Automorphic Representations, Shimura Varieties, and Motives. Ein Märchen*

R.P. Langlands

1. Introduction. It had been my intention to survey the problems posed by the study of zeta-functions of Shimura varieties. But I was too sanguine. This would be a mammoth task, and limitations of time and energy have considerably reduced the compass of this report. I consider only two problems, one on the conjugation of Shimura varieties, and one in the domain of continuous cohomology. At first glance, it appears incongruous to couple them, for one is arithmetic, and the other representation-theoretic, but they both arise in the study of the zeta-function at the infinite places.

The problem of conjugation is formulated in the sixth section as a conjecture, which was arrived at only after a long sequence of revisions. My earlier attempts were all submitted to Rapoport for approval, and found lacking. They were too imprecise, and were not even in principle amenable to proof by Shimura’s methods of descent. The conjecture as it stands is the only statement I could discover that meets his criticism and is compatible with Shimura’s conjecture.

The statement of the conjecture must be preceded by some constructions, which have implications that had escaped me. When combined with Deligne’s conception of Shimura varieties as parameter varieties for families of motives they suggest the introduction of a group, here called the Taniyama group, which may be of importance for the study of motives of CM-type. It is defined in the fifth section, where its hypothetical properties are rehearsed.

With the introduction of motives and the Taniyama group, the report takes on a tone it was not originally intended to have. No longer is it simply a matter of formulating one or two specific conjectures, but we begin to weave a tissue of surmise and hypothesis, and curiosity drives us on.

Deligne’s ideas are reviewed in the fourth section, but to understand them one must be familiar at least with the elements of the formalism of tannakian categories underlying the conjectural theory of motives, say, with the main results of Chapter II of [40].

* First appeared in Automorphic forms, representations, and L-functions, Proc. of Symp. in Pure Math. XXXIII (1977)

The present Summer Institute is predicated on the belief that there is a close relation between automorphic representations and motives. The relation is usually couched in terms of L-functions, and no one has suggested a direct connection. It may be provided by the principle of functoriality and the formalism of tannakian categories. This possibility is discussed in the highly speculative second section.

However there is a small class of automorphic representations which are certainly not amenable to this formalism. I have called them anomalous, and in order to make their significance clearer I have discussed an example of Kurokawa at length in the third section. Although the anomalous representations form only a small part of the collection of automorphic representations, they are frequently encountered, especially in the study of continuous cohomology, and so we have come full circle. Our long divagation has not been in vain, for we have acquired concepts that enable us to appreciate the global significance of the local examples described in the seventh section, which deals with the second of our original two problems.

At all events, I have exceeded my commission and been seduced into describing things as they may be and, as seems to me at present, are likely to be. They could be otherwise. Nonetheless it is useful to have a conception of the whole to which one can refer during the daily, close work with technical difficulties, provided one does not become too attached to it, but takes pains to ensure that it continues to conform to the facts, and is prepared to abandon it when that is called for. The views of this report are in any case not peculiarly mine. I have simply fused my own observations and reflections with ideas of others and with commonly accepted tenets.

I have also wanted to draw attention to the specific problems, on which expert advice would be of great help, and I hope that the report is sufficiently loosely written that someone familiar with continuous cohomology but not with arithmetic can turn to the seventh section, overlooking the first few pages, and see what the study of Shimura varieties needs from that theory, or that someone familiar with Shimura varieties but not automorphic representations or motives will be able to find the definition of the Serre group in the fourth section and the Taniyama group in the fifth, and then turn to read the sixth section.

Finally a word about what is not discussed in this report. The investigations of Kazhdan [26] and Shih [46] on conjugation of Shimura varieties are not reviewed; their bearing on the problem formulated here is not yet clear. L-indistinguishability is not discussed. Little is known, and that is described in another lecture [44]. Problems caused by noncompactness are ignored. There is a tremendous amount of material on compactification and on the cohomology of noncompact quotients, but no one has yet tried to bring it to bear on the study of the zeta-functions. The omission I regret most is that of a discussion of the reduction of the Shimura varieties modulo a prime [36]. Here there is a great deal to be said, especially about the structure of the set of geometric points in the algebraic closure of a finite field, starting with the work of Ihara on curves. I hope to report on this topic on another occasion.

2. Automorphic representations. Our present knowledge does not justify an attempt to fix a language in which the relations of automorphic representations with motives are to be expressed. Nonetheless that of tannakian categories [40] appears promising and it might be worthwhile to take a few pages to draw attention to the problems to be solved before it can be applied in the study of automorphic representations.

We first recall the rough classification of irreducible admissible representations of GL(n, F), F being a local field, and of automorphic representations of GL(n, A_{F}), F being a global field, reviewed in some of the other lectures ([3], [5], [35], [48], cf. also [7], [31]). In either case the representation π has a central character ω and z → |ω(z)| may be extended uniquely from Z(F), or Z(A_{F}), to a positive character ν of GL(n, F), or of GL(n, A_{F}).

If F is a local field then π is said to be cuspidal if for any K-finite vectors u and v, in the space of π and its dual, ν^{−1}(g)⟨π(g)u, v⟩ is square-integrable on the quotient Z(F)\GL(n, F). To construct an arbitrary irreducible admissible representation one starts from a partition {n_{1}, ⋯, n_{r}} of n and cuspidal representations π_{1}, ⋯, π_{r} of GL(n_{i}, F). If ω_{i} is the central character of π_{i}, there is a real number s_{i} such that |ω_{i}(z)| ≡ |z|^{s_{i}} if z lies in the centre of GL(n_{i}, F), a group isomorphic to F^{×}. Changing the order of the partition, one supposes that s_{1} ≧ ⋯ ≧ s_{r}. The partition defines a standard parabolic subgroup P of GL(n) and σ = ⊗π_{i} a representation of M(F), because the Levi factor M of P is isomorphic to GL(n_{1}) × ⋯ × GL(n_{r}). The representation σ yields in the usual way an induced representation I_{σ} of G(F). I_{σ} may not be irreducible, but it has a unique irreducible quotient, which we denote π_{1} ⊞ ⋯ ⊞ π_{r}. Every representation is of this form and π_{1} ⊞ ⋯ ⊞ π_{r} ≃ π′_{1} ⊞ ⋯ ⊞ π′_{s} if and only if r = s and after renumbering π_{i} ≃ π′_{i}. Thus every representation can be represented uniquely as a formal sum, in the sense of this notation, of cuspidal representations. The representation π_{1} ⊞ ⋯ ⊞ π_{r} is said to be tempered if all the s_{i} are 0. We can clearly define, in a formal manner, the sum of any finite number of representations.

We can in fact formally define an abelian category Π(F) whose collection of objects is the union over n of the irreducible, admissible representations of GL(n, F). If π = π_{1} ⊞ ⋯ ⊞ π_{r}, π′ = π′_{1} ⊞ ⋯ ⊞ π′_{s}, with π_{i} and π′_{j} cuspidal, we set

Hom(π, π′) = ⊕_{{i,j|π_{i}∼π′_{j}}} C.

The composition is obvious. The tempered representations form a subcategory Π^{∘}(F).

If F is a global field then π is said to be cuspidal if π is a constituent of the representation of GL(n, A_{F}) on the space of measurable cusp forms φ satisfying

(a) φ(zg) = ω(z)φ(g), z ∈ Z(A_{F}),

(b) ∫_{Z(A_{F})G(F)\G(A_{F})} ν^{−2}(g)|φ(g)|^{2}dg < ∞.

If n_{1} ⋯ n_{r} is a partition of n and π_{1}, ⋯, π_{r} cuspidal representations of GL(n_{i}, A_{F}) we may again change the order so that s_{1} ≧ ⋯ ≧ s_{r} and then construct the induced representation I_{σ}. Every automorphic representation is a constituent of some I_{σ}. For an adequate classification, one needs more. The following statement may eventually result from the investigations of Jacquet, Shalika, and Piatetski-Shapiro, but has not yet been proved in general.

A. If π is a constituent of I_{σ} and of I_{σ′} then the partitions {n_{1}, ⋯, n_{r}} and {n′_{1}, ⋯, n′_{s}} have the same number of elements, and, after a renumbering, n_{i} = n′_{i} and π_{i} ∼ π′_{i}.

If π_{i} = ⊗_{v}π_{i}(v) then one constituent of I_{σ} is the representation π = ⊗_{v}π(v) with local components π(v) = π_{1}(v) ⊞ ⋯ ⊞ π_{r}(v). This representation will be denoted π = π_{1} ⊞ ⋯ ⊞ π_{r}, but the notation is not justified until statement A is proved. The representations of this form will be called isobaric and can again be used to define an abelian category Π(F).

We agree to call π = π_{1} ⊞ ⋯ ⊞ π_{r} tempered if each of the cuspidal representations π_{i} has a unitary central character, that is, if each of the s_{i} is 0. However the language is only justified if we can prove the following statement, the strongest form of the conjecture of Ramanujan to which the examples of Howe-Piatetski-Shapiro and Kurokawa allow us to cling.

B. If π = ⊗π(v) is a cuspidal representation with unitary central character then each of the factors π(v) is tempered.

The tempered representations form a subcategory Π^{∘}(F) of Π(F).

It is clear that we have tried to define the categories Π(F) and Π^{∘}(F) in such a way that if v is a place of F there are functors Π(F) → {Π(F_{v}) and Π^{∘}(F) → Π^{∘}(F_{v}) taking π to its factor π(v). However without a natural definition of the arrows, there is no unique way to define Hom(π, π′) → Hom(π(v), π′(v)).

If I_{σ} is not irreducible it will have other constituents in addition to π_{1} ⊞ ⋯ ⊞ π_{r}. These automorphic representations will be called anomalous. Although the principle of functoriality may apply to them, there is considerable doubt that they can be fitted into a tannakian framework. Observe that statement A and the strong form of multiplicity one imply that if π is any automorphic representation there is a unique isobaric representation π′ such that π(v) ∼ π′(v) for almost all v.

If F is a global field the purpose of the tannakian formalism would be to provide us with a reductive group over C whose n-dimensional representations, or rather their equivalence classes, are to correspond bijectively to the isobaric automorphic representations of GL(n, A_{F}). This hypothetical group will have to be very large, a projective limit of finite-dimensional groups. We denote it by G_{Π(F)}.

The category Rep(G_{Π(F)}) of the finite-dimensional representations over C of the algebraic group G_{Π(F)} would certainly be abelian, but in addition it is a category in which tensor products can be defined. Moreover there is a functor to the category of finite-dimensional vector spaces over C. If (φ, X), consisting of the space X and the representation φ of G_{Π(F)} on it, belongs to Rep(G_{Π(F)}) one simply ignores φ. The tensor product satisfies certain conditions of associativity, commutativity, and so on, and the functor, called a fibre functor, is compatible with tensor products and other operations of the two categories.

A theorem of [40], but not the principal one, asserts that, conversely, an abelian category with tensor products and a fibre functor is equivalent to the category of representations of a reductive group, provided certain natural axioms are satisfied. Thus it appears that if we are to be able to introduce G_{Π(F)} we will have to associate to each pair consisting of a cuspidal representation π of GL(n, A) and a cuspidal representation π′ of GL(n′, A) an isobaric representation π ⊠ π′ of GL(nn′, A). In general, if π = π_{1} ⊞ ⋯ ⊞ π_{r} and π′ = π_{1} ⊞ ⋯ ⊞ π′_{s} we would set

π ⊠ π′ = ⊞_{i,j}(π_{i} ⊠ π′_{j}).

In addition, we will have to associate to each isobaric representation π of GL(n, A), n = 1, 2, ⋯, a complex vector space X(π) of dimension n, together with isomorphisms

X(π ⊞ π′) ≃ X(π) ⊕ X(π′) X(π ⊠ π′) ≃ X(π) ⊗ X(π′).

There are a large number of conditions to be satisfied, among them one which is perhaps worth mentioning explicitly. Suppose π and π′ are cuspidal and π ⊠ π′ = ⊞_{i=1}^{r}π_{i} with π_{i} cuspidal.

Then the set {π_{i}, ⋯, π_{r}} contains the trivial representation of GL(1, A) if and only if π′ is the contragredient of π, when it contains this representation exactly once.

At the moment I have no idea how to define the spaces X(π); indeed, no solid reason for believing that the functor π → X(π) exists. Even though the attempt to introduce the groups G_{Π(F)} may turn out to be vain the prize to be won is so great that one cannot refuse to hazard it. One would like to show, in addition, that if π and π′ are tempered then π ⊠ π′ is also, and thus be able to introduce a group G_{Π^{∘}(F)} classifying the tempered automorphic representations. If Ω_{F}^{+} is the group of multiplicative type whose module of rational characters is the module of positive characters of the topological group F^{×}\I_{F} then G_{Π(F)} will be a direct product G_{Π^{∘}(F)} × Ω_{F}^{+}.

One will also wish to introduce, by a similar process, groups G_{Π(F)} and G_{Π^{∘}(F)}, attached to a local field F and classifying the irreducible, admissible representations of GL(n, F), n = 1, 2, ⋯, and the tempered representations of GL(n, F). The formalism is clearly intended to be such that if F_{v} is a completion of F there are homomorphisms G_{Π(F_{v})} → G_{Π(F)} and G_{Π^{∘}(F_{v})} → G_{Π^{∘}(F)} dual to Π(F) → Π(F_{v}) and Π^{∘}(F) → Π^{∘}(F_{v}).

If F is a local field the conjectured classification of the representations of GL(n, F) [3], verified when F is archimedean, provides a concrete description of the category Π(F) with its product ⊠. If F is archimedean and W_{F} is the Weil group of F then Π(F) is equivalent to the category of continuous semisimple representations σ of W_{F} on complex vector spaces X. The tensor product is the usual one (σ, X) ⊗ (σ′, X′) = (σ ⊗ σ′, X ⊗ X′) and the fibre functor is (σ, X) → X.

Thus G_{Π(F)} is a kind of algebraic hull of the topological group W_{F}. In particular there is a homomorphism W_{F} → G_{Π(F)}(C) whose image is Zariski-dense. The subcategory corresponding to Π^{∘}(F) is obtained by taking only those (σ, X) for which the image of σ(W_{F}) is relatively compact.

If F is nonarchimedean one should take not the Weil group but a direct product W′_{F} = SL(2, C) × W_{F}.

Conjecturally at least, σ is to be replaced by a continuous, semisimple representation of W′_{F} whose restriction to SL(2, C) is complex analytic. To obtain a category equivalent to Π^{∘}(F) one should take only those σ for which σ(W_{F}) is relatively compact. Observe that in order to obtain a semisimple category we have replaced the group W D_{F} employed by Borel and Tate [47] by the group W′_{F}. If w → |w| is the usual positive character of the Weil group, there is an obvious homomorphism of W D_{F} into W′_{F} which takes w ∈ W_{F} ⊆ W D_{F} to

( \begin{matrix}|w|^{1/2} & 0 \\ 0 & |w|^{−1/2} \end{matrix} ) × w.

Notice that according to these classifications there are homomorphisms of algebraic groups

G_{Π(F)} → Gal(F̄/F) and G_{Π^{∘}(F)} → Gal(F̄/F),

the group on the right being a projective limit of finite groups. The principle of functoriality cannot be valid unless there are similar homomorphisms when F is a global field.

Let Ω_{F} , Ω_{F}^{+}, Ω_{F}^{0} , be the groups of multiplicative type whose modules of rational characters are, respectively, the module of all characters of F^{×}, or F^{×}\I_{F} if F is global, of all positive characters, or of all unitary characters. Then Ω_{F} = Ω_{F}^{+} × Ω_{F}^{0} and there will be homomorphisms

G_{Π(F)} → Ω_{F} , G_{Π^{∘}(F)} → Ω_{F}^{0} .

If F is nonarchimedean and local we may also define Ω_{un}, Ω_{un}^{+}, and Ω_{un}^{0}, by replacing the modules of characters of various types by the modules of unramified characters of the same type. The groups Ω_{un}, Ω_{un}^{+}, and Ω_{un}^{0} contain a distinguished point over C, the Frobenius Φ, which is simply the image of a uniformizing parameter in F^{×}. In any case the formalism will certainly allow us to introduce for any representation σ of the algebraic group G_{Π(F)} over C an L-function L(s, σ) and if σ corresponds to the representation π of GL(n, A_{F}) then L(s, σ) = L(s, π).

But the reasons for wishing to introduce the groups G_{Π(F)} and G_{Π^{0}(F)} and the associated formalism are not simply, or even primarily, aesthetic. There are problems which will be difficult to formulate exactly without them. Suppose, for example, that π = ⊗_{v}π_{v} is an isobaric representation of GL(n, A) and each of the factors π_{v} is tempered. For almost all v, π_{v} is unramified and associated to a conjugacy class {g_{v}} = {g(π_{v})} in GL(n, C). Since π_{ν} is supposed tempered this class meets the unitary group U(n) and I may, as I prefer, regard it as a conjugacy class in U(n). The general analytic analogue of the Tchebotarev theorem or the Sato-Tate conjecture would be a theorem or conjecture describing the asymptotic distribution of the classes {g_{v}}.

Suppose the formalism existed and π were associated to a representation σ of G_{Π^{∘}(F)}. The image σ(G_{Π^{∘}(F)}(C)) would be a reductive subgroup H(C) of GL(n, C) with a maximal compact subgroup K_{H}. For almost all v, σ_{v} would factor through G_{Π^{∘}(F_{v})} → Ω_{un}^{0} and σ_{v}(Φ_{v}) would be defined. Since its conjugacy class in H(C) would meet K_{H}, it would define a conjugacy class in K_{H}, which we denote {σ_{v}(Φ_{v})}. Of course {σ_{v}(Φ_{v})} ⊆ {g_{v}} and the asymptotic distribution of the classes {g_{v}} can be inferred from that of the classes {σ_{v}(Φ_{v})}. There is a natural probability measure on the space of conjugacy classes in K_{H}. If X is a set of conjugacy classes and if the set Y = ∪_{x∈X}x is measurable in K_{H}, one takes meas X = meas Y. It is natural to suppose that it is this measure which defines the asymptotic distribution of the classes {σ_{v}(Φ_{v})}. To verify the supposition, it will be necessary to establish that if ρ is any representation of G_{Π^{∘}(F)} over C then the order of the pole of L(s, ρ) at s = 1 is equal to the multiplicity with which the trivial representation of G_{Π^{∘}(F)} occurs in ρ. If the existence of G_{Π^{∘}(F)} were established, it would be easy enough to deduce this from the recent results of Jacquet and Shalika [25].

