊂ Q is a CM field containing F and Galois over Q, while S E F denotes the torus (S E ) ΣF . Any such extension gives rise to and is completely determined by a map and it is possible (see Proposition 6.1 for a precise statement) to write down certain conditions which, when satisfied, guarantee that such a map does indeed come from an extension of the above form.
For E as above let W E/Q,f denote the finite-adelic version of the Weil group of the extension E/Q. As with Langlands' construction, we pick a liftc ∈ W E/Q,f of complex conjugation and a set {w σ } of coset representatives for 3 . An embedding j : F ֒→ E induces a natural map j : Aut F (F ⊗ E ab ) ֒→ Aut E (E ⊗ E ab ) and hence a map j : Γ E ab /Q #Γ E ab /F → Γ E ab /Q #Γ E ab /E . This can be expressed in purely group theoretic terms, and as a result we have a similar map on the level of Weil groups. We view the torus S E F = (S E ) ΣF as a quotient of where j ⊗σ denotes the component on which F acts by j ∈ Σ F and E acts by σ ∈ Γ E/Q . The key construction in this paper is that of a particular map which we call the F -plectic universal Taniyama element. This is defined by sending g ∈ Aut F (F ⊗ E ab ) to the image of the element It should be pointed out explicitly that Nekovář reinterprets his results terms of a generalised Taniyama group which is somewhat different to the plectic Taniyama group defined by Theorem 1.2. Our approach, which is motivated by plectic Hodge theory and more closely mirrors Langlands' original construction, is inspired by Nekovář's work but proceeds in a different direction. 3 This is very slight simplification of the choices involved. See §5 for a more detailed exposition of the construction.
under T F ⊗E ։ S E F , whereα g is any lift of α g through the map W E/Q,f #W E/F,f ։ Γ E ab /Q #Γ E ab /F . To make it clear, here each j(α g ) is a element of the group for all w ∈ W E/Q,f , e ∈ A × E,f /E × we have α(we) = α(w)e} Any such element induces a permutation σ → j(α g )σ of the quotient Γ E/Q such that j(α g )(w σ −1 ) lies in the same coset as simply stores this collection of elements of A × E,f /E × for each j ∈ Σ F and σ ∈ Γ E/Q in an appropriate way. The content is that this becomes well-defined independent of our choices when we pass to the quotient T F ⊗E ։ S E F . What's more, this F -plectic universal Taniyama element goes on to satisfy the conditions of Proposition 6.1 and so defines an extension that we call the F -plectic Taniyama group. Theorem 1.2. For any totally real field F and CM number field E ⊂ Q which splits F : is well-defined independent of the choice of liftα g and the choices ofc and of coset representatives w σ for σ ∈ Γ E/Q .
(2) There is a unique extension This family of extensions pieces together produce an extension of the form together with a continuous finite-adelic splitting s F : We end by making clear the relationship between our F -plectic Taniyama group and Nekovář's extended half-transfer maps. Let K be a CM field with maximal totally real subfield F . Then given a CM type Φ on K, we can write Φ = {Φ j : j ∈ Σ F } where φ j is the unique element of Φ such that φ j | F = j. We can think of the cocharacter µ Φ = φ∈Φ [φ] as the restriction to the diagonal of the homomorphism µ Φ,F = (µ j ) j∈ΣF : G ΣF m,C → T K,C where for each j ∈ Σ F we have µ j = [φ j ] ∈ X * (T K ). This can be thought of as using the real multiplication by F to define an F -plectic Hodge structure on a CM abelian variety of type (K, Φ). If E ⊂ Q is a CM number field containing K and Galois over Q then there is a unique representation ρ j : T E → T K taking the canonical cocharacter µ E of T E to µ j . These piece together to give a representation E → T K whose restriction to the diagonal is the classical reflex norm. It important to note that ρ Φ,F itself need not factor through the quotient T F ⊗ Q E ։ S E F . Now the universal Taniyama element f E F was constructed by looking at the image of If instead of mapping down to this quotient, we instead look at the image of this element under ρ Φ,F , we obtain an element In general this element may depend both on the choice of liftα g of α g and possibly also on the choices we made in the construction. Regardless, it follows from Theorem 1.1 that for any ρ : K ֒→ E the image of the ρ component of ρ Φ,F (h E F (α g )) ∈ T K (A E,f )/T K (E) under the Artin reciprocity map A × E,f /E × ։ Γ ab E is always equal to the image of F Φ (g) ∈ Γ ab K under the map Γ ab K → Γ ab E induced by ρ. This is a direct analogue of the relationship between Langlands' universal Taniyama element and Tate's half-transfer maps.

Plectic Galois theory
Let F be any number field and write Σ F for the set Hom Q (F, Q). We will say a subfield k of Q splits F if it contains the image of F under any embedding j ∈ Σ F . In this case the map a ⊗ b → (j(a)b) j∈ΣF defines an isomorphism of F -algebras where the action of F on the j component of the right hand side is via the embedding j : F ֒→ k. When k is Galois over Q then we call the group Aut F (F ⊗ Q k) of all automorphisms of the F -algebra F ⊗ Q k the F -plectic Galois group of k. From the decomposition above it is clear that Aut F (F ⊗ Q k) acts on the set Σ F . When F is a subfield of k we can interpret the F -plectic Galois group Aut F (F ⊗ Q k) as a wreath product. In this case we have Γ k/F := Gal(k/F ) ⊂ Γ k/Q := Gal(k/Q) and we can identify the set X = Γ k/Q /Γ k/F with Σ F . If we consider the ring k X as an F -algebra where the F -algebra structure comes from F ⊂ k on each factor then there is a natural isomorphism (see [6], where the semidirect product is taken with respect to the natural permutation action of the symmetric group S X on Γ X k/F . This is given by mapping ) x∈X is an isomorphism of F -algebras and induces an isomorphism More generally the construction of plectic Galois groups is functorial in F in the sense that if F and F ′ are both split by k then a map F ֒→ F ′ of number fields induces a map of plectic Galois groups. When F ⊂ F ′ ⊂ k this can be described explicitly in terms of wreath products as follows: Proposition 2.1. Let k be Galois over Q and F ⊂ F ′ ⊂ k. As above let us write X for the set Γ k/Q /Γ k/F = Σ F but also let Proof. This follows from [6], Proposition 1.1.8 and (the proof of) Proposition 2.2.4.
Again we have the following explicit description: Proof. This is [6], Proposition 1.1.6 and Proposition 1.1.7(ii).
Plectic Galois groups can be understand in purely group-theoretic terms. To see this, let G be any group and H ⊂ G a subgroup of finite index. We write X for the set G/H of left cosets of H in G. Then H acts on the right on the set G by right translation and an orbit of H under this action is nothing more than an element of X.
Definition (Nekovář, Scholl -cf. [7], §3). The plectic group of the pair (G, H) is the group G#H . . = Aut sets-H G of automorphisms of G as a right H-set. More explicitly, this consists of the following collection of set-theoretic bijections Note there is a natural inclusion G ֒→ G#H given by mapping an element g ∈ G to the left translation by g map L g : G → G.
Any α ∈ G#H gives rise to a well defined permutation of the set X = G/H. We will denote this permutation by π = π(α) ∈ S X but where the context is clear we will often ease the notation by writing this simply as x → α(x), or alternatively as x → αx, for all x ∈ X. This plectic group can also be interpreted in terms of a wreath product. To see this, let us choose a collection s = (s x ) x∈X of coset representatives for H in G. The permutation π is defined by the fact that α(s x )H = π(s x H) = s αx H and so the elements After making these choices we can define an isomorphism ρ s : G#H → S X ⋉ H X by sending α ∈ G#H to (π, (h x (α)) x∈X ) ∈ S X ⋉ H X . As before, the semidirect product on the right hand side is taken with respect to the permutation action of S X on H X . If the choice of coset representatives s = (s x ) x∈X is changed to s ′ = (s x t x ) x∈X for some t = (t x ) x∈X ∈ H X then the maps ρ s and ρ s ′ are related by the formula ρ s ′ (α) = (1, t) −1 ρ s (α)(1, t) for all α ∈ G#H.
If H ′ ⊂ H ⊂ G are subgroups of finite index then G#H is a contained in G#H ′ . The following proposition describes this inclusion map explicitly in terms of wreath products: hxy h x t y as claimed.