Within this formalism, the principle of functoriality asserts that if F is a local field and G a reductive group over F then any L-packet of representations of G(F) is associated to a homomorphism φ : G_{Π(F)} → ^{L}G of algebraic groups over C for which

G_{Π(F)}  ⟶^{φ}  ^{L}G
↘  ↙
Gal(\overline{F} /F)

is commutative. If G_{Π(F)} is replaced by G_{Π^{∘}(F)}, the L-packet should be tempered. If F is global, some caution will have to be exercised. If the L-packet Π consists of π = ⊗π_{v} for which the π_{v} are always tempered, it should correspond to a φ : G_{Π^{∘}(F)} → ^{L}G. Otherwise this may not be so, for a reason which will perhaps be clearer after an example of Kurokawa [29] is discussed in the next section. If there is a representation

ψ : ^{L}G → GL(n) × Gal(F̄/F)

and if the image ψ_{∗}(Π) of Π given by the principle of functoriality is not isobaric then Π can be associated to no φ.

One may nonetheless hope to prove, both locally and globally, that to each φ : G_{Π(F)} → ^{L}G is associated an L-packet, provided φ commutes with the homomorphisms to the Galois group. If G is not quasi-split the local behaviour of φ with respect to parabolic subgroups will also have to be taken into account [3]. For archimedean fields one recovers the usual classification.

The few examples studied [44] suggest that questions about the multiplicity with which elements of Π occur in the space of automorphic forms will have to be answered in terms of φ.

The principal reason for wishing to define the group G_{Π(F)} is that it provides the only way visible at present to express completely the relation between automorphic forms and the conjectural theory of motives [40]. The category of motives over F, a local or a global field, is Q-linear and tannakian, but it does not always possess a fibre functor over Q and seldom a single naturally defined one. Thus tannakian duality associates to it not a group over Q but a group-like object, a “gerbe” in the rustic terminology which has become so popular in recent years. Over C this object becomes a group G_{Mot(F)}, and the relations between motives and automorphic representations will probably be adequately expressed by the existence of a homomorphism ρ_{F} : G_{Π(F)} → G_{Mot(F)} defined over C. The field F can be local or global. The local and global homomorphisms are to be compatible with each other and with the formation of L-functions. Both the image and the kernel of ρ_{F} will probably be rather large when F is a number field (cf. C.6.2 of [43]) but rather small when F is local.

3. Anomalous representations. Since the anomalous representations cause some difficulty in the study of the zeta-functions of Shimura varieties, it will be useful to acquire some feeling for them before going on. The brief remarks of the previous section suggest that an automorphic representation π of the reductive group G, or rather the L-packet Π containing it, should be called anomalous if for some homomorphism ψ : ^{L}G → GL(n, C)×Gal(F̄/F) the principle of functoriality takes π or Π to an anomalous representation of GL(n, A). It may be that the counter-examples to the Ramanujan conjecture of Howe-Piatetski-Shapiro [19] are anomalous in this sense.

Since their paper is not available to me as I write, I have to test this suggestion on another example, discovered by Kurokawa and quite explicit. The group G is to be the projective symplectic group in four variables over Q. The L-group is then the direct product of ^{L}G^{∘}, the symplectic group in four variables and the Galois group Gal(Q̄/Q). Since all the groups with which we shall deal in this section will be split, we may ignore the factor Gal(Q̄/Q).

Let G_{1} be the product of P GL(2) over Q with itself, so that ^{L}G_{1}^{∘} = SL(2, C) × SL(2, C) and let G_{2} be GL(4) over Q. Define φ_{1} : ^{L}G_{1}^{∘} → ^{L}G to be the homomorphism

( \begin{matrix}α_{1} & β_{1} \\ γ_{1} & δ_{1} \end{matrix} ) × ( \begin{matrix}α_{2} & β_{2} \\ γ_{2} & δ_{2} \end{matrix} ) → [ \begin{matrix}α_{1} & 0 & β_{1} & 0 \\ 0 & α_{2} & 0 & β_{2} \\ γ_{1} & 0 & δ_{1} & 0 \\ 0 & γ_{2} & 0 & δ_{2} \end{matrix} ]

and let φ_{2} be the standard imbedding of ^{L}G^{∘} in ^{L}G_{2}^{∘} which is GL(4, C).

In order to analyze Kurokawa’s example one must formulate his statements representationtheoretically. I state the facts necessary to this purpose, but have to ask that the reader understand the discrete series sufficiently well to verify them for himself. There is nothing to prove. It is simply a question of writing down explicitly for the special case of concern to us here some of the results of [17] and [18], and some definitions from [31].

An automorphic representation π = ⊗π_{v} of G(A) is associated to a holomorphic form of weight k in the classical sense of [39] if and only if π_{∞} is a member of the holomorphic discrete series and lies in an L-packet Π_{ψ(k)}, where the restriction of ψ(k) to C^{×} ⊆ W_{C/R} has the form

z → [ \begin{matrix}z^{λ}z̄^{−λ} \\ & z^{μ}z̄^{−μ} \\ & & z^{−λ}z̄^{λ} \\ & & & z^{−μ}z̄^{μ} \end{matrix} ]

with λ = (2k − 3)/2, μ = \frac{1}{2}. Since G is the projective group, k must be even. Notice that z^{λ}z̄^{−λ} is to be calculated as z^{2λ}(zz̄)^{−λ} = z̄^{−2λ}(zz̄)^{λ}. We will eventually take k = 10.

On the other hand an automorphic representation π = ⊗π_{ν} of P GL(2, A) is associated to a holomorphic form of weight 2k − 2 if and only if π_{∞} belongs to the discrete series and to an L-packet Π_{φ(k)}, where

φ(k) : z → ( \begin{matrix}z^{λ}z̄^{−λ} \\ & z^{−λ}z̄^{λ} \end{matrix} )

with λ as above. In particular φ_{1} takes the L-packet Π_{φ(k)} ⊗ Π_{φ(2)}, consisting in fact of a single representation, to Π_{ψ(k)}.

We want to apply the principle of functoriality to φ_{1} and a very special automorphic representation π of G_{1}(A). π must be a tensor product π′ ⊗ π′′ of two representations of P GL(2, A). π′_{∞} and π′′_{∞} will both be members of the discrete series, the first in Π_{φ(10)} and the second in Π_{φ(2)}. In fact π′ will be the automorphic representation associated to the cusp form of weight 18, but π′′ will be anomalous or, to be more exact, its pull-back π̃′′ to GL(2, A) will be anomalous. To construct it we begin with the partition {1, 1} of 2 and the two characters

η : x → |x|^{1/2}, ν : x → |x|^{−1/2}

of GL(1, A), and then construct the induced representation of GL(2, A) as in the preceding section. Any constituent π̃′′ of the induced representation factors through a representation π̃′′ of P GL(2, A). We so choose π̃′′ that π′′_{∞} ∈ Π_{φ(2)} while π̃′′_{p} = η_{p} ⊞ ν_{p} for all p. The representation π̃′′ is clearly anomalous.

The representation π is unramified. Thus the automorphic representation π^{∘} of G(A) lies in the L-packet φ_{1^{∗}}({π}) defined by π, φ_{1} and the principle of functoriality if π_{p}^{∘} is unramified for all p and π_{p}^{∘} ∈ φ_{1^{∗}}({π_{p}}), and π_{∞}^{∘} is in Π_{ψ(10)}. We take π^{∘} to be the representation defined by the cusp form χ_{10} of weight 10 [39]. Then π_{∞}^{∘} lies in Π_{ψ(10)}. According to the definitions [3] the relation π_{p}^{∘} ∈ φ_{1^{∗}}({π_{p}}) is a statement about eigenvalues of Hecke operators. These statements have been verified for small primes by Kurokawa [29]. The necessary equalities are too complicated to be merely coincidences, and we may assume with some confidence that π^{∘} ∈ φ_{1^{∗}}({π}).

In any case the representation π^{∘} certainly is a counterexample to Ramanujan’s conjecture. If φ = φ_{2} ∘ φ_{1} then the principle of functoriality yields the same L-packet when applied to π and φ as it does when applied to π^{∘} and φ_{2}. Since G_{2} is GL(4) the L-packet consists of a single representation. It is easily seen to be anomalous and to be equivalent almost everywhere to the isobaric representation π′⊞η⊞ν. Thus π^{∘} itself is anomalous in the sense described at the beginning of the section.

4. Shimura varieties. In this section we review the definition of Shimura varieties, taken for now over C, and their relations with motives. The point of view is Deligne’s and most of what follows has been taken from his papers [9], [10], or learned in conversation with him. Of course the book of Saavedra Rivano [40] has again been a basic reference; many of the facts and definitions below will be found in it.

Recall that the data needed to define a Shimura variety are a connected reductive group G over Q and a homomorphism h : R → G defined over R. The symbol R is used to denote the group Res_{C/R}GL(1). Thus we have a canonical isomorphism GL(1) × GL(1) ≃ R over C and we may speak of the restriction of h to the first or the second factor. It is not h which matters but the set

ℌ = {Ad g ∘ h | g ∈ G(R)}

and it will be best simply to let h denote an arbitrary element of ℌ.

Recall that the pair (G, h) is subject to three conditions [10, §1.5]:

(a) If w is the diagonal map GL(1) → GL(1)×GL(1) then the homomorphism h∘w is central.

(b) The Lie algebra 𝔊 of G(C) is a direct sum 𝔊 = 𝔭 + 𝔨 + 𝔭̄ and if (z_{1}, z_{2}) ∈ R(C) then

ad h(z)(X) = z_{1}^{−1}z_{2}X,  X ∈ 𝔭,
= X,  X ∈ 𝔨,
= z_{1}z_{2}^{−1}X,  X ∈ 𝔭̄.

In fact the summands 𝔭, 𝔨, and 𝔭̄ vary with h, and when it is useful to make the dependence on h explicit we write 𝔭_{h}, 𝔨_{h}, and 𝔭̄_{h}.

(c) The adjoint action of h(i, −i) on the adjoint group is a Cartan involution.

Since G(R) acts on the real manifold ℌ by conjugation every element of 𝔊 defines a complex vector field on ℌ. Let X_{h} be the value of the vector field associated to X at h ∈ ℌ. The complex structure on ℌ is so defined that the holomorphic tangent space at h is {X_{h} | X ∈ 𝔭̄} and the antiholomorphic tangent space is {X_{h} | X ∈ 𝔭}.

If A_{f} is the ring of adèles over Q whose component at infinity is 0 and K is an open compact subgroup of G(A_{f}) then G(A_{f})/K is discrete and X_{K} = ℌ × G(A_{f})/K is a complex manifold on which G(Q) acts to the left. If K is sufficiently small then any γ ∈ G(Q) with a fixed point lies in Z(Q) ∩ K and thus fixes the whole manifold. We shall always assume that K is sufficiently small and then

Sh_{K}(C) = G(Q)\ℌ × G(A_{f})/K

is a complex manifold, proved by Baily-Borel to be the set of complex points on an algebraic variety Sh_{K} = Sh_{K}(G, h) = Sh_{K}(G, ℌ) over C.

Deligne anticipates that Sh_{K} will often be a moduli space for a family of motives over C. This is sometimes so, the motives then being those attached to abelian varieties, but can certainly not yet be proved in general. Nonetheless there is a good deal to be learned from a rehearsal of the considerations that suggest such an interpretation of Sh_{K}. In essence one observes that Sh_{K}(C) is the parameter space for a family of polarized Hodge structures; the difficulty is to show that these Hodge structures all arise from motives.

A real Hodge structure V is a finite-dimensional vector space V_{R} over R together with a decomposition of its complexification V_{C} = ⊕_{p,q∈Z}V^{p,q}, satisfying V^{q,p} = V̄^{p,q}. The collection of real Hodge structures forms a tannakian category over R whose associated group is R. Indeed to a real Hodge structure V, one associates the representation σ of R defined by

(4.1)  σ(z_{1}, z_{2})v = z_{1}^{−p}z_{2}^{−q}v, v ∈ V^{p,q}.

The relations V^{q,p} = V̄^{p,q} imply that σ is defined over R. Conversely each representation of R that is defined over R yields a Hodge structure, the elements of V^{p,q} being defined by (4.1). The real Hodge structure V is said to be of weight n if V^{p,q} = 0 whenever p + q ≠ n. Certainly any Hodge structure is a direct sum V = ⊕_{n}V^{n} with V^{n} of weight n.

We are however interested in the category of polarized rational Hodge structures. A rational Hodge structure V is formed by a finite-dimensional vector space V_{Q} over Q, a direct sum decomposition V_{Q} = ⊕VQ^{n}, and real Hodge structures of weight n on VR^{n} = VQ^{n} ⊗_{Q} R. There is a distinguished object of weight −2, the Tate object Q(1), in the category of rational Hodge structures. The underlying rational vector space is Q(1)_{Q} = 2πiQ ⊆ C and, by definition,

Q(1)^{−1,−1} = Q(1)_{C}.

It seems to be customary to identify the underlying vector space of Q(n) = Q(1)^{⊗n} with (2πi)^{n}Q and Q(n)_{R} with (2πi)^{n}R ⊆ C. The factors 2πi have been chosen for reasons which need not concern us. It is no trouble to carry them along.

If V is a rational Hodge structure and σ the associated representation of R let C be σ(−i, i) acting on V_{R}. If V is of weight n, a polarization of V is a bilinear form P : V × V → Q(−n) satisfying:

(a) For all u and ν in V_{C} and all r ∈ R(C)

P (σ(r)u, σ(r)ν) = σ(r)P (u, ν).

Thus the form is compatible with the Hodge structures.

(b) P (v, u) = (−1)^{n}P (u, v).

(c) The real-valued form (2πi)^{n}P (u, Cv) on V_{R} is symmetric and positive-definite.

A rational Hodge structure is said to be polarizable if each of its homogeneous components admits a polarization, a polarization of the full structure being defined by polarizations of the homogeneous components. The category HOD(Q) of polarizable Hodge structures is tannakian, with a natural fibre functor ω_{Hod} : V → V_{Q} and an associated group G_{Hod}, reductive but overwhelmingly large. It does have factor groups of manageable size.

If V is a polarizable rational Hodge structure, one may take the tannakian category generated by V and Q(1) and the repeated formation of duals, sums, tensor products, and subobjects. The associated group is called the Mumford-Tate group of V and denoted by MT (V). It is finitedimensional and reductive, and there is a surjection G_{Hod} → MT (V) defined over Q. If σ is the representation of R attached to V then MT (V) is simply the smallest subgroup of the group of automorphisms of the rational vector space underlying V which contains σ(R) and is defined over Q [37]. It is consequently connected.

The polarizable rational Hodge structures for which MT (V) is abelian play a particularly important role in the study of Shimura varieties. They are said to be of CM type. The second description of the groups MT (V) shows that the category of such Hodge structures is closed under sums and tensor products and thus is a tannakian category. The associated group has been studied at length in [41] and is often called the Serre group. At the risk of making a comparison with [41] difficult, for Serre himself employs a different notation, we shall denote the group by S.

It is not difficult to describe S. Let Q̄ be the algebraic closure of Q in C and let ι ∈ Gal(Q̄/Q) be complex conjugation. To construct X^{∗}(S), the module of rational characters of S, we start with the module M of locally constant integral-valued functions on Gal(Q̄/Q). The Galois group acts by right translation and

X^{∗}(S) = {λ ∈ M | (σ − 1)(ι + 1)λ = (ι + 1)(σ − 1)λ = 0 ∀σ ∈ Gal(Q̄/Q)}.

In particular if λ ∈ X^{∗}(S),

(σ − 1)λ(1) + (σ − 1)λ(ι) = 0

because the left side if (ι + 1)(σ − 1)λ(1). The lattice of rational characters of R is canonically isomorphic to Z ⊕ Z and the homomorphism h_{0} : R → S dual to the homomorphism X^{∗}(S) → X^{∗}(R) which sends λ to (λ(1), λ(ι)) is defined over R. The composite h ∘ w : GL(1) → S is dual to the homomorphism X^{∗}(S) → Z taking λ to λ(1) + λ(ι) and is defined over Q.

To verify that the group S just defined in terms of its module of characters is the Serre group defined in terms of Hodge structures is easy enough. The existence of the two homomorphisms h_{0} and h_{0} ∘w implies that every representation of S defined over Q defines a rational Hodge structure. It is enough to show that these are polarizable when the representation is irreducible. To obtain the irreducible representations, one takes a λ ∈ X^{∗}(S) and defines the field F by

Gal(Q̄/F) = {σ ∈ Gal(Q̄/Q) | σλ = λ}.

The underlying space of the representation is the vector space over Q defined by F, and the representation r = r_{λ} is that defined symbolically by r_{λ}(s) : x ∈ F → λ(s)x. The weight of the associated Hodge structure is −(λ(1) + λ(ι)) = n. There is an α ∈ F such that ι(α) = (−1)^{n}α and (−1)^{λ(ι)}i^{n}α is totally positive. A possible polarization is

P (u, ν) = (2πi)^{−n}T r_{F/Q}uαι(ν).

Conversely suppose one has a rational Hodge structure V whose Mumford-Tate group MT (V) is abelian. Since there is a homomorphism R → MT (V), the coweight GL(1) → R of the group R defined by z → (z, 1) also defines a coweight ν^{∨} of MT (V). The lattice Y_{∗} of coweights of MT (V) is a Gal(Q̄/Q) module generated by ν^{∨}. We define an injective homomorphism of its lattice of rational characters V^{∗} into X^{∗}(S) by sending ν ∈ Y^{∗} to the element λ given by

λ(σ) = ⟨σν, ν^{∨}⟩, σ ∈ Gal(Q̄/Q).

The dual homomorphism S → MT (V) is surjective and V is defined by a rational representation of S.