Now fix u ∈ G and let H ′ denote the conjugate subgroup uHu −1 . Then u gives rises to a map [u] : G#H → G#H ′ given by α → α ′ where for all g ∈ G we have α ′ (g) = α(gu)u −1 . The map [u] only depends on the left coset uH and can be described in terms of wreath products as follows: The proof is a straightforward calculation using the the definitions.
From these computations, we conclude: Proof. The first observation is due to Nekovář and Scholl (cf. [7], §3). Indeed, we have already noted that if the choice of s = (s x ) x∈X is changed to s ′ = (s x t x ) x∈X for some t = (t x ) x∈X ∈ H X then we have both Suppose now that G is a topological group and H ⊂ G is an open subgroup of finite index. As discussed above, a choice of coset representatives s = (s x ) x∈X for H in G defines an isomorphism ρ s : G#H ∼ − → S X ⋉ H X . The two groups H X (with the product topology) and S X (with the discrete topology) are topological groups and the action of S X on H X is continuous. Thus S X ⋉ H X carries the structure of a topological group. We define a topology on G#H by declaring U ⊂ G#H to be open if and only is open if and only if ρ s (U ) is. It follows from the explicit calculations in Proposition 2.3 and Proposition 2.4 that with this topology: • the inclusion G ֒→ G#H is continuous;

Plectic CM theory
Let K ⊂ Q be a CM field and Σ K = Hom Q (K, Q) = Γ Q /Γ K . Let F = K + denote the maximal totally real subfield of K. In his work on hidden symmetries in the theory of complex multiplication [6], Nekovář has defined a natural map extending Tate's half-transfer F Φ : Γ Q → Γ ab K for any CM type Φ of K. Following Nekovář's notation let us write X = Σ F = Γ Q /Γ F and fix a section s given by x → s x to the restriction map Γ Q ։ X = Hom Q (F, Q). We note that {s x | K : x ∈ X} gives a distinguished CM type of K, and so we get the following data: • A bijection between (Z/2Z) X and the set of CM types of K sending a collection a = (a x ) x∈X to Proposition 3.1. (Nekovář) Fix s as above and let Φ be a CM type corresponding to a ∈ (Z/2Z) X . Define a map s F Φ : Then the composite map is independent of the choice of s and its restriction to Γ Q is F Φ .
Using the language developed in §2, we can give an alternative description of Nekovář's half-transfer maps. As with the classical case, we choose a section ρ → w ρ to the projection Γ Q ։ Hom Q (K, Q) such that w cρ = cw ρ for all ρ ∈ Σ K . We notice that Tate's original definition makes sense for any element α of the set Γ Q #Γ K := Aut sets-ΓK Γ Q of all automorphisms of Γ Q as a right Γ K -set. Indeed any such α induces a well-defined permutation ρ → αρ of Σ K and for any α ∈ Γ Q #Γ K the elements and it is clear that if we compose with the inclusion Γ Q ֒→ Γ Q #Γ K we recover Tate's half-transfer map F Φ .
K is independent of the choice of coset representatives ρ → w ρ .
Proof. Denote by F ′ Φ (α) the element of Γ ab K we would have obtained had we chosen another collection {w ′ ρ } of coset representatives. Then since each w ′ ρ is of the form w ′ ρ = w ρ e ρ for some e ρ ∈ Γ K we see that The conditions on w ρ and w ′ ρ force e cρ = e ρ and so e ρ only depends on ρ| F . The condition α ∈ Γ Q #Γ F ⊂ Γ Q #Γ K ensures that αΦ = {αφ : φ ∈ Φ} is another CM type for K. It follows that the product on the right hand side is trivial and F Φ (α) = F ′ Φ (α) as claimed. Exactly as with Tate's half-transfer maps, we have the following compatibilities: Proposition 3.3. Let K ⊂ Q be a CM field and let Φ ⊂ Σ K = Hom Q (K, C) be a CM type.
(i) Let K ′ ⊂ Q be a CM field containing K, The maps F Φ and F Φ ′ fit into a commutative diagram: The maps F Φ and F Φ ′ fit into a commutative diagram: where u * : Γ ab K → Γ ab K ′ is the map induced by the map Γ K → Γ K ′ given by g → ugu −1 . Proof. Both of these are straightforward calculations using the computations in Proposition 2.3 and Proposition 2.4 respectively.