We return to the Shimura varieties Sh_{K}(C) and show how one attaches families of rational Hodge structures to Sh_{K}(C). Let G_{0} be the largest quotient of G such that (σ − 1)(ι + 1)ν^{∨} = (ι + 1)(σ − 1)ν^{∨} = 0 for every coweight of the centre of G_{0}. Let ξ be a rational representation of G on V which factors through G_{0}.

To each x = (h, g) in X_{K} we associate a triple (V^{x}, 𝔨^{x}, φ^{x}). V^{x} is a rational Hodge structure whose underlying space is VQ^{x} = V_{Q}, the Hodge structure being defined by the representation ξ ∘ h of R. The third term φ^{x} is an isomorphism VA_{f}^{x} → V_{A_{f}} and is given by ν → ξ(g)^{−1}ν. It is only defined up to composition with an element of ξ(K). The homogeneous components of VQ^{x} are independent of x and may be written VQ^{n}. It follows from Lemma 2.8 of [11] that if x ∈ X_{K} there is at least one collection P^{x} of bilinear forms P_{n}^{x} : VQ^{n} × VQ^{n} → Q(−n)_{Q} which is invariant under the derived group of G and is a polarization of V^{x}. Let 𝔓^{x} denote the collection of all such polarizations. If g ∈ G(Q), P^{x} ∈ 𝔓^{x}, and x′ = γx then the collection P^{x′} given by P_{n}^{x′} : (u, v) → P_{n}^{x}(ξ(γ^{−1})u, ξ(γ^{−1})v) lies in 𝔓^{x′}.

We must also verify that the family {V^{x}} over X_{K} is a family of rational Hodge structures in the sense of [9]. Otherwise it could not possibly be attached to a family of motives. There are two points to be verified. Let V_{p,q}^{x} be the subspace of VC^{x} of type p, q and set V_{p}^{x} = ⊕_{p′≧p}V_{p′,q′}^{x}. The space V_{p}^{x} ⊆ V_{C} must be shown to vary holomorphically with x. In other words if v(x) is any local section of V_{p}^{x} and Y any antiholomorphic vector field then Y v(x) also takes values in V_{p}^{x}. The condition of transversality must also be established, to the effect that for the same v(x) and any holomorphic vector field Y the values of Y v(x) lie in V_{p−1}^{x}.

There is certainly no harm in supposing that v(x) takes values in V_{p,q}^{x}. Then one has to show that if x^{∘} = (h^{∘}, g_{f}^{∘}) is fixed and Y is the vector field defined by X ∈ 𝔊 then Y v(x^{∘}) lies in V_{p}^{x} when X ∈ 𝔭_{h^{∘}} and in V_{p−1}^{x} when X ∈ 𝔭̄_{h^{∘}}. Let K_{h^{∘}} be the stabilizer of h^{∘} in G(R). We represent X_{K} as the quotient G(A)/K_{h^{∘}}K and lift v(x) to a function on G(A) which we write as

(g, g_{f}) → ξ(g)u(g, g_{f}), g ∈ G(R), g_{f} ∈ G(A_{f}).

The function u takes values in the constant space V_{p,q}^{x^{∘}}. Moreover

Y v(x^{∘}) = ξ(X)u(1, g_{f}^{∘}) + Xu(1, g_{f}^{∘}).

The second term lies in V_{p,q}^{x^{∘}} for all X ∈ 𝔊. If X ∈ 𝔭 then ξ(X)V_{p,q}^{x^{∘}} ⊆ V_{p+1,q−1}^{x^{∘}} and if X ∈ 𝔭̄ then ξ(X)V_{p,q}^{x^{∘}} ⊆ V_{p−1,q+1}^{x^{∘}}.

If γ ∈ G(Q) and x′ = γx then v → ξ(γ)v provides an isomorphism between (V^{x}, 𝔓^{x}, φ^{x}) and (V^{x′}, 𝔓^{x′}, φ^{x′}). Thus if s ∈ S_{K}(C), any two elements of {(V^{x}, 𝔓^{x}, φ^{x}) | x → s} are canonically isomorphic, and we may take (V^{s}, 𝔓^{x}, φ^{s}) to be any one of them, and redefine our family as a family of rational Hodge structures, with supplementary data, over the base S_{K}(C). The locally constant sheaf F_{ξ}(Q) of rational vector spaces underlying this family is the quotient of V_{Q} × X_{K} by the action γ : (v, x) → (ξ(γ)v, γx) of G(Q). For this quotient to be well defined, the group K must be sufficiently small, for G(Q) ∩ K_{h}K is then contained in the kernel of ξ for all h (cf. II.A.2 of [41]). It is here that the condition that ξ factors through G_{0} intervenes.

When dealing with motives, one does not need to introduce the polarizations explicitly as part of the moduli problem, but in order to introduce a Hodge structure on the cohomology groups of the sheaves F_{ξ}(Q) one must verify that on each connected component of S_{K}(C) a locally constant section s → P^{s} can be defined. Let G^{0}(R) be the connected component of G(R) and choose x^{∘} = (h, g_{f}) in X_{K}. If

X_{K}^{∘} = {(Ad g ∘ h, g_{f}k) | g ∈ G^{∘}(R), k ∈ K}

then the image of X_{K}^{∘} in S_{K}(C) is open. If x and x′ lie in X_{K}^{∘} and x′ = γx then γ lies in

(4.2)  G(Q) ∩ G^{∘}(R)K_{h}g_{f}Kg_{f}^{−1}.

All we need do is find a collection P = {P_{n}} such that P ∈ 𝔓^{x} for all x ∈ X_{K}^{∘} and P_{n}(ξ(γ)u, ξ(γ)ν) = P_{n}(u, ν) if γ lies in the group (4.2). Choose any P in 𝔓^{x^{∘}}. Then P ∈ 𝔓^{x} for all x ∈ X_{K}^{∘}. There are certainly homomorphisms λ_{n} of G into the general linear group of V^{n} such that

P_{n}(ξ(γ)u, ξ(γ)v) = P_{n}(λ_{n}(γ)u, v).

The eigenvalues of λ(γ) are positive if γ lies in G^{∘}(R)K_{h}. Moreover λ is trivial on the derived group of G and factors through G_{0}. It therefore follows from the results of II.A.2 of [41] that each λ_{n} is the identity on all of (4.2).

Although we have defined the families (V^{s}, 𝔓^{s}, φ^{s}) for any G, it is clear that S_{K}(C) is not going to appear as a moduli space unless G is equal to G_{0}, and so for the rest of this section we assume this. The moduli problem is best formulated completely in the language of tannakian categories. We can drop the polarizations and retain only the pairs (V^{s}, φ^{s}), or (V^{x}, φ^{x}), but we now have to emphasize that (V^{x}, φ^{x}) is defined for every (finite-dimensional) representation ξ of G over Q, and so we write (ξ, V (ξ)) for the representation and (V^{x}(ξ), φ^{x}(ξ)) for the pair (V^{x}, φ^{x}).

On the category REP(G) of finite-dimensional representations of G we have the natural fibre functor ω_{Rep(G)} : (ξ, V (ξ)) → V (ξ)_{Q} and η^{x} : (ξ, V (ξ)) → V^{x}(ξ) is a ⊗-functor from REP(G) to HOD(Q) which satisfies ω_{Hod} ∘ η^{x} = ω_{Rep(G)}. Since HOD(Q) and REP(G_{Hod}) are the same categories, η^{x} defines a homomorphism [40, II.3.3.1] φ^{x} : G_{Hod} → G and η^{x} may be defined by (ξ, V (ξ)) → (ξ ∘ φ^{x}, V (ξ)). When we emphasize η^{x}, φ^{x} appears as an isomorphism of two fibre functors

φ^{x} : ω Hod^{A_{f}} ∘ η^{x} → ωA_{f}_{Rep(G)}.

However, when we emphasize φ^{x}, as we shall, then these two fibre functors are the same, for they are both obtained from ω_{Hod} ∘ η^{x} = ω_{Rep(G)} by tensoring with A_{f}, and φ^{x} may be interpreted as an isomorphism of ωA_{f}_{Rep(G)}. Such an isomorphism is given by a g^{−1} ∈ G(A_{f}) [40, II.3]. This is the g appearing in x = (h, g). Only the coset gK ⊆ G(A_{f}) is well defined.

We have arrived, by a rather circuitous route, at the conclusion that X_{K} parametrizes pairs (φ, g), φ being a homomorphism from G_{Hod} to G defined over Q, and g in G(A_{f}) being specified only up to right multiplication by an element of K. In addition, φ is subject to the following constraint:

H. The composition of φ with the canonical homomorphism R → G_{Hod} lies in ℌ.

If γ ∈ G(Q) the pairs (φ, g) and (ad γ ∘ φ, γg) will be called equivalent. The variety Sh_{K}(C) parametrizes equivalence classes of these pairs.

One of the important tannakian categories is the category MOT (k) of motives over a field k. It cannot be constructed at present unless one assumes certain conjectures in algebraic geometry, referred to as the standard conjectures [40]. It is covariant in k, and rational cohomology together with its Hodge structure yields a ⊗-functor h_{BH} : MOT (C) → HOD(Q). MOT (C) together with the fibre functor ω_{Mot(C)} of rational cohomology also defines a group G_{Mot(C)} over Q and h_{BH} is dual to a homomorphism h_{BH}^{∗} : G_{Hod} → G_{Mot(C)} defined over Q.

Implicit in Deligne’s construction is the hope that any homomorphism φ′ : G_{Hod} → G satisfying H is a composite φ′ = φ ∘ h_{BH}^{∗}. According to the Hodge conjecture, φ would be uniquely determined [40, VI.4.5] and Sh_{K}(C) would appear as the moduli space for pairs (φ, g), with g as before, but where φ is now a homomorphism from G_{Mot(C)} to G defined over Q and satisfying:

H′. The composition of φ with the canonical homomorphism R → G_{Mot(C)} lies in ℌ.

This may be so but it will not be a panacea for all the problems with which the study of Shimura varieties is beset. So far as I can see, we do not yet have a moduli problem in the usual algebraic sense, and, in particular, no way of deciding over which field the moduli problem is defined. We can be more specific about this difficulty.

Suppose τ is an automorphism of C. Then τ^{−1} defines a ⊗-functor η(τ) : MOT (C) → MOT (C). Let G_{Mot(C)}^{τ} be the group defined by MOT (C) and the fibre functor ω_{Mot(C)} ∘ η(τ). The dual of η(τ) is then an isomorphism over Q:

φ(τ) : G_{Mot(C)} → G_{Mot(C)}^{τ}.

The homomorphism φ has a dual, a ⊗-functor η : REP(G) → MOT (C) and the fibre functor ω_{Mot(C)} ∘ η(τ) ∘ η defines a group G^{τ,φ} over Q. The ⊗-functor η then defines a dual

φ^{τ} : G_{Mot(C)}^{τ} → G^{τ,φ}, and φ′ = φ^{τ} ∘ φ(τ) : G_{Mot(C)} → G^{τ,φ}.

Moreover the two fibre functors ωA_{f}_{Mot(C)} and ωA_{f}_{Mot(C)} ∘ η(τ) are canonically isomorphic. As a consequence, there is a canonical isomorphism G(A_{f}) → G^{τ,φ}(A_{f}). Let g′ be the image of g.

The pair (φ′, g′) seems once again to define a solution to our moduli problem. The difficulty is that G^{τ,φ} may not be the group G or, even if it is, the composition of φ′ with the canonical homomorphism may not lie in ℌ. One of the purposes of the next two sections is to discover what G^{τ,φ} is likely to be.

5. The Taniyama group. There is one type of Shimura variety which is very easy to study, that obtained when G is a torus T. Then the set ℌ reduces to a single point {h}. For each open compact subgroup U of T (A_{f}) the manifold Sh_{U}(C) consists of a finite set and Sh_{U} = Sh_{U}(T, h) is zero-dimensional. In general a special point of (G, ℌ) will be a pair (T, h) with T ⊆ G and h ∈ ℌ. If U = K ∩ T (A_{f}) then Sh_{U}(C) is a subset of Sh_{K}(C), the points of which have traditionally been referred to as special points, and I shall continue this usage. But it is best to give priority to the pair (T, h) rather than to the points of Sh_{K}(C) it defines. There are a number of unsolved problems about Shimura varieties and their special points that I want to describe in the next section. To formulate them some Galois cocycles have to be defined. Deligne has shown me that my original construction gave, in particular, a specific extension of Gal(Q̄/Q) by the Serre group, S, an extension I venture to call the Taniyama group and denote by T . Since the cocycles needed are, as Rapoport observed, often easily defined in terms of T , I begin by constructing it.

The group S is an algebraic group over Q, and T will also be defined over Q. Thus we will have an exact sequence

(5.1)  1 → S → T → Gal(Q̄/Q) → 1.

Recall that X^{∗}(S) is a module of functions on Gal(Q̄/Q), and that the Galois action on X^{∗}(S) giving the structure of S as a group over Q is defined by right translation. We are still free to use left translation to define an algebraic action of Gal(Q̄/Q) on S, and it is this action which is implicit in (5.1). The extension will not split over Q but it will be provided with canonical splittings over each ℓ-adic field Q_{ℓ}, Gal(Q̄/Q) → T (Q_{ℓ}), which will fit together to give Gal(Q̄/Q) → T (A_{f}).

Rather than attempting to work directly with S, I choose a finite Galois extension L of Q, let S^{L} be the quotient group of S whose lattice of rational characters consists of all functions in X^{∗}(S) invariant under G(Q̄/L), and define extensions

(5.2)  1 → S^{L} → T^{L} → Gal(L^{ab}/Q) → 1,

afterwards lifting to Gal(Q̄/Q), and then passing to the limit.

To motivate the construction we suppose that the extension is defined and that there is a section τ → a(τ) of T^{L} → Gal(L^{ab}/Q) with a(τ) ∈ T^{L}(L). Let a(τ_{1})a(τ_{2}) = d_{τ_{1},τ_{2}}a(τ_{1}τ_{2}) with d_{τ_{1},τ_{2}} ∈ S^{L}(L), and with

(5.3)  τ_{1}(d_{τ_{2},τ_{3}})d_{τ_{1},τ_{2}τ_{3}} = d_{τ_{1},τ_{2}}d_{τ_{1}τ_{2},τ_{3}}.

Observe that the elements of the Galois group play two different roles. They are first of all elements of a quotient group of T^{L}, and secondly they are automorphisms of L^{ab} and thus act on T^{L}(L^{ab}), since T^{L} is defined over Q. In the first role they will be denoted by τ, perhaps with a subscript added, and in the second by ρ or σ.

We have ρ(a(τ)) = c_{ρ}(τ)a(τ) with c_{ρ}(τ) ∈ S^{L}(L). Certainly

(5.4)  c_{ρσ}(a(τ)) = ρ(c_{σ}(τ))c_{ρ}(τ).

In addition

(5.5)  d_{τ_{1},τ_{2}}c_{ρ}(τ_{1})τ_{1}(c_{ρ}(τ_{2})) = ρ(d_{τ_{1},τ_{2}})c_{ρ}(τ_{1}τ_{2}).

Conversely if we have collections {c_{ρ}(τ)} and {d_{τ_{1},τ_{2}}} satisfying (5.3), (5.4) and (5.5), we can construct T^{L} over Q, together with the section a.

Any splitting Gal(L^{ab}/Q) → T (A_{f}) will be of the form τ → b(τ)a(τ) with b(τ) ∈ S^{L}(A_{f}(L)). In order that it be a splitting, we must have

(5.6)  b(τ_{1})τ_{1}(b(τ_{2}))d_{τ_{1},τ_{2}} = b(τ_{1}τ_{2}).

If the b(τ)a(τ) are to lie in T^{L}(A_{f}) we must have

(5.7)  ρ(b(τ))c_{ρ}(τ) = b(τ).

Again any collection {b(τ)} satisfying (5.6) and (5.7) defines a splitting, and it is our task to construct {b(τ)}, {c_{ρ}(τ)}, and {d_{τ_{1},τ_{2}}}.

The group S is a quotient of G_{Hod} and thus is provided with a canonical homomorphism h : R → S. Over C the group R is canonically isomorphic to GL(1) × GL(1). Restricting h to the first factor we obtain a coweight μ of S, the canonical coweight. If λ ∈ X^{∗}(S) then ⟨λ, μ⟩ = λ(1). Since S^{L} is a quotient of S, μ also defines a coweight of S^{L}, which for convenience will also be denoted by μ. If ν is any coweight of S^{L} and x any invertible element of L or of A_{f}(L) then x^{ν} will be the element of S^{L}(L), or S^{L}(A_{f}(L)), satisfying λ(x^{ν}) = x^{⟨λ,ν⟩} for all λ ∈ X^{∗}(S^{L}). Recall that we have two actions of Gal(L/Q), or Gal(L^{ab}/Q), on X^{∗}(S^{L}) or on

X_{∗}(S^{L}) = Hom(X^{∗}(S^{L}), Z),

the lattice of coweights. That defined by right translation we write ν → σν, and that defined by left translation we write ν → ντ. Thus ⟨λ, σν⟩ = λ(σ^{−1}) while ⟨λ, μτ⟩ = λ(τ), because an inverse intervenes in the action by left translation, and μτ^{−1} = τμ. Moreover σ(x^{μ}) = σ(x)^{σμ} while τ(x^{μ}) = x^{μτ}. These aspects of the notation have to be emphasized because at some points our convention of distinguishing between ρ, σ on the one hand, and τ on the other, fails us.

For the study of Shimura varieties, it is best to take Q̄ to be the algebraic closure of Q in C, and we shall do this. Thus we provide ourselves with an extension of the infinite valuation on Q to Q̄. There is a property of the Weil groups that will play a prominent role in our discussion. Let v be a valuation of Q̄ and hence of Q, and let Q_{v} and Q̄_{v} be the completions of Q and Q̄ with respect to v. Eventually v will be defined by the inclusion Q̄ ⊆ C, and Q_{v} will be R and Q̄_{v} will be C. In any case, the data provide us with imbeddings

(5.8)  F_{v}^{×} ↪ C_{F}
(5.9)  Gal(F_{v}/Q_{v}) ↪ Gal(F/Q),

if F is any finite Galois extension of Q in Q̄. The local and global Weil groups W_{F_{v}/Q_{v}} and W_{F/Q} are defined as extensions

1 → F_{v}^{×} → W_{F_{v}/Q_{v}} → Gal(F_{v}/Q_{v}) → 1

and

1 → C_{F} → W_{F/Q} → Gal(F/Q) → 1.

We may imbed the arrows (5.8) and (5.9) in a commutative diagram

1 → F_{v}^{×} → W_{F_{v}/Q_{v}} → Gal(F_{v}/Q_{v}) → 1
↓ ↓I_{F} ↓
1 → C_{F} → W_{F/Q} → Gal(F/Q) → 1

Moreover we may so choose the central arrows that they are compatible with field extensions and upon passage to the limit yield I: W_{Q_{v}} → W_{Q}. It is the image of W_{Q_{v}} in W_{Q} that will be fixed, and I_{F} may be changed to w → xI_{F} (w)x^{−1} where x ∈ C_{F} and xσ(x)^{−1} ∈ F_{v}^{×} for all σ ∈ Gal(F_{v}/Q_{v}).

Now let v be the valuation given by Q̄ ⊆ C. Let F_{∞}^{×} = ∏_{w|v} F_{w}^{×}. The natural map F_{∞}^{×} → C_{F} is an imbedding, and we sometimes regard F_{∞}^{×} as a subgroup of C_{F}. If we take an element τ of Gal(Q̄/Q), lift to W_{Q}, and then project to W_{L/Q} we obtain an element w = w(τ) of W_{L/Q} which is well defined modulo the connected component, and in particular modulo the closure of L_{∞}^{×}.

We choose a set of representatives w_{σ}, σ ∈ Gal(L/Q), for the cosets of C_{L} in W_{L/Q} in such a way that the following conditions are satisfied:

(a) w_{1} = 1.

(b) If σ ∈ Gal(L_{v}/Q_{v}) then w_{σ} ∈ W_{L_{v}/Q_{v}}.

(c) If ρ ∈ Gal(L/Q) and σ ∈ Gal(L_{v}/Q_{v}) then w_{ρ}w_{σ} = a_{ρ,σ}w_{ρσ} with a_{ρ,σ} ∈ L_{∞}^{×}.

To arrange the final condition we may choose a collection of 𝔣 of representatives η for the cosets Gal(L/Q)/Gal(L_{v}/Q_{v}) and set w_{ησ} = w_{η}w_{σ} if σ ∈ Gal(L_{v}/Q_{v}). We suppose that 𝔣 contains 1. With this choice we also have:

(d) If {a_{ρ,σ}} is the cocycle defined by w_{ρ}w_{σ} = a_{ρ,σ}w_{ρσ} then a_{η,ρ} = 1 for η ∈ 𝔣 and σ ∈ Gal(L_{v}/Q).

If w ∈ W_{L/Q}, let w_{σ}w = c_{σ}(w)w_{σ}, c_{σ}(w) ∈ C_{L}. If w = w(τ) we set

b_{0}(τ) = ∏_{σ∈ Gal(L/Q)} c_{σ}(w)^{σμ}.

It lies in C_{L} ⊗ X_{∗}(S^{L}), but is not well defined, because w is not. However we can show that it is well defined if taken modulo L_{∞}^{×} ⊗ X_{∗}(S^{L}), and that, in addition, it behaves properly under extensions of the field L.

The ambiguity in w now has no effect, for we are only free to replace w by uw where u = lim_{n} u_{n} and u_{n} lies in the image of L_{∞}^{×}U, where U is a subgroup of the group of units of L defined by a strong congruence condition. But [41, II.A.2],

∏_{σ∈ Gal(Q^{}¯ /Q)} σ(u)^{σμ} = 1

for all u ∈ U. Consequently

∏_{σ} c_{σ}(uw)^{σμ} = (∏_{σ} σ(u)^{σμ}) (∏_{σ} c_{σ}(w)^{σμ})

is congruent modulo L_{∞}^{×} ⊗ X_{∗}(S^{L}) to ∏_{σ} c_{σ}(w)^{σμ}.

Suppose the representatives w_{σ} are replaced by e_{σ}w_{σ}, and, hence, a_{ρ,σ} by

a′_{ρ,σ} = e_{ρ}ρ(e_{σ})e_{ρσ}^{−1}a_{ρ,σ}.

If σ ∈ Gal(L_{v}/Q_{v}) then e_{σ} ∈ L_{v}^{×} and ρ(e_{σ}) ∈ L_{∞}^{×}. Since a_{ρ,σ} and a′_{ρ,σ} must then both be in L_{∞}^{×}, we infer that e_{ρ} ≡ e_{ρσ}(mod L_{∞}^{×}) when σ ∈ Gal(L_{v}/Q_{v}). Moreover c_{σ}(w) is replaced by c′_{σ}(w) = e_{στ} e_{σ}^{−1}c(w) and b_{0}(τ) by

b′_{0}(τ) = {∏_{σ} e_{στ}^{σμ}e_{σ}^{−σμ}} b_{0}(τ).

The factor may be written

∏_{σ} e_{σ}^{σ(τ^{−1}−1)μ} ≡ ∏_{τ∈𝔣} ∏_{σ∈Gal(L_{v}/Q_{v})} e_{η}^{ησ(τ^{−1}−1)μ}.

Since

∑_{σ∈ Gal(L_{v}/Q_{v})} σ(1 − τ^{−1})μ = (1 + ι)(1 − τ^{−1})μ = 0,

this change has no effect on b_{0}(τ). In this argument we have denoted the image in Gal(L/Q) of τ ∈ Gal(Q̄/Q) by the same symbol, a practice we shall continue to indulge in.

If we modify I then w is replaced by xwx^{−1} with x ∈ C_{L} and xσ(x)^{−1} ∈ L_{v}^{×} for all σ ∈ Gal(L_{v}/Q). Then c_{σ}(xwx^{−1}) = σ(x)στ(x^{−1})c_{σ}(w) and

∏_{σ} σ(x)^{−σμ}στ(x)^{σμ} = ∏_{σ} σ(x)^{σ(τ^{−1}−1)μ}.

Since σ(x) ≡ x(mod L_{∞}^{×}) if σ ∈ Gal(L_{v}/Q_{v}), the same argument as before shows that b_{0}(τ) is unchanged.

Finally suppose that L ⊆ L′. Then b′_{0}(τ) ∈ C_{L} ⊗ X_{∗}(S^{L′}) and b_{0}(τ) ∈ C_{L} ⊗ X_{∗}(S^{L}) are both defined, and we must verify that b′_{0}(τ) is taken to b_{0}(τ) by the canonical mapping of the first group to the second. Either L_{v} = R or L_{v} = C, and the two cases must be treated separately. Suppose first that L_{v} = C and hence that Gal(L′/L) ∩ Gal(L′_{v}/Q_{v}) = {1}. Since both b_{0}(τ) and b′_{0}(τ) are independent of the choices of coset representatives, we may choose those which make it easiest to verify that b_{0}(τ) is the image of b′_{0}(τ) is the image of b′_{0}(τ). Let 𝔢 be a set of representatives for the cosets Gal(L′/L)\Gal(L′/Q)/Gal(L′_{v}/Q_{v}) containing 1. Every elelment σ of Gal(L′/Q) may be written uniquely as a product σ = ζηρ, ζ ∈ Gal(L′/L), η ∈ 𝔢, ρ ∈ Gal(L′_{v}/Q_{v}). We may suppose that w′_{σ} = w′_{ρ}w′_{η}w′_{ρ} with w′_{1} = 1 and w′_{ρ} ∈ w_{L′_{v}/Q_{v}}. Let w′ be w(τ) with respect to L′, and w be w(τ) with respect to L. Then under the canonical map π : W_{L′/Q} → W_{L/Q} the element w′ maps to w. If σ ∈ Gal(L/Q) it lifts to a unique element of Gal(L′/Q) of the form ηρ. We suppose that w_{σ} is the image of w′_{η}w′_{ρ}. Thus if

w′_{η}w′_{ρ}w′ = d_{η,ρ}(w′)w′_{η′}w′_{ρ′}

with d_{η,ρ}(w′) ∈ W_{L′/L} then c_{σ}(w) = π(d_{η,ρ}(w′)). On the other hand if σ_{1} = ζηρ lies in Gal(L′/Q) then

w′_{ζ}d_{η,ρ}(w′) = c_{σ_{1}}(w′)w′_{ζ}.

Consequently, by the very definition of π,

c_{σ}(w) = ∏_{σ_{1}→σ} c_{σ_{1}}(w′).

It follows immediately that b_{0}(τ) is the image of b′_{0}(τ).

If L_{v} = R then X^{∗}(S^{L}) ≃ Z and the Galois group acts trivially. Suppose we replace w_{σ} by e_{σ}w_{σ} with e_{σ} ∈ C_{L}. Then c_{σ}(w) is replaced by e_{σ}e_{στ}^{−1}. Since

∏_{σ}(e_{σ}e_{στ}^{−1})^{σμ} = ∏_{σ}(e_{σ}e_{στ}^{−1})^{μ} = 1,

this has no effect on b_{0}(τ), and when defining b_{0}(τ) we need not suppose that the collection {w_{σ}} is subject to the constraints (a), (b), and (c). If we want to define b′_{0}(τ), we may still need to be careful about the choice of the coset representatives w′_{σ}, σ ∈ Gal(L′/Q). However, since we are only interested in the image of b′_{0}(τ) in S^{L}, we may again ignore (a), (b), and (c). We choose a set 𝔢 of representatives for the cosets Gal(L′/L)\Gal(L′/Q), write σ = ρη, ρ ∈ Gal(L′/L), η ∈ 𝔢, and take w′_{σ} = w′_{ρ}w′_{η}. If σ ∈ Gal(L/Q) is the image of η, we take w_{σ} = π(w′_{η}). The argument can now proceed as before.

Let b̃(τ) be a lift of b_{0}(τ) to I_{L} ⊗ X_{∗}(S^{L}) = S^{L}(A(L)) and let b(τ) be the projection of b̃(τ) on S^{L}(A_{f}(L)). The element b(τ) is well defined modulo S^{L}(L) and, as we shall see, this bit of ambiguity will cause us no difficulty. But we have to fix one choice. The first point to verify is that

d_{τ_{1},τ_{2}} = b(τ_{1})τ_{1}(b(τ_{2}))b(τ_{1}τ_{2})^{−1}

lies in S^{L}(L). When verifying this, we may choose the liftings b̃(τ_{1}), b̃(τ_{2}), and b̃(τ_{1}τ_{2}) in any way we like. We choose liftings c̃_{σ}(w_{1}) and c̃_{σ}(w_{2}) of c_{σ}(w_{1}) and c_{σ}(w_{2}) to I_{L} and take

b̃(τ_{1}) = ∏_{σ} c̃_{σ}(w_{1})^{σμ}, b̃(τ_{2}) = ∏_{σ} c̃_{σ}(w_{2})^{σμ}.

Since c_{σ}(w_{1}w_{2}) = c_{σ}(w_{1})c_{στ_{1}}(w_{2}), we may take c̃_{σ}(w_{1}w_{2}) to be c̃_{σ}(w_{1})c̃_{στ_{1}}(w_{2}). Because

τ_{1}^{−1}(b̃(τ_{2})) = ∏_{σ} c̃_{σ}(w_{2})^{στ_{1}^{−1}μ} = ∏_{σ} c̃_{στ_{1}}(w_{2})^{σμ},

the element d_{τ_{1},τ_{2}} will then be 1.

Finally we have to establish that the elements c_{ρ}(τ) defined by equation (5.7) lie in S^{L}(L), for equations (5.4) and (5.5) will then follow immediately. It will suffice to show that for any w ∈ W_{L/Q} and any ρ ∈ Gal(L/Q)

(5.10)  {∏_{σ} c_{σ}(w)^{σμ}} {∏_{σ} ρ(c_{σ}(w))^{−ρσμ}}

lies in L_{∞}^{×} ⊗ X_{∗}(S^{L}). Suppose w = w_{1}w_{2} and w_{1} projects to τ_{1} ∈ Gal(L/Q). Then c_{σ}(w) = c_{σ}(w_{1})c_{στ_{1}}(w_{2}) and (5.10) is equal to

{∏_{σ} c_{σ}(w_{1})^{σμ}ρ(c_{σ}(w_{1}))^{−ρσμ}} τ_{1} {∏_{σ} c_{σ}(w_{2})^{σμ}ρ(c_{σ}(w_{2}))^{−ρσμ}} .

Consequently we need only verify that (5.10) lies in L_{∞}^{×} ⊗ X_{∗}(S^{L}) for w in C_{L} and for w = w_{τ}. If w lies in C_{L} then c_{σ}(w) = σ(w) and ∏_{σ} σ(w)^{σμ} = ∏_{σ} ρσ(w)^{ρσμ}. The expression (5.10) is therefore equal to 1. If w = w_{τ} then c_{σ}(w) = a_{σ,τ} and

(5.11)  ∏_{σ} a_{σ,τ}^{σμ} ρ(a_{σ,τ})^{−ρσμ} = ∏_{σ} a_{ρ,σ}^{ρσ(τ^{−1}−1)μ}.

However it follows from condition (c) that a_{ρ,σ} ≡ a_{ρ,σι}(mod L_{∞}^{×}). Since (1 + ι) · (1 − τ^{−1})μ = 0, the right side of (5.11) lies in L_{∞}^{×} ⊗ X_{∗}(S^{L}).

Since b(τ), although defined for τ ∈ Gal(Q̄/Q), depends only on the image of τ in Gal(L^{ab}/Q), the groups T^{L} and T are now completely defined. The ambiguity in the b(τ) is easily seen to correspond to the ambiguity in the choice of the section a(τ).

If E ⊆ Q̄ is any finite extension of Q, we let T_{E} be the inverse image of Gal(Q̄/E) in T . If E ⊆ L^{ab} we may also introduce T_{E}^{L}. The group T_{L}^{L} has been introduced by Serre [41], who uses it to formulate some ideas of Taniyama. He makes its arithmetic significance quite clear, but his definition is sufficiently different from that given here that an explanation of the reasons for their equivalence is in order.

If x is an idèle of L then ψ(x) = ∏_{Gal(L/Q)} σ(x)^{σμ} is an element of S^{L}(A(L)). By II.4 of [41] there is an open subgroup U of the group of idèles I_{L} such that ψ(x) = 1 if x ∈ U ∩ L^{×}. The standard map of I_{L} onto Gal(L^{ab}/L) restricts to U and if τ = τ(x) is the image of x we may take w = w(τ) to be the image of x in C_{L}. Then ψ(x) is a lifting of b_{0}(τ) to S^{L}(A(L)). However ψ(x) depends only on τ, and thus we may take b̃(τ) = ψ(x). Then d_{τ_{1},τ_{2}} = 1 and c_{ρ}(τ) = 1 if τ, τ_{1}, τ_{2} lie in the image Gal(L^{ab}/F) of U. Here F is the finite extension of L defined by U. The elements c_{ρ}(τ) are in fact 1 for all τ ∈ Gal(L^{ab}/L). If we choose a set of representatives 𝔢 for Gal(L^{ab}/F)\Gal(L^{ab}/L) and set b̃(τη) = b̃(τ)b̃(η), τ ∈ Gal(L^{ab}/F), η ∈ 𝔢, then, in general, d_{τ_{1},τ_{2}} will depend only on the images of τ_{1}, τ_{2} in Gal(F/L), and T_{L}^{L} may be obtained by pulling back an extension

1 → S^{L} → T_{U} → Gal(F/L) → 1

to Gal(L^{ab}/L). Here T_{U} is the quotient of T_{L}^{L} by the normal subgroup {a(τ) | τ ∈ Gal(L^{ab}/F)}. It is the extension T_{U} that Serre defines directly. He denotes it by the symbol S_{𝔪}.

The map ψ defines a homomorphism of L^{×}/L^{×} ∩ U ≃ L^{×}U/L^{×} into S^{L}(L) and to verify that T_{U} is the group studied by Serre, we have only to verify that it can be imbedded in a commutative diagram

1 → L^{×}U/U → T_{L}/U → I_{L}/L^{×}U → 1
↓ ↓ ↓≀
1 → S^{L}(Q) → T_{U}(Q) → Gal(F/L) → 1

The right-hand arrow is x → τ(x)^{−1}. We do not have to pass to the quotient but may define the homomorphism

(5.12)  I_{L}/L^{×} → T_{L}^{L}(L)

directly. The central arrow is then obtained by composing with the projection T_{L}^{L}(L) → T_{U}(L). The homomorphism (5.12) is

x → {∏_{σ} σ(x)^{σμ}} b̃(τ)^{−1}a(τ)^{−1}

if τ is the image of x in Gal(L^{ab}/L). The composition of our splitting Gal(L^{ab}/L) → T_{L}^{L}(Q_{ℓ}) with T_{L}^{L}(Q_{ℓ}) → T_{U}(Q_{ℓ}) is either Serre’s ϵ_{ℓ} or its inverse, presumably its inverse, for we are so arranging matters that the eigenvalues of the Frobenius elements acting on the cohomology of algebraic varieties are greater than or equal to 1.

The cocycle ρ → c_{ρ}(τ) certainly becomes trivial at every finite place, but is not necessarily trivial at the infinite place, v. Indeed under the isomorphism

H^{1}(Gal(L_{v}(Q_{v}), S^{L}(L_{v})) ≃ H^{−1}(Gal(L_{v}/Q_{v}), X_{∗}(S^{L}))

given by the Tate-Nakayama duality it corresponds to the element of the group on the right represented by (1 − τ^{−1})μ. If, as has been our custom, we denote the image of τ in Gal(L/Q) again by τ we may suppose that w(τ) = w_{τ}, for the class of {c_{ρ}(τ)} depends only on this image. According to the discussion of formula (5.11)

∏_{σ} a_{ρ,σ}^{ρσ(τ^{−1}−1)μ} = ∏_{η∈𝔣}(a_{ρ,ηι}a_{ρ,η}^{−1})^{ρη(τ^{−1}−1)μ} = e_{ρ}(τ)

lies in L_{∞}^{×} ⊗ X_{∗}(S^{L}) = S^{L}(L_{∞}). By definition, the classes of {c_{ρ}(τ)} and {e_{ρ}(τ)} are inverse to one another in H^{1}(Gal(L/Q), S^{L}(A(L))). Thus all we need do is calculate the projection of e_{ρ}(τ) on S^{L}(L_{v}) for ρ ∈ Gal(L_{v}/Q_{v}).