In §2 we defined a canonical isomorphism Aut F (F ⊗ Q Q) ∼ − → Γ Q #Γ F . The following theorem tells us that Nekovář's half-transfer maps F Φ correspond under this isomorphism to the very natural maps F Φ defined above.
Proof. Let us make choices as in the definition of F Φ . In particular we have chosen a section s = (s x ) x∈X to the projection Γ Q ։ X = Σ F and this gives us a section ρ → w ρ to Γ Q ։ Σ K = Hom Q (K, Q) such that w cρ = cw ρ by setting w c b sx|K = c b s x . We identify the CM type Φ with a = (a x ) x∈X ∈ (Z/2Z) X . To ease the notation write α for α g and let us write ρ s (α) = (π, (h x ) x∈X ) = β s (g). Recall this means that π is the permutation of X = Σ F induced by α and we have h We can make the following computations To see this recall that by definition h x = s −1 π(x) α(s x ) so the right hand side of the first claim is just x c ax s x | K = c ax | K and in particular s −1 x c ax s x | F = 1, i.e. s −1 x c ax s x ∈ Γ F . Since α ∈ Γ Q #Γ F we can take this inside the brackets to get α(s x s −1 x c ax s x ) = α(c ax s x ) = α(w φx ) For the second claim, the fact that α(w φx ) = s π(x) h x s −1 x c ax s x implies that α(w φx )| K = s π(x) h x c ax | K = c ax+hx s π(x) | K (using again the fact that K is CM). This shows that w αφx = c ax+hx s π(x) | K and so by definition w αφx = c ax+hx s π(x) .
With this established, we compute straight from the definition that as claimed.

The plectic Serre group
Let E ⊂ Q be any CM field and for each σ ∈ Hom Q (E, Q) let [σ] denote the associated character of the torus T E = Res E/Q G m . These characters form a Z-basis for the character group X * (T E ) and the arithmetic action of τ ∈ Gal(Q/Q) is given in terms of this basis by [σ] → [τ σ]. The level E Serre group is the quotient S E of T E = Res E/Q G m with character group : there is some w ∈ Z such that n σ + n cσ = w for all σ } ⊂ X * (T E ) When E is Galois over Q the action of the Galois group Γ E/Q on E × gives rise to an algebraic action Γ E/Q ֒→ Aut(T E ). The algebraic action of an element τ ∈ Γ E/Q can be described in terms of characters by [σ] → [σ • τ ] and induces a well-defined action of Γ E/Q on the quotient S E . As discussed in the introduction, for such E Langlands' has constructed a particular extension of the profinite group Γ E ab /Q by the torus S E . The action of Γ E ab /Q on S E implicit in this level E Taniyama extension is given by precomposing this algebraic action of Γ E/Q defined above with the natural projection Γ E ab /Q ։ Γ E/Q . Now let F be any totally real number field and suppose E is a CM number field which is big enough to split F . In this case we can identify the set Σ F = Hom Q (F, Q) with Hom Q (F, E). When E is Galois over Q, we can define an action of the F -plectic Galois group Aut and in this way we can view S E F as a quotient of T F ⊗ Q E . We have an isomorphism where here j ⊗ σ signifies that the action of F is via j : F ֒→ Q and then action of E is via σ : E ֒→ Q. If we let [j ⊗ σ] denote the projection onto the j ⊗ σ) component of the above decomposition then the arithmetic action of τ ∈ Gal(Q/Q) is given in terms of this character basis by In this way we identify the characters [j, σ] of T ΣF E and [σ • j ⊗ σ] of T F ⊗ Q E . Now we make use of the language developed in §2. Recall that an embedding j : F ֒→ E defines a map j : We will try to be careful to to distinguish this from the element σ • j(g) ∈ S Γ E/Q . That this is equivariant for the arithmetic action of Γ Q on X * (T F ⊗ Q E ) follows from Theorem 2.5 which implies that if j ′ = τ • j then j ′ (g)(σ) = j(g)(στ ) · τ −1 .