Observe first of all that if we agree to choose coset representatives satisfying (d), then a_{ρ,ηι}a_{ρ,η}^{−1} = ρ(a_{η,ι}^{−1})a_{ρη,ι} = a_{ρη,ι}. Again if ρη = η_{1}ρ_{1} then

a_{ρη,ι} = a_{η_{1}ρ_{1},ι} = η_{1}(a_{ρ_{1},ι})a_{η_{1},ρ_{1}ι}a_{η_{1},ρ_{1}}^{−1} = η_{1}(a_{ρ_{1},ι}).

Since a_{ρ_{1},ι} ∈ L_{v}^{×}, the term on the right has a projection on L_{v}^{×} different from 1 only if η_{1} = 1. Then η too equals 1, and so the projection of e_{ρ}(τ) on S^{L}(L_{v}) is

a_{ρ,ι}^{ρι(τ^{−1}−1)μ} = ∏_{Gal(L_{v}/Q_{v})} a_{ρ,σ}^{ρσ(τ^{−1}−1)μ},

in conformity with our assertion.

Suppose T is a torus over Q, provided with a coweight μ such that

(5.13)  (1 + ι)(τ − 1)μ = (τ − 1)(1 + ι)μ = 0

for all τ ∈ Gal(Q̄/Q). Then there exists a unique homomorphism ψ : S → T such that the composition of ψ with the canonical coweight of S is μ. We can transport the cocycles ρ → c_{ρ}(τ) from S to T, obtaining cocycles {c_{ρ}(τ, μ)} as well as b(τ, μ) ∈ T (A_{f}(L)). However if T and h : R → T define a Shimura variety and μ is the restriction of h to the first factor of R the condition (5.13) will not necessarily be satisfied. Nonetheless, we may repeat the previous construction and define b(τ, μ) and {c_{ρ}(τ, μ)} for all τ such that (1+ι)(τ^{−1}−1)μ = 0. This generalization is necessary for the treatment of those Shimura varieties Sh(G, h) for which G is not equal to G_{0}. It should perhaps be observed that b(τ, μ) is not insensitive to the ambiguity in the choice of w = w(τ), although {c_{ρ}(τ, μ)} is. However, if Z is the centre of G the ambiguity all lies in Z(A_{f}) ∩ K, and may be ignored.

If E ⊆ C the motives over E whose associated Hodge structure is of CM type are themselves said to be of CM type. They form a tannakian category CM(E) with a natural fibre functor ω_{CM(E)}, given by rational cohomology. Since one expects that the natural functor CM(Q̄) → CM(C) is an equivalence, we may as well suppose that E ⊆ Q̄. According to the hopes expressed at the end of the previous section there should be an equivalence η : CM(Q̄) → REP(S) and an isomorphism ω_{Rep(S)} ∘ η → ω_{CM(Q^{}¯ )}, which would enable us to identify S with the group G_{CM(Q^{}¯ )}, defined by the category CM(Q̄) and the functor ω_{CM(Q^{}¯ )}. If E ⊆ Q̄, one hopes that in the same way it will be possible to identify T_{E} with G_{CM(E)}.

There are properties which this identification whould have, and it is necessary to describe them explicitly. First of all, if F ⊆ E the diagram

\begin{matrix}S ⟶ & T_{E} ⟶ & T_{F} \\ ↑ & ↕ & ↕ \\ G_{CM(Q^{}¯ )} ⟶ & G_{CM(E)} ⟶ & G_{CM(F)} \end{matrix}

should be commutative.

The other properties are more complicated to describe. Suppose τ ∈ Gal(Q̄/Q) and τ takes E to E′. Its inverse then naturally defines a ⊗-functor η(τ) from CM(E′) to CM(E). Let G_{CM(E′)}^{τ} be the group defined by the category CM(E′) and the functor ω_{CM(E)} ∘ η(τ). The dual of η(τ) is then an isomorphism φ(τ) : G_{CM(E)} → G_{CM(E′)}^{τ}. In terms of representations η(τ) associates to every representation (ξ′, V (ξ′)) of G_{CM(E′)} a representation (ξ, V (ξ)) of G_{CM(E)}. Since ω_{CM(E′)} and ω_{CM(E)} ∘ η(τ) become isomorphic over Q̄ there is a family of homomorphisms, one for each ξ′, ψ(ξ′) : V (ξ′)Q^{}¯ → V (ξ)Q^{}¯ , compatible with sums and tensor products. If ψ′(ξ) is another possible family then there is a t ∈ T_{E′}(Q̄), such that ψ′(ξ′) = ψ(ξ′)ξ′(t). Finally since the two functors ωA_{f}_{CM(E′)} and ωA_{f}_{CM(E)} ∘ η(τ) are canonically isomorphic, arising as they do from the ℓ-adic cohomology, there is a canonical family of isomorphisms ψ_{A_{f}} (ξ′) : V (ξ′)_{A_{f}} → V (ξ)_{A_{f}}.

On the other hand, suppose a(τ) ∈ T (Q̄) maps to τ. Let σ(a(τ)) = c_{σ}(τ)a(τ). We may use a(τ) to associate to every representation (ξ′, V (ξ′)) of T_{E′} a representation (ξ, V (ξ)) of T_{E}. The space V (ξ) is obtained by twisting V (ξ′) by the cocycle {ξ′(c_{σ}(τ)^{−1})}. The representation ξ is t → ξ′(a(τ)ta(τ)^{−1}). This functor (ξ′, V (ξ′)) → (ξ, V (ξ)) is to be (isomorphic to) that obtained from η(τ) by identifying T_{E} and G_{CM(E)} and T_{E′} and G_{CM(E′)}. Moreover one possible choice for ψ(ξ′) is to be the isomorphism ψ_{0}(ξ′) : V (ξ′)Q^{}¯ → V (ξ)Q^{}¯ implicit in the definition of V (ξ). If b(τ)a(τ) is the image of τ under the canonical splitting Gal(Q̄/Q) → T (A_{f}), then ψ_{A_{f}} (ξ′) is to be ψ_{0}(ξ′) ∘ ξ′(b(τ))^{−1}.

If τ ∈ Gal(Q̄/E) then E′ = E and η(τ) : CM(E′) → CM(E) is the identity functor. Thus the functor (ξ′, V (ξ′)) → (ξ, V (ξ)) from Rep(T_{E′}) = Rep(T_{E}) to Rep(T_{E}) must be canonically isomorphic to the identity. A final property, which seems to be independent of the preceding ones, is that this isomorphism should be given by ξ′ → ξ and ψ_{0}(ξ′) ∘ ξ′(a(τ)) : V (ξ′) → V (ξ). Observe that a(τ) is now in T_{E} and these transformations are defined over Q.

I assume that all this is so, just to see where it leads, and especially to see what it suggests about the groups G^{τ,φ} introduced in the previous section. But there are some lemmas to be verified first. I conclude the present section by describing a property of the Taniyama group whose significance was pointed out to me by Casselman. It will be needed to show that the zeta-functions of motives, and especially abelian varieties, of CM type can be expressed as products of the L-functions associated to representations of the Weil group.

The point is that there is a natural homomorphism φ of the Weil group W_{Q} of Q into T (C) and thus for any finite extension F of Q a homomorphism φ_{F} : W_{F} → T_{F} (C). To define it we work at a finite level, defining W_{L/Q} → T^{L}(C), and afterwards passing to the limit.

Fix for now a set of coset representatives w_{σ} which satisfies (a), (b), and (c). If w ∈ W_{L/Q} we define b_{0}(w) = ∏_{σ∈Gal(L/Q)} c_{σ}(w)^{σμ}. If τ is the image of w in Gal(L^{ab}/Q) then b_{0}(w) ≡ b_{0}(τ)(mod L_{∞}^{×} ⊗ X^{∗}(S^{L})), and we may lift b_{0}(w) to b(w) in S^{L}(A(L)) in such a way that the projection of b(w) in S^{L}(A_{f}(L)) is b(τ). However a simple calculation shows that if τ_{1}, τ_{2} are the images of w_{1}, w_{2} then b(w_{1})τ_{1}(b(w_{2}))b(w_{1}w_{2})^{−1} ∈ S^{L}(L). Since its projection on S^{L}(A_{f}(L)) is equal to d_{τ_{1},τ_{2}}, it is itself equal to d_{τ_{1},τ_{2}}. If b_{v}(w) is the projection of b(w) in S^{L}(L_{ν}) = S^{L}(C), we may define φ by φ : w → b_{v}(w)a(τ). If the coset representatives w_{σ} are changed then φ is replaced by φ′ = ada ∘ φ, a ∈ S^{L}(C), but this is of no importance.

If G_{W_{F}} is the group over C defined by the tannakian category of continuous, finite-dimen-sional, complex, semisimple representations of W_{F} then φ_{F} is the composite of the imbedding W_{F} → G_{W_{F}} (C) and an algebraic homomorphism ψ_{F} : G_{W_{F}} → T_{F}. Moreover if the principle of functoriality is valid, there is a surjection G_{Π(F)} → G_{W_{F}} and we can expect to have a diagram

\begin{matrix}G_{Π(F)} ⟶^{ρ_{F}} & G_{Mot(F)} \\ ↓ & ↓ \\ G_{W_{F}} ⟶_{ψ_{F}} & T_{F} \end{matrix}

whose two composite arrows differ by ads, s ∈ S(C) ⊆ T_{F} (C).

6. Conjugation of Shimura varieties. The principal purpose of this section is to formulate a conjecture about the conjugation of Shimura varieties, a conjecture whose first justification is that it is a simple statement which implies what we need for the study of the zeta-functions at archimedean places and is compatible with all that we know. Some lemmas are necessary before it can be stated, and we shall see that these lemmas together with the hypothetical properties of the Taniyama group suggest an answer to the question that arose at the end of the fourth section. This answer in its turn throws new light on the conjecture, so that we can weave a consistent pattern of hypotheses, and our task will be ultimately to show that it has some real validity.

We need a construction, which we make in sufficient generality that it applies to all Shimura varieties and not just those associated to motives. Suppose the pair (G, ℌ) defines a Shimura variety, T and T̄ are two Cartan subgroups of G defined over Q, and h : R → T, h̄ : R → T̄ both lie in ℌ. Let μ and μ̄ be the coweights of T and T̄ obtained by restricting h and h̄ to the first factor of R, and choose a finite Galois extension L which splits T and T̄. Let τ ∈ Gal(Q̄/Q); then the coweights (1 + ι)(τ^{−1} − 1)μ and (1 + ι)(τ^{−1} − 1)μ̄ are both central and they are equal. Choose w = w(τ) as before, and set

b_{0}(τ, μ) = ∏_{σ} c_{σ}(w)^{σμ}, b_{0}(τ, μ̄) = ∏_{σ} c_{σ}(w)^{σμ̄}.

Let b̃(τ, μ) and b̃(τ, μ̄) be liftings to T (A(L)) and T̄(A(L)) and let B(τ) = B(τ, μ, μ̄) be the projection of b̃(τ, μ̄)^{−1}b̃(τ, μ) on G(A_{f}(L)). Although we may not be able to define the cocycle {c_{ρ}(τ, μ)}, we can define {c_{ρ}(τ, μ_{ad})} if μ_{ad} is the composition of μ with the projection to the adjoint group.

It may be as well to check that B(τ) is indeed independent of the choice of w and of the coset representatives w_{σ}, provided the usual conditions (a), (b), and (c) are satisfied. If w_{σ} is replaced by e_{σ}w_{σ} then, apart from a factor in L_{∞}^{×} ⊗ X_{∗}(T), b_{0}(τ, μ) is multiplied by ∏_{η∈𝔣} e_{η}^{η(1+ι)(τ^{−1}−1)μ}, and b_{0}(τ, μ̄) by ∏_{η∈𝔣} e_{η}^{η(1+ι)(τ^{−1}−1)μ̄}. Since these two terms are central and equal by assumption, the change has no effect on B(τ). If w is replaced by xwx^{−1} with xσ(x^{−1}) ∈ L_{v}^{×} for σ ∈ Gal(L_{v}/Q_{v}), then b_{0}(τ, μ) is modified by the product of an element in L_{∞}^{×} ⊗ X_{∗}(T) and ∏_{η} x^{η(1+ι)(τ^{−1}−1)_{μ}}τ. Since b_{0}(τ, μ̄) undergoes a similar modification, B(τ) is not affected. One can also show easily that B(τ) is not changed when L is enlarged; the argument is once again basically the same as that used to treat b(τ).

B(τ) does depend on the choice of b̃(τ, μ) and b̃(τ, μ̄). These choices made, we will use them consistently to define c_{ρ}(τ, μ_{ad}), c_{ρ}(τ, μ̄_{ad}), and when

(1 + ι)(τ^{−1} − 1)μ = (1 + ι)(τ^{−1} − 1)μ̄ = 0,

c_{ρ}(τ, μ), c_{ρ}(τ, μ̄). Thus c_{ρ}(τ, μ_{ad}) is to be the projection of ρ(b̃(τ, μ)^{−1})b̃(τ, μ) in G_{ad}(A_{f}(L)). With these conventions, the ambiguity in B(τ) will cause no harm in the construction to be given next.

Let G^{τ,μ} and G^{τ,μ̄} be the groups obtained from G by twisting with the cocycles {c_{ρ}(τ, μ_{ad})^{−1}} and {c_{ρ}(τ, μ̄_{ad})^{−1}}. We are going to verify the following:

First Lemma of Comparison. (i) If c_{ρ} = c_{ρ}(τ, μ_{ad}) then

γ_{ρ} = B(τ)adc_{ρ}^{−1}(ρ(B(τ)^{−1}))

lies in G^{τ,μ}(L).

(ii) The cocycle {γ_{ρ}} in G^{τ,μ}(L) bounds.

(iii) If (1 + ι)(τ^{−1} − 1)μ = (1 + ι)(τ^{−1} − 1)μ̄ = 0 then

γ_{ρ} = c_{ρ}(τ, μ̄)^{−1}c_{ρ}(τ, μ).

The third assertion is clear; it is the other two with which we must deal. The element γ_{ρ} can be obtained by projecting

(6.1)  b̃(τ, μ̄)^{−1}ρ(b̃(τ, μ̄))ρ(b̃(τ, μ)^{−1})b̃(τ, μ)

on G(A_{f}(L)). This makes it perfectly clear that γ_{ρ} is not affected if w = w(τ) is replaced by xw(τ) with x ∈ C_{L}. Thus we may assume that w = w_{τ}, where τ is here also used to denote the image of τ in Gal(L/Q). Then we have to show that

b̃(τ, μ̄)^{−1}ρ(b̃(τ, μ̄)) ≡ b̃(τ, μ)^{−1}ρ(b̃(τ, μ)) (mod G(L_{∞})G(L)).

It follows easily from (5.11) that if ã_{ρ,η} is a lift of a_{ρ,η} to I_{L}, then both sides are congruent to

∏_{η} ã_{ρ,η}^{ρη(1+ι)(1−τ^{−1})μ} = ∏_{η} ã_{ρ,η}^{ρη(1+ι)(1−τ^{−1})μ̄}.

The desired equality follows.

To prove the second assertion we shall apply Hasse’s principle, but for this we need a group G whose derived group G_{der} is simply connected. Let G_{sc} be the simply connected covering of G_{der}. The Cartan subgroup T defines T_{der} and T_{sc}. We have an imbedding X_{∗}(T_{sc}) → X_{∗}(T) and G_{der} = G_{sc} if and only if the quotient is torsion-free. If we can construct a diagram of Gal(L/Q)-modules

0 ⟶ X_{∗}(T_{sc}) ⟶  Q ⟶ P ⟶ 0
∥ ↓ ↓
0 ⟶ X_{∗}(T_{sc}) ⟶ X_{∗}(T) ⟶ M ⟶ 0

in which P is torsion-free and P → M is a surjection whose kernel is a free Gal(L/Q)-module, then we can use it to define a central extension G′ of G with X_{∗}(T′) = Q. We will have G′_{der} = G′_{sc} = G_{sc}, and G′(R) → G(R) will be surjective. To construct the diagram, we choose an exact sequence of Gal(L/Q)-modules

0 ⟶ N ⟶ P ⟶ M ⟶ 0,

with P torsion-free and N free. Then Q is the set of all (x, ρ) in X_{∗}(T) ⊕ P for which x and ρ have the same image in M.

We lift μ to μ′ = (μ, ν) with ν ∈ P. If μ̄ = adg∘μ and g is the image of g′, we set μ̄′ = adg′∘μ′. If the assertion is valid for μ′, μ̄′, and G′ it is valid for μ, μ̄, and G. Consequently we may suppose that G_{der} is simply-connected.

There are two types of ambiguity in B(τ). It can be changed to tB(τ) with t ∈ T̄(L). Then {γ_{ρ}} is replaced by {(tγ_{ρ}adc_{ρ}^{−1}(ρ(t)^{−1})}, and its cohomology class is not affected. We can also change B(τ) to B(τ)t with t ∈ T (L). Then {c_{ρ}} is replaced by {c′_{ρ}} with c′_{ρ} = ρ(t_{ad}^{−1})c_{ρ}t_{ad}, t_{ad} being the image of t in G_{ad}, and {γ_{ρ}} is replaced by {γ′_{ρ}}, with

γ′_{ρ} = γ_{ρ}adc_{ρ}^{−1}(ρ(t^{−1}))t.

If γ_{ρ} = δadc_{ρ}^{−1}(ρ(δ^{−1})) then

γ′_{ρ} = (δt)ad^{−1}c′_{ρ}(ρ(δt)^{−1}).

Consequently the ambiguity in B(τ) has no effect on the assertion (ii).

Rather than prove (ii) directly for a given choice of the pair μ, μ̄, we want to prove it for a succession of pairs. For this one should first check that the validity of (ii) defines an equivalence relation. If μ and μ̄ are interchanged then {γ_{ρ}} is replaced by {γ_{ρ}^{−1}}, and if γ_{ρ} = δadc_{ρ}^{−1}(ρ(δ^{−1})) then

γ_{ρ}^{−1} = δ^{−1}adc̄_{ρ}^{−1}(ρ(δ)).

To show the transitivity, we introduce a new notation, denoting γ_{ρ} by γ_{ρ}(μ̄, μ), and c_{ρ} by c_{ρ}(μ). Suppose

\begin{aligned}γ_{ρ}(μ_{2}, μ_{3}) & = tadc_{ρ}^{−1}(μ_{3})(ρ(t^{−1})), \\ γ_{ρ}(μ_{1}, μ_{2}) & = sadc_{ρ}^{−1}(μ_{2})(ρ(s^{−1})). \end{aligned}

Observing that

γ_{ρ}(μ_{2}, μ_{3})adc_{ρ}^{−1}(μ_{3})(ρ(s^{−1}))γ_{ρ}(μ_{2}, μ_{3})^{−1} = adc_{ρ}^{−1}(μ_{2})(ρ(s^{−1}))

and that γ_{ρ}(μ_{1}, μ_{3}) = γ_{ρ}(μ_{1}, μ_{2})γ_{ρ}(μ_{2}, μ_{3}), one deduces with little effort that

γ_{ρ}(μ_{1}, μ_{3}) = stadc_{ρ}(μ_{3})(ρ(st)^{−1}).

Transitivity established, we return to the original notation. If μ̄ is conjugate to μ under G(Q), say μ̄ = adx ∘ μ then

γ_{ρ} = xadc_{ρ}^{−1}(x^{−1}) = xadc_{ρ}^{−1}(ρ(x^{−1})),

and certainly bounds. In general μ̄ and μ are not conjugate under G(Q), but they are conjugate under G(R). Since G(Q) is dense in G(R) we may take advantage of the transitivity and assume that they are conjugate under G_{der}(R).

It is now that the assumption that G_{der} is simply connected intervenes. If we are careful in our choice of b̃(τ, μ) and b̃(τ, μ̄), defining them by liftings of c_{σ}(w) to I_{L}, then B(τ) and the γ_{ρ} will lie in G_{der}^{τ,μ}. Moreover the cocycle {γ_{ρ}} in G_{der}^{τ,μ}(L) certainly bounds at every finite place. Since we are applying Hasse’s principle, we need only verify that it bounds at infinity as well.

One begins with a calculation similar to the one made while studying the cocycle {c_{ρ}(τ)}. If ρ ∈ Gal(L/Q), set

e_{ρ}(τ, μ) = ∏_{η∈𝔣} a_{ρη,ι}^{ρη(τ^{−1}−1)μ}

and define e_{ρ}(τ, μ̄) in a similar fashion. The projection of {γ_{ρ}} on G^{τ,μ}(L_{∞}) is cohomologous to e_{ρ}(τ, μ̄)e_{ρ}(τ, μ)^{−1}. If ρ ∈ Gal(L_{v}/Q_{v}) the projection of e_{ρ}(τ, μ) on G^{τ,μ}(L_{v}) is

f_{ρ}(τ, μ) = ∏_{σ∈Gal(L_{v}/Q_{v})} a_{ρ,σ}^{ρσ(τ^{−1}−1)μ}.

Thus all we need do is show that f_{ρ}(τ, μ̄)f_{ρ}(τ, μ)^{−1} bounds in G^{τ,μ}(L_{v}). Recall that the cocycle defining G^{τ,μ}(L_{v}) is

h_{ρ} = ∏_{σ∈Gal(L_{v}/Q_{v})} a^{ρσ(τ^{−1}−1)μ_{ad}}.

For brevity, we write {∏_{σ} a_{ρ,σ}^{ρσν}} = {α_{ρ}(ν)}. If x ∈ G(R) we write x(μ) = adx ∘ μ. If x(μ̄) = μ then

xf_{ρ}(τ, μ̄)f_{ρ}(τ, μ)^{−1}adh_{ρ}(ρ(x^{−1})) = α_{ρ}((xτ^{−1}x^{−1} − τ^{−1})μ),

because ρ(x) = x and

f_{ρ}(τ, μ)^{−1}adh_{ρ}(x^{−1}) = x^{−1}f_{ρ}(τ, μ)^{−1}.

If w lies in the normalizer of T_{der} in G_{der}(L_{v}) then

(6.2)  wadh_{ρ}(ρ(w^{−1})) = wρ(w^{−1})α_{ρ}((w − 1)(τ^{−1} − 1)μ).

However {wρ(w^{−1})} is a cohomology class in T_{der}(L_{v}) or T_{der}^{τ,μ}(L_{v}), the two groups being equal, and it is shown in [45] that it is equivalent to {α_{ρ}((w − 1)μ)}. Thus the class of (6.2) in G_{der}^{τ,μ}(L_{v}) is {α_{ρ}((w − 1)τ^{−1}μ)}. Since we may take w such that its action on the weights of T is the same as that of xτ^{−1}x^{−1}τ, the verification is complete.

There is another fact that we should verify while this proof is fresh in our minds. Suppose there is an ω in the Weyl group such that

(6.3)  ωμ = τ^{−1}μ.

Then (1 + ι)(τ^{−1} − 1)μ = 0 and {c_{ρ}^{−1}(τ)} = {c_{ρ}^{−1}(τ, μ)} is defined. It bounds in G(L). Once again it is enough to verify this when G_{der} is simply connected, although this time the modifications made to arrive at a simply-connected derived group would be different. It is no longer important that P → M be surjective, or that its kernel be free, but there must be a ν ∈ P which maps to μ and is fixed by Gal(Q̄/E).

The set of all τ which satisfy (6.3) for at least one ω form a group. Let E = E(G, h) = E(G, ℌ) be its fixed field. Then E ⊆ L. In order to apply Hasse’s principle we have to arrange that {c_{ρ}(τ)} lies in G_{der}(L) and that it obviously bounds in G_{der}(A_{f}(L)). Since c_{ρ}(τ) only depends on the image of τ in Gal(L/Q), we may calculate it when w = w_{τ}. If 𝔖 is a set of coset representatives for Gal(L/Q)/Gal(L/E), then

b_{0}(τ) = ∏_{ν∈𝔖} ∏_{σ∈Gal(L/E)} a_{νσ,τ}^{νσμ} .

We write a_{νσ,τ} = ν(a_{σ,τ})a_{ν,στ} a_{ν,σ}^{−1} and νσμ = νμ + ν(σ − 1)μ. Now for σ ∈ Gal(L/E), ν(σ − 1)μ is a weight of the derived group and ∏_{ν} ∏_{σ} a_{νσ,τ}^{ν(σ−1)μ} may be lifted to T_{der}(A(L)). On the other hand ∏_{ν} ∏_{σ} a_{ν,στ}^{νμ} a_{ν,σ}^{−νμ} = 1. If a = ∏_{σ} a_{σ,τ} then a lies in C_{E} and lifts to a′ in I_{E}. Moreover ∏_{ν} ν(a)^{νμ} lifts to ∏_{ν} ν(a′)^{νμ} which lies in T (A). If ā_{νσ,τ} is a lifting of a_{νσ,τ} to I_{L} then we so choose c_{ρ}(τ) that

(6.4)  c_{ρ}(τ) ≡ {∏_{ν,σ} ρ(ā_{νσ,τ})^{−ρν(σ−1)ν}} {∏_{ν,σ} ā_{νσ,τ}^{ν(σ−1)μ}}

modulo T (L_{∞}).

It remains to verify that the {c_{ρ}^{−1}(τ)} so defined bounds at the infinite place. The projection of the right side of (6.4) on C_{L} ⊗ X_{∗}(T_{der}) is

∏_{ν,σ} a_{ρ,νσ}^{ρνσ(τ^{−1}−1)μ} = ∏_{σ∈Gal(L/Q)} a_{ρ,σ}^{ρσ(τ^{−1}−1)μ}.

Thus, restricting {c_{ρ}^{−1}(τ)} to Gal(L/Q_{v}) and projecting on T_{der}(L_{v}), we obtain a class cohomologous to {α_{ρ}((τ^{−1} − 1)μ)} = {α_{ρ}((ω − 1)μ)}. We have observed already that it is shown in [45] that the right side bounds in G_{der}(L_{v}).

Although there is one more consequence to be derived from (6.3), there are some things that must first be said about the general (T, h), or (T, μ).

We drop the assumption (6.3) for a while, and return to it later. We have associated to the pair (T, μ) and τ a twisted form G^{τ,μ} of G. The twisting of T in G is trivial, and T^{τ} = T is a Cartan subgroup of G^{τ,μ}. Let μ^{τ} be τ^{−1}μ. There is a unique homomorphism h^{τ} : R → T^{τ} whose restriction on the first factor is μ^{τ} and which is defined over R.

The pair (G^{τ,μ}, h^{τ}) defines a Shimura variety.

The roots {γ} of T in G are the same as the roots of T^{τ} in G^{τ,μ}. However the classification into compact and noncompact differs for the two pairs. The root γ of T in G is compact or noncompact according as (−1)^{⟨τ,μ⟩} is 1 or −1. In order for (G^{τ,μ}, h^{τ}) to define a Shmura variety, γ must be compact or noncompact as a root of T^{τ} according as (−1)^{⟨γ,τ^{−1}μ⟩} is 1 or −1. However the ideas used in the proof of Lemma A.8 of [34] show that the type of γ is changed on passing from T, G to T^{τ} , G^{τ,μ} if and only if (−1)^{⟨γ,τ^{−1}μ−μ⟩} = −1.

We are now almost ready to formulate a conjecture about the conjugation of Shimura varieties. After discussing the conjecture and its consequences, we shall show how it can be heuristically justified in terms of the Taniyama group and motives.

Recall that Sh_{K}(C) = G(Q)\ℌ × G(A_{f})/K. Thus if g ∈ G(A_{f}) and K_{1} = g^{−1}Kg, then right multiplication by g defines a morphism

𝔉(g) : Sh_{K}(C) ⟶ Sh_{K_{1}}(C).

It is algebraic and is called a Hecke correspondence. If τ is an automorphism of C, we set Sh_{K}^{τ}(G, h) = Sh_{K}^{τ} = Sh_{K} ⊗_{τ^{−1}} C, the Shimura varieties being at the moment only defined over C. Then 𝔉^{τ} (g) : Sh_{K}^{τ} → Sh_{K_{1}}^{τ}.

If G = T is a torus and K = U then Sh_{U} is 0-dimensional. Let τ also denote the element of Gal(Q̄/Q) defined by τ. If we set U^{τ} = U then

Sh_{U}(C) = T (A_{f})/U = T^{τ} (A_{f})/U^{τ} = Sh_{U^{τ}} .

This gives us an isomorphism Sh_{U} = Sh_{U}(T, h) → Sh_{U^{τ}} = Sh_{U^{τ}} (T^{τ} , h^{τ}). In addition there is the natural map τ from complex points of Sh_{U}^{τ}(T, h) to complex points of Sh_{U}(T, h). Define φ_{τ} = φ_{τ} (U, T, h) by the commutativity of the diagram

Sh_{U}(T, h) ↗^{τ}

Sh_{U}^{τ}(T, h)  ↓

↘^{φ_{τ}} Sh_{U^{τ}} (T^{τ} , h^{τ})

In general G^{τ,μ} and G are different. But G^{τ,μ} is defined by the cocycle {c_{ρ}^{−1}(τ, μ_{ad})} and in G_{ad}(A_{f}(L))

c_{ρ}^{−1}(τ, μ_{ad}) = b(τ, μ_{ad})^{−1}ρ(b(τ, μ_{ad})).

Thus

g ⟶ g^{τ} = adb(τ, μ_{ad})^{−1}(g)

defines an isomorphism of G(A_{f}) with G^{τ,μ}(A_{f}).

Conjecture. There is a family of biregular maps φ_{τ} = φ_{τ} (K, G, h), K ⊆ G(A_{f}) defined over C, taking Sh_{K}^{τ}(G, h) to Sh_{K^{τ}} (G^{τ,μ}, h^{τ}), and rendering the following diagrams commutative:

(a)

Sh_{U}(T, h)  ⊂⟶  Sh_{K}(G, h)
τ↑  ↑τ
Sh_{U}^{τ}(T, h)  ⊂⟶  Sh_{K}^{τ}(G, h)
φ_{τ} ↓  ↓φ_{τ}
Sh_{U^{τ}} (T^{τ} , h^{τ})  ⊂⟶  Sh_{K^{τ}} (G^{τ,μ}h^{τ})

Here U = T (A_{f}) ∩ K and the φ_{τ} in the left column is φ_{τ} (U, T, h).

(b)

Sh_{K}^{τ}(G, H) ⟶^{𝔉^{τ} (g)} Sh_{K_{1}}^{τ}(G, h)
φ_{τ} ↓ ↓φ_{τ}
Sh_{K^{τ}} (G^{τ,μ}, h^{τ}) ⟶_{𝔉(g^{τ})} Sh_{K_{1}^{τ}} (G^{τ,μ}, h^{τ})

The conjecture in this form refers to a specific T and a specific h factoring through T. If we choose another pair (T̄, h̄) we obtain another group G^{τ,μ̄} and another collection {φ̄_{τ} = φ̄_{τ} (K; G, h̄)}. The conjecture is inadequate as it stands, and must be supplemented by a statement relating φ_{τ} and φ̄_{τ}. Observe that there can be at most one family {φ_{τ} } satisfying the conditions (a) and (b).

We have already associated to the two pairs (T, h) and (T̄, h̄) a cocycle {γ_{ρ}} which bounds in G^{τ,μ}(L). Let γ_{ρ} = uadc_{ρ}^{−1}(ρ(u^{−1})). Then g → ugu^{−1} defines an isomorphism of G^{τ,μ}(Q) with G^{τ,μ̄}(Q) or of G^{τ,μ}(A_{f}) with G^{τ,μ̄}(A_{f}). Set ^{u}K^{τ} = uK^{τ} u^{−1}.

Second Lemma of Comparison. The homomorphism adu ∘ h^{τ} is conjugate under G^{τ,μ̄}(R) to h̄^{τ}.

If this statement is true when u is replaced by a v in G(L_{v}) = G(C) which also trivializes {γ_{ρ}} restricted to Gal(L_{v}/Q_{v}) then it is true for u. An examination of the proof of the second property of {γ_{ρ}} shows that we can take v to lie in x^{−1}wT (L_{v}), x and w being as in that proof. Since x^{−1}w(τ^{−1}μ) = τ^{−1}μ̄, we have adν ∘ h^{τ} = h̄^{τ}.

We infer that adu carries ℌ to ℌ̄^{τ} and hence defines a bijection

G^{τ,μ}(Q)\ℌ^{τ} × G^{τ,μ}(A_{f})/K^{τ} ⟶ G^{τ,μ̄}(Q)\ℌ̄^{τ} × G^{τ,μ̄}(A_{f})/^{u}K^{τ}

and an isomorphism

ψ : Sh_{K^{τ}} (G^{τ,μ}, h^{τ}) ⟶ Sh_{μ_{K^{τ}}} (G^{τ,μ̄}, h̄^{τ}).

Since u and B(τ) both trivialize {γ_{ρ}} in G^{τ,μ}(A_{f}(L)) there is a y in G^{τ,μ}(A_{f}) with u = B(τ)y. Let φ be the composite ψ ∘ 𝔉(y).

Supplement to the conjecture. The diagrams

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are commutative.

The conjecture as it stands certainly implies that the conjugate of a Shimura variety is again a Shimura variety. Together with its supplement, it implies the usual form of Shimura’s conjecture [10]. To verify this one applies the Weil criterion [49] for descent of the field of definition. For this we need families of isomorphisms f_{ρ} : Sh_{K}^{ρ}(G, h) ⟶ Sh_{K}(G, h) defined for automorphisms ρ of C over E(G, h) and satisfying f_{σρ} = f_{ρ}f_{σ}^{ρ}.

Choose a Cartan subgroup T and an h which factors through it. We know that when τ fixes E(G, h) the cocycle {c_{ρ}^{−1}(τ, μ)} is defined and bounds in G(L). Let c_{ρ}^{−1}(τ, μ) = νρ(ν^{−1}). Then g → νgν^{−1} is an isomorphism of G(Q) with G^{τ,μ}(Q). Methods which we have already used show easily that

The composite adν ∘ h is conjugate under G^{τ,μ}(R) to h^{τ}.

Consequently ad ν defines an isomorphism ℌ → ℌ^{τ} and then, as before, we obtain from

G(Q)\ℌ × G(A_{f})/K ⟶ G^{τ,μ}(Q)\ℌ^{τ} × G^{τ,μ}(A_{f})/^{ν}K

an isomorphism

S_{K}(G, h) ⟶ S_{ν_{K}} (G^{τ,μ}, h^{τ}).

On the other hand zν = b(τ, μ)^{−1} with z ∈ G^{τ,μ}(A_{f}). We define f_{τ} by the commutativity of

Sh_{K}^{τ}(G, h) ⟶^{f_{τ}}  S_{K}(G, h)
φ_{τ} ↓  ↓
Sh_{K^{τ}} (G^{τ,μ}h^{τ}) ⟶_{𝔉(z)} Sh_{ν_{K}} (G^{τ,μ}, h^{τ})

I omit the calculations, lengthy but routine, by which it is deduced from the conjecture and its supplement that f_{τ} does not depend on the choice of T and h and that the cocycle condition f_{σρ} = f_{ρ}f_{σ}^{ρ} is satisfied.

Up to now the S_{K} have been taken as varieties over C, but by the criterion for descent we may now define them over E(G, h) in such a way that the f_{τ} are simply the identity maps. It has to be verified that the models thus obtained are canonical, but the construction is clearly such that only the case that G is a torus T need by considered. Let a be the transfer of w = w_{τ} to C_{E} and a′ a lifting of a to I_{E}. The proof that in this case c_{ρ}(τ, μ) is trivial shows in fact that we may take it to be 1 and b = b(τ, μ) to be ∏_{Gal(L/Q)/Gal(L/E)} ν(a′)^{νμ}. We take ν to be 1 and z to be b^{−1} = b(τ, μ)^{−1}. The composition

Sh_{U}(T, h) ⟶^{τ^{−1}} Sh_{U}(T, h) ⟶^{𝔉(b)} Sh_{U}(T, h)

is then the identity for all τ fixing E(T, h), and this is just the condition that Sh(T, h) be the canonical model.