It's easy to check this defines an action of Aut F (F ⊗ Q E) on T F ⊗ Q E (and hence on T ΣF E ), which we denote on points by t → g ⋆ t so that on Q-points we have g * χ(t) = χ(g ⋆ t). The following proposition implies that this action stabilises X * (S E F ) ⊂ X * (T ΣF E ) = X * (T F ⊗ Q E ) and thus induces a well-defined algebraic action of where here for each r = 1, . . . , d we let j r = σ r • j 0 . We can translate between the two matrices by an appropriate permutation of each row (i.e. j → σ • j in row σ). For any g ∈ Aut F (F ⊗ Q E) the permutation j r (g) lies in the subgroup Γ E/Q #Γ E/jr (F ) ⊂ S Γ E/Q and so maps Since the action of g on X * (T F ⊗E ) acts on the (j r ⊗ −) column above by the right action of j r (g) we deduce that g * χ is of the form More precisely we have shown that for σ ∈ Γ E/Q such that σ • j ′ 0 = j we have c j ′ 0 ,σ = a j0,τ where τ ∈ Γ E/Q is such that τ j(g) = σ. Since F is totally real, this implies that Φ ′ = {σ ∈ Γ E/Q : c j ′ 0 ,σ = 1} is a CM type of E.

The plectic universal Taniyama element
Let F ⊂ Q denote a totally real number field and E ⊂ Q a CM field which is Galois over Q and large enough to contain F . In this section we ape Langlands' construction of the universal Taniyama element to produce a map Let W E/Q,f denote the finite-adelic version of the Weil group of the extension E/Q, which sits in a diagram in which the columns are surjective. The implicit action of σ ∈ Γ E/Q on A × E,f /E × is via e → e σ . We need to make some choices which will turn out to play no role: • Corresponding to the place v of Q coming from the embedding into C (which we have fixed) there is a continuous map i v : W R → W Q (uniquely defined up to an inner automorphism of W Q by an element of Ker(ϕ)) and a diagram inducing the inclusion C × ֒→ C E ≃ W ab E coming from the place v. The map i v gives a well defined lift of complex conjugation c ∈ Γ E ab /Q to an elementc ∈ W E/Q,f of order 2 (which changes by conjugation by an element of Ker(ϕ f ) if we change i v -see [10], 1.6.1). We choose one such map i v and letc denote the associated complex conjugation.
• Having fixed i v we then pick elements w σ ∈ W E/Q,f such that σ → w σ is a section to the surjection W E/Q,f ։ Γ E/Q and further we have the relation w cσ =cw σ for all σ ∈ Γ E/Q To construct the F -plectic universal Taniyama element we exploit the canonical isomorphism Aut which, in terms of plectic groups, is now precisely the composite map where here u ∈ Γ E ab /Q is any element with u| F = j. We have a similar map on Weil groups and if we decompose with j ∈ Σ F and σ ∈ Γ E/Q we can define the F -plectic universal Taniyama element as follows: Here W E/F,f denotes the subgroup of W E/Q,f consisting of elements whose image in Γ E/Q lies in Γ E/F . The surjection ϕ f : Then h E F (α g ) simply stores this collection of elements of A × E,f /E × for each j ∈ Σ F and σ ∈ Γ E/Q in an appropriate way.
Remark. Let K be a CM field with maximal totally real subfield F . Then if E contains K we have a representation ρ Φ,F : T F ⊗ Q E → T K from §1 whose restriction to the diagonal is the classical reflex norm but which is general will not factor through the quotient T F ⊗ Q E ։ S E F . This representation can be described on character groups by the map might depend both on the choice of liftα g of α g and also on the choices we made in the construction, but can nevertheless be related to Nekovar's plectic-half transfer maps. More precisely, together Proposition 3.3 and Theorem 3.4 imply that for any i : under the Artin reciprocity map A × E,f /E × ։ Γ ab E is always equal to the image of F Φ (g) ∈ Γ ab K under the map Γ ab K → Γ ab E induced by i. Our next theorem justifies the choice of notation by showing that the F -plectic universal Taniyama element becomes well-defined independent of choices when we pass to the quotient T F ⊗E ։ S E F . The proof occupies most of the rest of this section, but is at heart a relatively straightforward generalisation of the calculations presented in [5] for Langlands' Taniyama element.
independent of the choice of liftα g ) and does not depend on the choice of i v and of coset representatives w σ for σ ∈ Γ E/Q .
Proof. Let us first fix a map i v as above, and letc ∈ W E/Q,f denote the associated lift of complex conjugation.