Suppose E(G, h) ⊆ R. If we take the canonical model for Sh_{K} then the complex conjugation defines an involution θ of the complex manifold Sh_{K}(C). It is necessary to have a concrete description of this involution in terms of the representation

Sh_{K}(C) = G(Q)\ℌ × G(A_{f})/K,

and one purpose of the conjecture is to provide it.

Choose some special point (T, h). If E(G, h) ⊆ R then we may define b(ι, μ) and {c_{ρ}(ι, μ)}. However the condition (c) on the coset representatives w_{σ} used to define b(ι, μ) implies that b(ι, μ) = 1. We may also take c_{ρ}(ι, μ) to be 1, and then ν may be taken to be 1 as well. It follows that h and h^{ι} are conjugate in G(R). Since K_{h^{ι}} = K_{h}, η : adg ∘ h^{ι} → adg ∘ h is a well-defined map of ℌ to itself.

Consequence of the conjecture. The involution θ may be realized concretely as the mapping (h, g) → (η(h), g) of G(Q)\ℌ × G(A_{f})/K to itself.

Since we are comparing two continuous mappings which commute with the 𝔉(g), g ∈ G(A_{f}), it is enough to see that they coincide on the point in S_{U}(T^{ι}, h^{ι}) represented by (h^{ι}, 1). Since f_{ι} is the identity and ν = z = 1, φ_{ι} takes this point to the point in Sh_{K^{ι}}(G^{ι,μ}, h^{ι}) = Sh_{K}(G, h) represented again by (h^{ι}, 1). It follows immediately from condition (a) of the conjecture that ι applied to the point represented by (h^{ι}, 1) is (h, 1).

For each T and μ let ν(μ) be the fibre functor from REP(G) to REP(G^{τ,μ}) which takes (ξ′, V (ξ′)), to (ξ, V (ξ)) with ξ = ξ′, and with V (ξ) being the space obtained from V (ξ′) by changing the Galois action to σ : x → ξ′(c_{σ}(τ, μ)^{−1})σ(x). If (T̄, h̄) is another special point, and if ν(μ, μ̄) is defined by the diagram

\begin{matrix}REP(G^{τ,μ}) & _{↖_{ν(μ)}} \\ ↓_{ν(μ,μ̄)} & REP(G) \\ REP(G^{τ,μ̄}) & ↙_{ν(μ̄)} \end{matrix}

then, according to the first lemma of comparison, the two fibre functors ω_{Rep(G^{τ,μ})} and ω_{Rep(G^{τ,μ̄})} ∘ ν(μ, μ̄) are isomorphic.

On the other hand, we associated, at the end of the fourth section, to each pair (φ′, g) a group G^{τ,φ} and a pair (φ′, g′). The groups G and G^{τ,φ} are associated to the same tannakian category, and there is thus an equivalence of categories ν(φ) : REP(G) → REP(G^{τ,φ}), determined up to isomorphism. If φ factors through the Serre group, then it can be factored through a Cartan subgroup T̄ of G and defines a coweight μ̄ of T. The hypothetical properties of the Taniyama group imply that G^{τ,φ} may be taken to be G^{τ,μ̄} with ν(φ) being ν(μ̄). Thus we have a diagram

\begin{matrix}REP(G^{τ,μ}) & _{↖_{ν(μ)}} \\ ↓_{ν(μ,φ)} & REP(G) \\ REP(G^{τ,φ}) & ↙_{ν(φ̄)} \end{matrix}

which is commutative up to isomorphism of functors. Moreover the two fibre functors ω_{Rep(G^{τ,μ})} ∘ ν(μ, φ) are isomorphic. If we choose an isomorphism between them, we obtain [40, II.3.3] an isomorphism over Q, G^{τ,φ} ⟶ G^{τ,μ}. Composing with φ′ we obtain a homomorphism φ′′ : G_{Mot(C)} ⟶ G^{τ,μ}. Since there are so many special points, it is not unreasonable to hope that ν(μ, φ) exists for all φ, and that ω_{Rep(G^{τ,μ})} and ω_{Rep(G^{τ,φ})} ∘ ν(μ, φ) are always isomorphic.

The second lemma of comparison in conjunction with the hypothetical properties of the Taniyama group implies that the composition of φ′′ with the canonical homomorphism R → G_{Mot(C)} lies in ℌ^{τ} with φ factors through the Serre group, and once again we may surmise or hope that this will be so in general.

If φ_{τ} is the biregular map appearing in the conjecture then the composition φ_{τ} ∘ τ^{−1} defines a map from the set of complex points on Sh_{K}(G, h) to the set of complex points on Sh_{K}^{τ}(G^{τ,μ}, h^{τ}). The idea is that φ_{τ} ∘ τ^{−1} will take (φ, g) to a pair (φ′′, g′′), by a process which can be defined within the moduli problem. We have just seen how to obtain φ′′, at least at the hypothetical level at which we are working.

To obtain g′′ we observe that we have two homomorphisms

(6.5)  ωA_{f}_{Rep(G^{τ,φ})} ∘ ν(μ, φ) ⟶ ωA_{f}_{Rep(G^{τ,μ})}.

One is obtained from the chosen isomorphism over Q by extending scalars to A_{f}. The other is obtained by a lengthy composition. The g from which we start provides an isomorphism ωA_{f}_{Mot(C)} ∘ η → ωA_{f}_{Rep(G)}. The canonical isomorphism ωA_{f}_{Mot(C)} ∘ η(τ) → ωA_{f}_{Mot(C)} can be composed with η. Finally the definitions provide an isomorphism from ω_{Rep(G^{τ,φ)}} ∘ ν(φ) to ωA_{f}_{Mot(C)} ∘ η(τ) ∘ η. Putting these all together, we obtain an isomorphism

(6.6)  ωA_{f}_{Rep(G^{τ,φ)}} ∘ ν(φ) ⟶ ωA_{f}_{Rep(G)}.

However we also have an isomorphism

(6.7)  ωA_{f}_{Rep(G)} ⟶ ωA_{f}_{Rep(G^{τ,μ)}} ∘ ν(μ).

It is given by the isomorpphism x → ξ′(b(τ, μ)^{−1})x of V (ξ′) with V (ξ). Composing (6.6) and (6.7) we obtain a second isomorphism between the two fibre functors figuring in (6.5). According to general principles, it can be obtained by composing the first with an element (g′′)^{−1} in G^{τ,μ}(A_{f}) [40, §II].

If one can establish, in some way or another, that the map ψ_{τ} : (φ, g) → (φ′′, g′′) is really defined, then to prove the conjecture and its supplement one will only need to verify that the composite ψ_{τ} ∘ τ is complex analytic. However our purpose here has been to see how the wheels mesh, not to find the mainspring.

7. Continuous cohomology. If G = G_{0} then, according to the principles of the fourth section, we should be able to attach to each point of Sh_{K}(C) an equivalence class of pairs (φ, g). Here φ is a homomorphism from G_{Mot(C)} to G defined over Q and if η is the associated θ-functor REP(G) → MOT (C), then g defines an isomorphism

ωA_{f}_{Mot(C)} ∘ η → ωA_{f}_{Rep(G)}.

In general we have mappings Sh_{K}(G, h) → Sh_{K_{0}}(G_{0}, h_{0}), and by pulling back we can associate to each point of Sh_{K}(C) a pair (φ, g) where g is again in G(A_{f}), but φ now takes G_{Mot(C)} to G_{0}.

If (ξ, V (ξ)) is a representation of G_{0} over Q or, what is the same, a representation of G factoring through G_{0}, then to each point s of Sh_{K}(C) we may associate the motive M(s, ξ) defined by ξ ∘ ϕ, together with the isomorphism

ωA_{f}_{Mot(C)}(M(s, ξ)) ≃ V (ξ)_{A_{f}}

defined by g. The variety Sh_{K}(G, h) should be defined over E = E(G, h) and so should this family of motives. Suppose now that the variety Sh_{K}(G, h) is proper. In the best of all possible worlds, one might be able to form the cohomology groups M′, 0 ≦ i ≦ 2 dim Sh_{K}, of the family M(·, ξ), which would again be motives, now over E, and thus correspond to representations σ^{i} of G_{Mot(E)}. If the formalism of the second section were established, one could compose σ^{i} with ρ_{F} to obtain representations ρ^{i} = σ^{i}(G_{Π(E)}). Then the basic problem would simply be to describe the image ρ^{i}(G_{Π(E)}).

But we do not have all this formalism, and one of the principal reasons for studying Shimura varieties is the hope that by grappling with the specific arithmetic problems they pose we will obtain an insight that will help us with its construction. Informed by the general principles and hypotheses we are attempting to establish, we can try to formulate questions that are, at least in part, tractable and which if answered will confirm or, if the answer is other than expected, perhaps refute these principles.

In the present context we can first observe that even if the M^{i} remained undefined, the zetafunction Z(z, M^{i}) can be defined directly in terms of the data at our disposal. It is a product over the places of E, ∏_{v} Z_{v}(z, M^{i}). At a nonarchimedean place it can be defined by the ℓ-adic representation of the Galois group on the ith cohomology group of the ℓ-adic sheaf F_{ξ}(Q_{ℓ}) associated to ξ, as in the papers [30] and [34]. Since our principal concern now is with the factors for the archimedean places, we need not enter into details.

The field E is contained in C, and the archimedean places are obtained by applying automorphisms of C, or of Q̄, to E. We first define the factor Z_{v}(z, M^{i}) for the place v given by E ⊆ C. We have seen that we can associate to ξ a locally constant sheaf F_{ξ}(Q) over Sh_{K}(C). Moreover, we have an analytic family of polarized Hodge structures on F_{ξ}(C) = F_{ξ}(Q) ⊗ C. By a construction of Deligne [12], [50] this defines a Hodge structure on the cohomology groups H^{i} = H^{i}(Sh_{K}(C), F_{ξ}(C)).

In accordance with the ideas of Serre [42] the factor Z_{v}(z, M^{i}) will be defined as L(s, ρ^{i}) where ρ^{i} is a representation of the Weil group W_{C/E_{v}} on H^{i}. The Hodge structure on H^{i} defines a representation of R(R). Since C^{×} = R(R), this can be used to define ρ^{i} on C^{×} ⊆ W_{C/E_{v}}. If E_{v} is equal to C, this defines ρ^{i} completely. If E_{v} = R then to define ρ^{i} completely we also have to define ρ^{i}(w), if w is the element of the Weil group which projects to ι and has square −1.

Since Sh_{K} is defined over E, ι also defines an involution on Sh_{K}(C) which we denoted by θ. What we need is a map of order two

ψ : θ^{∗}F ⟶ F, F = F_{ξ}(C),

such that on each fibre

θ^{∗}F_{s}^{p,q} = F_{θ(s)}^{p,q} ⟶ F_{s}^{q,p}.

The associated map ι^{∗} on cohomology takes H^{p,q} to H^{q,p} and we set ρ^{i}(w) equal to (−1)^{p}ι^{∗}.

To define ψ we have to assume that the consequence of the conjecture which was described in the previous section is valid. It can be proved directly in several cases. Then θ can be obtained by taking the map (h, g) → (η(h), g) and passing to the quotient. Given (h, g), the fibres at the image of (h, g) and (η(h), g) may both be identified with V (ξ)_{C} and ψ is simply the identity map.

If we replace the imbedding E ⊆ C by τ^{−1} : E → C, τ being an automorphism, then the complex manifold Sh_{K}(C) is replaced by Sh_{K}^{τ}(C), which we may identify by means of φ_{τ} with Sh_{K^{τ}} (C), the manifold associated to Sh_{K^{τ}} (G^{τ,μ}, h^{τ}). Thus the factor of the zeta-function defined by the place associated to τ^{−1} : E = E(G, h) → C can be calculated by replacing G by G^{τ,μ}, h by h^{τ}, and E by τ^{−1}(E) = E(G^{τ,μ}, h) with the place defined by its inclusion in C. The space V (ξ) has also to be twisted by the cocycle {ξ(c_{σ}(τ, μ)^{−1})}.

The function Z(z, M^{i}) defined, the immediate problem is to show that it can be expressed as a product of L-functions associated to automorphic representations

(7.1)  Z(z, M^{i}) = ∏_{j} L(z − a_{j}, π_{j}, r_{j}).

Here a_{j} ∈ C is a translation, π_{j} is an automorphic representation of some group H_{j}, and r_{j} is a representation of the L-group ^{L}H_{j}. The first step is to decide which H_{j}, which π_{j}, and which r_{j} intervene in the product.

The first step is to use the theory of continuous cohomology to compute the cohomology groups of the sheaves F_{ξ}(C) together with their Hodge structure, and thus to compute Z_{v}(z, M^{i}), v being again defined by E ⊆ C. Using this together with an analysis of the L-packets of automorphic representations of G [44], one searches for an identity (7.1) which is at least valid when both sides are replaced by their factors at v. An example is discussed in detail in [34]. The identity found, it must be verified for the local factors at the other places v′. If v′ is an archimedean place, then the theory of continuous cohomology will allow us to compute Z_{v′}(z, M^{i}) in terms of the automorphic representations of a G^{τ,μ}, a group which differs from G by an inner twisting. To make the comparison it will be necessary to have established the principle of functoriality for the pair G and G^{τ,μ}, and to have understood in detail how it manifests itself. This is bound up with the study of L-packets and is primarily an analytic problem, for we expect that the trace formula will give us a good purchase on it [23].

At the finite places the identity (7.1) is difficult to treat as it stands, and for reasons familiar from topology one replaces the left side by

(7.2)  Z(z) = ∏_{i} Z(z, M^{i})^{(−1)^{i}}

modifying the right side accordingly. The right side is then analyzed by the trace formula, at least if there is no ramification or, at worst, a mild sort [8], [14], [30]. I do not see at the moment any general way of dealing with a truly nonabelian situation, although a rather curious method has been discovered by Deligne for treating the group GL(2) [13].

If there is no ramification, the factor of (7.2) at a finite place can be analyzed by the fixed point formulae of ℓ-adic cohomology. Apart from combinatorial difficulties [28] the critical factor is to have a reasonably explicit description of the set of geometric points on Sh_{K}(G, h) in κ̄_{𝔭}, the algebraic closure of the residue field at a prime 𝔭 of E, together with the action of the Frobenius on it [32]. This is an idea first applied by Ihara [20], who has since intensively studied the structure of this set for Shimura curves [21], [22].

Not much has been done when there is ramification. The first thing is to analyze in reasonably simple cases the manner in which the variety reduces badly. Some interesting discoveries have been made for curves [8], [14], [15], but higher dimensional varieties behave in a more complicated manner. However tools are available for studying their reduction, and it is time to begin.

None of these steps will be easy to carry out. The study of L-packets is in an embryonic stage, and even the combinatorial problems will demand considerable ingenuity in their solution [27]. There is still a great deal to be learned from the study of specific examples.

The theory of continuous cohomology is itself in its infancy, and my purpose in this section is to draw attention to some problems which arise in the study of Shimura varieties and which the mature theory should resolve.

I begin by introducing a representation r of the L-group ^{L}G which will play a fundamental role in the discussion. The group ^{L}G is a semidirect product ^{L}G^{∘} × Gal(Q̄/Q). If T is a Cartan subgroup of G over Q then there is an isomorphism of X_{∗}(T), the lattice of coweights of T, with X^{∗}(^{L}T^{∘}), the lattice of weights of ^{L}T^{∘}, defined up to an element of the Weyl group. In particular if (T, h) is a special point then μ defines an orbit Θ in X^{∗}(^{L}T^{∘}), and Θ is independent of (T, h). Let r^{∘} be the representation of ^{L}G^{∘} whose set of extreme weights is Θ. The group Gal(Q̄/Q) acts on the weights of ^{L}T^{∘} and preserves the set of dominant weights. The group Gal(Q̄/E) fixes the set Θ.

Thus Gal(Q̄/E) fixes the dominant element μ^{∨} in Θ, and we may extend r^{∘} to ^{L}G^{∘} × Gal(Q̄/E) in such a way that Gal(Q̄/E) acts trivially on the weight space of μ^{∨}. The extended representation will also be called r^{∘}, and we define r to be the induced representation

r = Ind(^{L}G,^{L}G^{∘} × Gal(Q̄/E), r^{∘}).

Let d be the dimension of Sh_{K}(G, h). One expects that the function (7.2) will be equal to a product of functions

(7.3)  L(z − d/2, π, ρ).

Here π is an automorphic representation of one of the groups H attached to G in [33]. There is, in general, an imbedding φ : ^{L}H ↪ ^{L}G and ρ is a subrepresentation of r ∘ φ.

If w is any place of Q, let r_{w} be the restriction of r to the L-group ^{L}G_{w}, which equals LG^{∘} × Gal(Q̄_{w}/Q_{w}). Implicit in this notation is an imbedding Q̄ ⊆ Q̄_{w}. Then the double cosets in Gal(Q̄/E)\Gal(Q̄/Q)/Gal(Q̄_{w}/Q_{w}) parametrize the places of E dividing w, and r_{w} = ⊕_{v|w}r_{v}, with

r_{v} = Ind(^{L}G_{w},^{L}G^{∘} × Gal(Q̄_{w}/E_{v}), r_{v}^{∘})

and r_{v}^{∘}(τ) = r^{∘}(σ_{v}τσ_{v}^{−1}) if σ_{v} is some element in the coset defining v. The representation ρ will also be a direct sum ρ = ⊕_{v|w}ρ_{v}, and the function (7.3) will be a product ∏_{w} ∏_{v|w} L(z − d/2, π_{w}, ρ_{v}). The factor corresponding to the place v is L(z − d/2, π_{w}, ρ_{v}).

Since we shall only be interested in the place v given by E ⊆ C, we shall write r and ρ instead of r_{v}. Moreover we shall write an automorphic representation of G(A) (or of H(A)) as π ⊗ π_{f}, π being a representation of G(R) and π_{f} of G(A_{f}).