We need to show that f E F (α) only depends on α and not onα or the choices of i v and w σ for σ ∈ Γ E/Q . First we can show: does not depend of the choice of coset representatives w σ satisfying such that w cσ =cw σ for all σ ∈ Γ E/Q .
where b σ•j⊗σ = a j,σ , i.e. b j⊗σ = a σ −1 •j,σ . From §4, we have an algebraic action of W E/Q,f #W E/F,f on S E F via the quotient Γ E/Q #Γ E/F , and let χ ′ denote the character (α −1 ) * χ of S E F . If we write then by the definition of the algebraic action we have d j⊗(j(α)σ −1 ) −1 = b j⊗σ , i.e. d j⊗σ = b j⊗(j(α) −1 σ −1 ) −1 . If w = (w σ ) is replaced by w ′ = (w ′ σ ) with w ′ σ = w σ e σ and e σ ∈ A × E,f /E × then we have h ′ σ (j(α)) = (w ′ j(α)σ ) −1 j(α)(w ′ σ ) = e −1 j(α)σ e σ h σ (j(α)) and we note here that the conditions on w σ and w ′ σ (with respect toc) mean that e σ only depends on σ| E + , where E + denotes the maximal totally real subfield of E. Thus We want to show that this product is trivial for all χ. In order to prove the claim we may assume χ is the pull back of a character of S E attached to a CM type Φ ⊂ Γ E/Q via the projection S E F = (S E ) ΣF ։ T E coming from an embedding j 0 : F ֒→ E. In other words, we have all a j,σ ∈ {0, 1} and a j,σ is always zero if j = j 0 and we have a j0,σ + a j0,cσ = 1 for all σ ∈ Γ E/Q . Thanks to Proposition 4.1, we know that in this case if j ′ 0 : F ֒→ E is such thatαj ′ 0 = j 0 then we have c j,σ = 0 for all j = j ′ 0 and c j ′ 0 ,σ + c j ′ 0 ,cσ = 1 for all σ ∈ Γ E/Q . Thus if we let Φ and Φ ′ denote the CM types attached to χ and χ ′ = (α −1 ) * χ we have and similarly d σ = Φ ′ (σ). We deduce that χ(f E F (α)) changes by and this is just 1 since both Φ and Φ ′ are CM types of E and e σ only depends on σ| E + .
From Proposition 5.2, we can deduce: Here we are consider the arithmetic action of Gal(E/Q) coming from the action on A E,f points. Let τ be in Gal(E/Q). If we write T ΣF f /E × then the arithmetic action of τ on the left hand side translates into the action (e j,σ ) j,σ → (e τ j,τ −1 σ ) j,σ on the right hand side.
We define a new set of coset representatives w ′ σ by w ′ σ = w στ −1τ and by Proposition 5.2 we know f E F (α) can be computed as the image of We use a topological argument to conclude that the map f E F factors through Γ E ab /Q #Γ E ab /F . The group W E/Q,f #W E/F,f inherits a naturally topology coming from the wreath product structure for which the map f E F is continuous (this follows from our remarks right at the end of §2). It follows from Chevalley's theorem on units that the induced topology on S E (Q) ⊂ S E (A Q,f ) is the discrete topology (see [5], §1.2). As a consequence, S E (Q) is closed in S E (A Q,f ) and the quotient S E (A Q,f )/S E (Q) is totally disconnected.
Similarly, if E is Galois over Q, then the space S E (A E,f )/S E (E) Gal(E/Q) is totally disconnected, and this plays an important role in showing that Langlands' Taniyama element is well-defined (see [2], §9.9). It immediately follows that the product group S E F (A E,f )/S E F (E) Gal(E/Q) is also totally disconnected, and so it suffices to establish that the kernel of W E/Q,f #W E/F,f ։ Γ E ab /Q #Γ E ab /F is connected. If we fix a collections = (s j ) j∈ΣF of coset representatives for W E/F,f in W E/Q,f and let s = (s j ) j∈ΣF = (ϕ f (s j )) j∈ΣF be the induced collection of coset representatives for Γ E ab /F in Γ E ab /Q then the maps ρ s and ρs fit into a commutative diagram where the bottom line is induced by ϕ f on each component. It is clear that the kernel of the bottom line is connected with respect to the natural topology on S ΣF ⋉ W ΣF E/F,f and so by definition so is the kernel of the top line. We conclude that the map f E F does indeed factor through the surjection W E/Q,f #W E/F,f ։ Γ E ab /Q #Γ E ab /F . Finally we can show: Proof. Let i v , i ′ v be two continuous maps W R → W Q which corresponding to the place v of Q coming from the fixed embedding into C. Then i v and i ′ v differ by an inner automorphism of W Q by an element of Ker(ϕ) , and so ifc andc ′ the associated lifts of complex conjugation to W E/Q,f we know that we havec ′ = ucu −1 F,c and f E F,c ′ the maps constructed usingc andc ′ respectively. By definition f E F,c (α) is equal to the image of (w −1 j(α)σ · j(α)(w σ )) j⊗σ −1 whereα ∈ W E/Q,f #W E/F,f is any lift of α and the w σ are any coset representatives such that w cσ =cw σ .
If we set This competes the proof of Theorem 5.1.
We end this section by studying the properties of the F -plectic universal Taniyama element f E F as we vary the fields F and E.
Proposition 5.5. Let F ⊂ Q be a totally real number field and E ⊂ Q be CM, Galois over Q and containing F . Then: where the map u * on the right hand side is induced by the bijection Σ F ′ → Σ F given by j ′ → j = j ′ • u| F .
Proof. The inclusion Aut F (F ⊗ Q E ab ) ֒→ Aut F ′ (F ′ ⊗ Q E ab ) corresponds to the natural inclusion of plectic groups. Let α be an element of Γ E ab /Q #Γ E ab /F ⊂ Γ E ab /Q #Γ E ab /F ′ andα be a lift of α to an element of W E/Q,f #W E/F,f , which can view as a subgroup of The map u * on the right hand side of the diagram sends [j ′ ⊗ σ] → [j ⊗ σ] where j = j ′ • u| F . The map 4 Throughout this paper we have taken our totally real field F to be an embedded subfield of Q. Although this simplifies the exposition, it is not necessary. Indeed, by Proposition 5.5(ii), if F is any abstract totally real field and E ⊂ Q is CM, Galois over Q and splits F then the composite is independent of the choice of embedding j : F ֒→ E and so defines a canonical F -plectic universal Taniyama element.
[u] : Aut F (F ⊗ Q E ab ) → Aut F ′ (F ′ ⊗ Q E ab ) on the left hand side is induced by [ũ] on plectic groups, and the Proposition 5.6. Fix F and suppose E ′ ⊃ E is also CM and Galois over Q. There is a commutative diagram Proof. In terms of plectic groups, we need to show that we have a commutative diagram To this end let W Q ⊃ W E ⊃ W E ′ be the relevant Weil groups and let We are in the setting of Proposition 2.3 and in that notation we have X = G/H = Γ E/Q and X ′ = G/H ′ = Γ E ′ /Q and we set Y = H/H ′ . The transfer map [10], §1). Our fixed map i v : W C/R → W Q gives us an elementc ∈ W E/Q,f lifting complex conjugation c ∈ Γ E ′ /Q and whose image in W E/Q (which we'll denotec) does likewise for E. If we pick coset representatives w σ for H in G such that for each σ ∈ Γ E/Q we have w cσ =cw σ and any set s y for y ∈ Y of coset representatives for H ′ ⊂ H then • The set {w σ ′ } = {w σ s y : σ ∈ Γ E/Q , y ∈ Y } give a set of coset representatives for H ′ in G satisfying the relevant criterion forc (indeed w cσ s y =cw σ s y ). • The the images w σ in W E/Q,f = G/[H, H] give a set of coset representatives for and hence by Proposition 2.3 (we're using that j(α) The result then follows since if we let α ∈ Γ E ab /Q #Γ E ab /F be the reduction of α then the (j ⊗ σ −1 ) component of f E F (α) (computed with the coset representatives w σ ) is given by h j,σ = h σ (j(α)) ∈ A × E,f /E × .