Any irreducible representation π of G(R) lies in some L-packet Π_{φ} where φ is a homomorphism from W_{C/R} to ^{L}G. If α is the character of W_{C/R} obtained by composing W_{C/R} → R with the absolute value, let ψ_{1}(π) = α^{−d/2} ⊗ (r ∘ φ). Then L(s − d/2, π, r) = L(s, ψ_{1}(π)).

On the other hand, suppose π ⊗ π_{f} is an automorphic representation of G(A). Let it act on the subspace U ⊗ U_{f} of the space of automorphic forms. If h ∈ ℌ then, according to the principles of continuous cohomology [4], its contribution to the cohomology of Sh_{K}(C) with values in F_{ξ}(C) in dimension i is

(7.4)  Hom_{K_{h}}(Λ^{i}𝔤/𝔨 ⊗ Ṽ , U) ⊗ U_{f}^{K}.

Here 𝔨 is the Lie algebra of K_{h} and Ṽ the dual of V (ξ). The space U_{f}^{K} is the space of vectors in U_{f} fixed by K.

The action of z ∈ C^{×} = R(R) defining the Hodge structure sends φ ⊗ u to φ′ ⊗ u with

φ′(X ⊗ ν̃) = φ(η(z)X ⊗ ξ̃(h(z^{−1}))ν̃).

Here X → η(z)X is the action which multiplies the exterior product of p holomorphic and q antiholomorphic vectors by z^{−p}z̄^{−q}. Thus η(z) = ad h(z^{−1}) ∘ η(z̄) and

φ′(X ⊗ ν̃) = π(h(z^{−1}))φ(η(z̄)X ⊗ ν̃).

If E ⊆ R we may extend this action of C^{×} to an action of W_{C/R} = W_{C/E_{v}}. Let n ∈ G(R) be such that ad n ∘ h = h^{ι}. Then the element of w which projects to ι and has square −1 sends φ ⊗ u to φ′ ⊗ u with

φ′(X ⊗ ν̃) = π(n)φ(ad n^{−1}(X) ⊗ ξ̃(n^{−1})ν̃).

In either case the representation of W_{C/E_{v}} on (7.4) factors as ψ^{i}(π) ⊗ 1, where ψ^{i}(π) acts on Hom_{K_{h}}(Λ^{i}𝔊/𝔨 ⊗ Ṽ , U). Let ψ_{2}(π) be the element in the representation ring of W_{C/R} defined by

ψ_{2}(π) = ⊕(−1)^{i} Ind (W_{C/R}, W_{C/E_{v}}, ψ^{i}(π)).
Let m(π_{f}, K) be the dimension of U_{f}^{K}. If for all π we had
(7.5)  ψ_{2}(π) = m(π)ψ_{1}(π)

we could expect a relation

(7.6)  Z(z) = ∏_{π} L(z − d/2, π, r)^{m(π_{∞})m(π_{f} ,K)}.

Here, as a single exception, we have taken π to be a representation of G(A), its component at infinity being denoted π_{∞}, and r to be the representation of ^{L}G. However (7.5) is not always valid, and it is the true form of the relation between ψ_{1}(π) and ψ_{2}(π) that we must discover, for it is the clue to the correct expression of (7.2) as a product of L-functions associated to automorphic representations.

Let Π(ξ) = {π_{1}, ⋯, π_{r}} be the set of discrete series representations with the same central and infinitesimal characters as ξ̃. Then Π(ξ) = Π is an L-packet Π_{φ} and the representations ψ_{1}(π_{i}) are all equal. We denote them by ψ_{1}(Π). The continuous cohomology of the representations π_{i} is completely understood [4], and it is a simple exercise to prove the following lemma.

Lemma ⊕_{j=1}^{r}ψ_{2}(π_{j}) = (−1)^{d}ψ_{1}(Π).

Thus, in this case, the relation (7.5) fails when the L-packet has more than one element. In order to correct (7.6) one has to replace r by a subrepresentation. However r is in general irreducible as a representation of ^{L}G, and so we have to introduce the groups ^{L}H of [33], and begin the study of L-indistinguishability.

If we accept L-indistinguishability, but expect no other difficulties with the correction of (7.6), then we have to be prepared to prove that every irreducible component of ψ_{2}(π) is a component of ψ_{1}(π). But we will again be deceived. There is another difficulty.

It appears already in the simplest of the examples considered by Casselman [6] and Milne [36], although they had no occasion to draw attention to it. Suppose G is the group associated to a quaternion algebra over Q which is split at infinity but not at p. Let π_{f} = π_{p} ⊗ π^{p}, and suppose π ⊗ π_{f} is trivial on the centre, π_{p} is one-dimensional and trivial on the maximal compact subgroup K_{p} of G(Q_{p}), and π is either one-dimensional or the first element of the discrete series. ξ is taken to be trivial. If K = K^{p}K_{p} then L(s− \frac{1}{2}, π ⊗π_{f}, r) should appear in the zeta-function Z(s, Sh_{K}) with the exponent ±m(π^{p}, K^{p}). Here m(π^{p}, K^{p}) is the multiplicity with which the trivial representation of K^{p} occurs in π_{f}^{p}, and the sign is positive if π is one-dimensional and negative if it is the first element of the discrete series. As Casselman and Milne show in their lectures, this is so locally almost everywhere.

One can probably show without great difficulty that the local statement is correct at p as well when π belongs to the discrete series, for π then contributes to the cohomology in dimension one and π_{p} = π(σ_{p}) where σ_{p} is a special representation of the thickened Weil group. In particular

L(z − \frac{1}{2}, π_{p}, r) = \frac{1}{1 − ϵ/p^{z}} , |ϵ| = 1.

However if π is one-dimensional then π contributes to the cohomology in dimensions zero and two and the corresponding local contribution to the zeta-function should be

\{\frac{1}{(1 − ϵ/p^{z})(1 − ϵp/p^{z})}\}^{m(π^{p},K^{p})} .

The factor inside the brackets is not L(z − \frac{1}{2}, π_{p}, r).

The difficulty is resolved if we realize that when π, and hence π ⊗ π_{f}, is one-dimensional we should not be using L(z − \frac{1}{2}, π ⊗ π_{f}, r) at all but rather L(z − \frac{1}{2}, π′ ⊗ π′_{f}, r) where π′ ⊗ π′_{f} is the one-dimensional representation of G′(A) = GL(2, A) defined by the same character of the idèle-class group as π ⊗ π_{f}. Since G′(Q_{v}) ∼ G(Q_{v}) and π′_{v} ∼ π_{v} for almost all places v, the error of using π ⊗ π_{f} instead of π′ ⊗ π′_{f} is not detected when one only considers the local zeta-function almost everywhere.

The significance of the considerations of the second and third sections begins to appear. The representation π ⊗ π_{f} and the representation π′′ ⊗ π′′_{f} of G′(A) associated to it by the principle of functoriality are anomalous, because π′′_{v} is one-dimensional for almost all places v while π′′_{p} is infinite-dimensional. The isobaric representation equivalent to π′′⊗π′′_{f} almost everywhere is π′⊗π′_{f}. It was implicit in the discussion of the second section that anomalous representations would have nothing to do with motives, and so it should come as no surprise now that we must discard π ⊗ π_{f} and replace it by π′ ⊗ π′_{f}.

In this example π itself was not changed for G(R) ∼ G′(R) and π ∼ π′. However in general we must expect that π itself will have to be modified. Thus the proper factor will not be L(z − d/2, π ⊗ π_{f}, r) but L(z − d/2, π′ ⊗ π′_{f}, r), where π′ ⊗ π′_{f} is an automorphic representation of a group G′ obtained from G by an inner twisting. Again π′_{v} will have to be equivalent to π_{v} almost everywhere.

Since at the moment we are primarily interested in the infinite place, we simply ask whether it is possible to find a candidate for π′ or, rather, for an L-packet {π′} = Π′. There are apparently two conditions to be satisfied, the first arising from the compatibility of functional equations.

(a) Let π ∈ Π_{φ} and let {π′} = Π_{φ′}. For any additive character ψ of R and any representation σ of ^{L}G = ^{L}G′,

ϵ′(z, σ ∘ φ, ψ) = ϵ(z, σ ∘ φ, ψ)\frac{L(1 − z, σ ∘ φ)}{L(z, σ ∘ φ)}

is equal to

ϵ′(z, σ ∘ φ′, ψ) = ϵ(z, σ ∘ φ′, ψ)\frac{L(1 − z, σ ∘ φ′)}{L(z, σ ∘ φ′)}.

(b) It is possible to find a summand ψ_{0}(π′) of ψ_{1}(π′) which is such that ψ_{2}(π) = αψ_{0}(π′), α ∈ Z. These conditions are only tentative, and may have to be modified in the course of time, but

they will serve for the explanation of our problem.

The first condition involves only φ and φ′ and we begin by constructing some pairs that satisfy it. Fix an element w of W_{C/R} that projects to ι and satisfies w^{2} = −1. We may suppose that φ(w) = a × ι with a in the normalizer of ^{L}T^{∘} in ^{L}G^{∘}. Then φ(w) also normalizes ^{L}T^{∘}. Let φ(ι) denote the transformation of X^{∗}(^{L}T^{∘}) or of X_{∗}(^{L}T^{∘}) defined by φ(w). We may also suppose that φ takes C^{×} to ^{L}T^{∘} and that φ(z) = z^{Λ}z̄^{φ(ι)Λ} with Λ ∈ X_{∗}(^{L}T^{∘}) ⊗ C and Λ − φ(ι)Λ ∈ X_{∗}(^{L}T^{∘}).

The representation φ′ will be defined in a similar way. Thus φ′(w) = a′ × ι with a′ in the normalizer of ^{L}T^{∘}, and φ′(z) = z^{Λ}z̄^{φ′(ι)Λ}. Notice that Λ is to be the same for φ′ as for φ. However a, which is given, is replaced by a′, which we must now define.

We suppose that φ(ι) sends every root to its negative, and choose λ in X_{∗}(^{L}T^{∘}) such that λ^{∨}(a) = e^{2πi⟨λ,λ^{∨}⟩} for any weight λ^{∨} of ^{L}T^{∘} which is orthogonal to all roots. We shall take a′ to lie in ^{L}T^{∘} and to be such that λ^{v}(a′) = e^{2πi⟨λ′,λ^{∨}⟩} when λ^{∨} is orthogonal to all roots. Here λ′ is still to be defined. If we also denote by a the operator on X_{∗}(^{L}T^{∘}) ⊗ C defined by a and if we let q be one-half the sum of the positive roots then λ′ is to be given by the equation

λ′ = \frac{1 + a}{2}λ − \frac{(1 − a)(1 + φ′(ι))}{8}Λ + \frac{q}{2}.

Observe that the action of φ′(ι) is the same as that of ι.

We are assuming that φ is a given, well-defined homomorphism, and hence [31] that

λ + φ(ι)λ ≡ \frac{Λ − φ(ι)Λ}{2} − q(mod X_{∗}(^{L}T^{∘})).

In order to show that φ′ is also well defined we must verify that

(7.7)  λ′ + φ′(ι)λ′ ≡ \frac{Λ − φ(ι)Λ}{2}(mod X_{∗}(^{L}T^{∘})).

We begin with the equations aφ′(ι) = φ′(ι)a = φ(ι) and a(1 + φ(ι)) = 1 + φ(ι), remarking also that the square of both φ(ι) and φ′(ι) is the identity. We infer that the left side of (7.7) is equal to

\frac{(1 + φ′(ι))(1 + φ(ι))λ}{2} − \frac{(1 − a)(1 + φ′(ι))}{4}Λ + q,

because (1 + φ′(ι))q/2 = q. The sum is in turn congruent to

\frac{1 − φ(ι)}{2}Λ − \frac{(1 − a)(1 + φ′(ι))}{4}Λ = \frac{1 − φ′(ι)}{2}Λ.

Consequently the homomorphism φ′ can be constructed whenever φ is defined and φ(ι) sends every root to its negative. The following lemma is valid in this generality.

Lemma. For any representation σ of the Weil form of ^{L}G and any nontrivial character ψ of R,

ϵ′(τ, σ ∘ φ, ψ) = ϵ′(τ, σ ∘ φ′, ψ).

The proof is a computation based on the proof of Lemma 3.2 of [31] and on Chapters 5 and 6 of [24], but it is rather lengthy, and not worth including here.

Observe that we could have started with φ′, defined by an a′ in ^{L}T^{∘}, and reversing the process, passed to λ and a. More generally if ^{L}M and ^{L}M′ are two parabolic subgroups of ^{L}G, and φ : W_{C/R} → ^{L}M has an image which lies in no proper parabolic subgroup of ^{L}M, then we can use the process to pass to a φ′′ whose image lies in the minimal parabolic of ^{L}M, and thus of LG or ^{L}M′, and afterwards reverse it to pass from φ′′ to a φ′ : W_{C/R} → ^{L}M′ whose image lies in no proper parabolic subgroup of ^{L}M′.

My intention now is simply to show, by means of a few examples, how for a given π in some Π_{φ} one can choose one of the φ′ just described so that the condition (b) is satisfied for the elements π′ of Π_{φ′}. Of course the problem is to decide if such a choice is always possible. Without more examples or a general theorem, we cannot be at all confident that this is so.

If all the continuous cohomology of ξ ⊗ π is zero there is no difficulty satisfying (b). We take φ′ = φ and α = 0. The simplest nontrivial example is obtained by taking ξ trivial and π trivial. Let π ∈ Π_{φ}. Then φ(z) = z^{q}z̄^{φ(ι)q}, with q equal again to one-half the sum of the positive roots. If ^{L}M is the parabolic subgroup of ^{L}G corresponding to the minimal parabolic of G over R, then the image of φ lies in ^{L}M and φ(ι) takes every root of ^{L}T^{∘} in ^{L}M to its negative. Define φ′ as above, with a′ ∈ ^{L}T^{∘}. G′ can be taken to be the quasi-split form of G over R. The continuous cohomology of π is all in even dimensions and all of type (p, p) for some p. To compute it one observes that it is the same as the cohomology of the compact dual, which can be computed by using Schubert cells. One verifies without difficulty that for π′ ∈ Π_{φ′} the representation ψ_{1}(π′), which depends in reality only on φ′, is equivalent to ψ_{2}(π).

If G is not quasi-split over R then φ′ is different from φ. If G is not quasi-split over R then it is certainly not quasi-slit over Q, and the trivial representation of G(A) is anomalous. Once again we see that the passage from π to π′ is the local expression of the passage from an anomalous representation to one which is not anomalous.

Other interesting examples are the representations π = J_{i,j} of P SU(n, 1) discussed in Chapter XI of the notes of Borel-Wallach [4]. Take ξ trivial. In this case ψ_{2}(π) is (−1)^{i+j} times a representation induced from C^{×}, the representation of C^{×} used having the weights

(7.8)  z^{−i}z̄^{−j}, z^{−i−1}z̄^{−j−1}, ⋯, z^{−(n−j)}z̄^{−(n−i)}.

Here 0 ≦ i + j ≦ n − 1 and 0 ≦ i, j. Borel and Wallach lapse into vagueness at one point, and it may be that the roles of i and j should be reversed, but that is of little consequence.

The group ^{L}G^{∘} is SL(n + 1, C) and ^{L}T^{∘} may be taken to be the group of diagonal matrices. The representation r^{∘} is the standard representation of SL(n + 1, C). It is easy enough to deduce from [4] that if π ∈ Π_{φ} then

φ(z) = z^{Λ}z̄^{φ(ι)Λ}, z ∈ C^{×},

with Λ being equal to (n/2 − i, n/2, n/2 − 1, ⋯, n/2 − i + 1, n/2 − i − 1, ⋯, −n/2 + j + 1, −n/2 + j − 1, ⋯, −n/2, −n/2 + j). The numbers occurring here are n/2, n/2 − 1, ⋯, −n/2, but the order is somewhat unusual. The transformation φ(ι) is given by

(x_{1}, ⋯, x_{n+1}) ⟶ (−x_{n+1}, −x_{2}, ⋯, −x_{n}, −x_{1}).

We are of course using the obvious representation of the elements of X_{∗}(^{L}T^{∘}) ⊗ C as sequences of n + 1 complex numbers whose sum is 0.

Suppose, to be definite, that i ≦ j. We will choose φ′ to be such that the transformation φ′(ι) takes (x_{1}, ⋯, x_{n+1}) to (−x_{n+1}, −x_{n}, ⋯, −x_{n−i+1}, −x_{n−j}, ⋯, −x_{i+2}, −x_{n−j+1}, ⋯, −x_{n−i}, −x_{i+1}, ⋯, −x_{1}). The indices within the gaps decrease or increase regularly by one. If π′ ∈ Π_{φ′} then the representation ψ_{1}(π′) is induced from a representation of C^{×} with weights

(7.9)  \begin{matrix}z^{−i}z̄^{−j}, 1, z^{−1}z̄^{−1}, ⋯, z^{−i+1}z̄^{−j+1}; \\ z^{−i−1}z̄^{−j−1}, ⋯, z^{−(n−j−1)}z̄^{−(n−i−1)}; \\ z^{−(n−j+1)}z̄^{−(j−1)}, ⋯, z^{−(n−i)}z̄^{−i}; \\ z^{−(n−i+1)}z̄^{−(n−i+1)}, ⋯, z^{−n}z̄^{−n}, z^{−(n−j)}z̄^{−(n−i)}. \end{matrix}

Happily the set (7.8) is a subset of (7.9) and the condition (b) is satisfied.

It should be observed that the representation ψ_{0}(π′) that is chosen to satisfy (b) will have to be, except for some degenerate values of i and j, a proper subrepresentation of ψ_{1}(π′). This phenomenon will, I hope, be taken into account by L-indistinguishability. For example if ϵ is the element of ^{L}T^{∘} with diagonal entries

1, \overbrace{−1, −1, ⋯, −1}^{i} 1, ⋯, 1, \overbrace{−1, ⋯, −1,}^{j} 1

then ϵ commutes with φ′(W_{C/R}) and ψ_{0}(π′) may be taken to be the restriction of ψ_{1}(π′) to the +1 eigenspace of r(ϵ).

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