A plectic Taniyama group
The data of Langlands' universal Taniyama element f E : Γ E ab /Q → S E (A E,f )/S E (E) (for E ⊂ Q a CM number field which is Galois over Q) is equivalent to the data of the level E Taniyama group together with its continuous finite-adelic splitting s E : Γ E ab /Q → T E (A Q,f ). This is a special case of the following more general formalism: Proposition 6.1. Let S be any torus split over a Galois extension E/Q and Γ a profinite group. Suppose we are given an 'algebraic' action Γ → Aut(S) which factors through a finite quotient of Γ and an extension inducing the given algebraic action of Γ on S together with a continuous finite-adelic splitting s : Γ → T (A Q,f ). Pick a section a : Γ → T /E that is a morphism of pro-algebraic varieties (this exists as S is split over E), and define , is independent of the choice of a and satisfies the following three properties: (1) b is a cocycle for the algebraic action of Γ on S( where ⋆ denotes the given algebraic action is locally constant Conversely any map b : Γ → S(A E,f )/S(E) satisfying (1), (2) and (3) arises from a unique extension of Γ by S (together with a continuous finite-adelic splitting).
Proof. This is proven in [5] Proposition 2.7 for Γ = Γ E ab /Q acting on S = S E by the algebraic action defined in §4. However, as noticed by Fargues [2] and Nekovář [6], the argument is not specific to that case.
In the above setting, if S ′ is another torus split by E and S → S ′ is any homomorphism then the extension defined by the composite map Γ Similarly the pull back of T via a homomorphism Γ ′ → Γ corresponds to the composite [2], §9).
Example. Let E ⊂ Q be a CM field which is Galois over Q. Under Proposition 6.1, Langlands' level E Taniyama extension which can be checked to satisfy the properties (1), (2) and (3).
Previously, Serre [9] had constructed an extension 1 → S E → ǫ E → Γ ab E → 1 together with a continuous finite-adelic splitting which was related to the theory of algebraic Hecke characters. Under Proposition 6.1 this corresponds to the homomorphism One can check from the definition that f E | Γ ab E = N E , i.e. the Serre extension is the pull back of the Taniyama extension via Γ ab E ֒→ Γ E ab /Q . Similarly, if F ⊂ E is totally real and g = (g j ) j∈ΣF lies in the subgroup Γ ΣF E ab /E ⊂ Aut F (F ⊗ Q E ab ) then α g : Γ E ab /Q → Γ E ab /Q is the bijection given by left multiplication by g j on {g ∈ Γ E ab /Q : g| F = j}. If we let N E F denote the componentwise map N E F = (N E , . . . , N E ) : Γ ΣF E ab /E → S E F (A Q,f )/S E F (Q) one can check straight from the definition that if g ∈ Γ ΣF E ab /E then f E F (g) = N E F (g).
It follows immediately from the above proposition that b E F satisfies property (1) of Proposition 6.1. On the other hand property (2) follows immediately from Proposition 5.3. Therefore, it only remains to prove: Proof. In [5], it is shown that there is a finite abelian extension E ′ /E and a continuous homomorphism b : Γ E ab /E ′ → S E (A Q,f ) fitting in a diagram The map (b, . . . , b) : Γ ΣF E ab /E ′ → S E F (A Q,f ) ⊂ S E F (A E,f ) can be extended to a continuous map b E F : Aut F (F ⊗ E ab ) → S E F (A E,f ) lifting b E F by, for example, picking coset representatives (see [5], §2.10). This b E F satisfies the condition of the proposition as when restricted to the open subgroup Γ ΣF E ab /E ′ (which acts trivially) it is a homomorphism.
This completes the proof of Theorem 6.2.
Finally we can construct the full F -plectic Taniyama group. It follows from Proposition 5.6 and Proposition 6.1 that whenever E ⊂ E ′ we have a transition map T E ′ F → T E F which sits in a diagram and respects the finite-adelic splittings. We can pull back each extension T E F by Aut F (F ⊗ Q Q) ։ Aut F (F ⊗ Q E ab ), and in the limit we obtain an extension of the form together with a continuous finite-adelic splitting s F : Aut F (F ⊗ Q Q) → T F (A Q,f ), where here S F denotes the F -plectic Serre group S ΣF . When F = Q this is precisely the definition of Langlands' Taniyama group T . Moreover, Proposition 5.5 ensures that the construction is functorial in F . A map F → F ′ of totally real fields induces a homomorphism T F → T F ′ restricting to the natural map on Serre groups and inducing the inclusion Aut F (F ⊗ Q) ֒→ Aut F ′ (F ′ ⊗ Q). In particular, there is a canonical homomorphism from Langlands' Taniyama group T to the the F -plectic Taniyama group T F for any totally real number field F .