Some Contemporary Problems with Origins in the Jugendtraum

R.P. Langlands

The twelfth problem of Hilbert reminds us, although the reminder should be unnecessary, of the blood relationship of three subjects which have since undergone often separate developments. The first of these, the theory of class fields or of abelian extensions of number fields, attained what was pretty much its final form early in this century. The second, the algebraic theory of elliptic curves and, more generally, of abelian varieties, has been for fifty years a topic of research whose vigor and quality shows as yet no sign of abatement. The third, the theory of automorphic functions, has been slower to mature and is still inextricably entangled with the study of abelian varieties, especially of their moduli.

Of course at the time of Hilbert these subjects had only begun to set themselves off from the general mathematical landscape as separate theories and at the time of Kronecker existed only as part of the theories of elliptic modular functions and of cyclotomic fields. It is in a letter from Kronecker to Dedekind of 1880,^{1} in which he explains his work on the relation between abelian extensions of imaginary quadratic fields and elliptic curves with complex multiplication, that the word Jugendtraum appears. Because these subjects were so interwoven it seems to have been impossible to disintangle the different kinds of mathematics which were involved in the Jugendtraum, especially to separate the algebraic aspects from the analytic or number theoretic. Hilbert in particular may have been led to mistake an accident, or perhaps necessity, of historical development for an “innigste gegenseitige Berührung.” We may be able to judge this better if we attempt to view the mathematical content of the Jugendtraum with the eyes of a sophisticated contemporary mathematician.

An elliptic curve over a field k is a curve A in some projective space P^{n} defined, say, by equations

g_{i}(x_{0}, …, x_{n}) = 0

together with a rational map

z_{j} = f_{j}(x_{0}, …, x_{n}; y_{0}, …, y_{n})

from A × A to A which turns the set of points on A into a group. Roughly speaking — the adverb is to be taken seriously — an elliptic curve over an arbitrary commutative ring R, which we always take to be noetherian, is defined in the same way except that the coefficients of f_{j} and g_{i} are to lie in R. If one has an elliptic curve over B_{1} and a homomorphism φ : B_{1} → B_{2} then replacing the coefficients of f_{j} and g_{i} by their images under φ we obtain an elliptic curve over B_{2}. In this way the sets A(B) of isomorphism classes of elliptic curves over a commutative noetherian ring B become a covariant functor on the category of such rings.

In the theory of complex multiplication one introduces a subfunctor. Take E to be an imaginary quadratic field and let O be the ring of integers in E. We are now interested only in rings B together with a homomorphism ψ : O → B and maps B_{1} → B_{2} for which

\begin{matrix}O \\ ψ_{1} ↙ ↘ ψ_{2} \\ B_{1} ⟶ B_{2} \end{matrix}

is commutative. The tangent space T (A) to an elliptic curve over B at the zero is a B-module. We are interested in abelian varieties A over B together with an action of the elements of O as endomorphisms of A such that the associated action of x ∈ O on T (A) is just multiplication by ψ(x) ∈ B. This gives us a new functor B → A^{O}(B). If n is a positive integer and if we consider only rings B in which n is invertible,

1 Gesammelte Werke, Bd V. we can introduce a refinement. We can let A_{n}(B) be the points of A with coefficients from B whose order divides n and introduce as additional datum an isomorphism of O-modules

λ : O/nO → A_{n}(B).

This defines a new function B → A_{n}^{0}(B).

The methods of contemporary algebraic geometry, with which the present author is as yet only superficially acquainted, allow one to prove the existence of a universal object for this functor. This is a ring B_{n}, a homomorphism O → B_{n}, an abelian variety A′ over B_{n}, an action of O on A′, and an isomorphism

λ′ : O/nO → A′_{n}(B_{n})

such that the conditions imposed above are satisfied and such that for any B any element of A_{n}^{O}(B) is obtained by functoriality from A′, λ′ and a uniquely determined homomorphism B_{n} → B. This is not quite true for small n but the difficulty can be obviated by some technical considerations and is not worth stressing here.

The methods not only establish the existence but also allow one to read off properties of the ring B_{n} from properties of the functor A_{n}^{O}, that is, of elliptic curves over rings. For example, the notion of smoothness or, in the language of algebraic number theory, lack of ramification, is translated into a notion of deformability. The deformation theory of elliptic curves, and of abelian varieties, is well understood, and one can show that F_{n} = B_{n} ⊗_{O} E is a finite direct sum ⊕E_{i} of finite algebraic extensions of E unramified away from the primes dividing n and that if O_{i} is the ring of integers in E_{i} then B_{n}, a subring of F_{n}, is equal to

⊕O_{i}[\frac{1}{n}].

Here O_{i} \frac{1}{n}] is the subring of E generated by O_{i} and \frac{1}{n}.

If we imbed E into Q̄ ⊆ C then the algebra F_{n} is determined by the action of 𝔊(Q̄/E) on the set of its E-homomorphisms into C, which is also A_{n}^{O}(Q̄) = A_{n}^{O}(C). If the action is transitive the algebra is a field. Before investigating it we introduce some automorphisms of F_{n}. These are defined by automorphisms of the functor restricted for the moment to rings in which every positive integer is invertible. This means that B_{n} is to be replaced by F_{n}.

Let I_{f} be those ideles of E whose component at infinity is 1. We may imbed E^{×} in I_{f}. We are going to define an action of I_{f} on the functor A_{n}^{O}. Let O_{f} be the ring of adeles which are integral everywhere and have component 1 at infinity. Suppose first that g ∈ I_{f} ∩ O_{f}. There is a positive integer m and an h ∈ I_{f} ∩ O_{f} such that gh = m. Suppose {A, λ} in A_{n}^{O}(B) is given. There is an extension B′ of B and an isomorphism (of sheaves!)

λ′ : O/n′O ⟶ A_{n′}(B′)

such that

mλ′(x) = λ(x).

g acts on O/n′O and we define a new elliptic curve A_{1} by dividing by

{λ′(gx) | x ∈ nO}.

There is then an isogeny ψ : A → A_{1} with this kernel and we define λ_{1} by

λ_{1}(x) = ψ(λ′(gx)).

The pair {A_{1}, λ_{1}} actually defines an element of A_{n}^{O}(B). The action of g takes A, λ to A_{1}, λ_{1}. Since elements of O are easily seen to act trivially we can extend the action to all of I_{f} by letting that of ℓg, with ℓ a positive integer, be the same as that of g.

The action on A_{n}^{O}(C) can easily be made explicit. If g ∈ I_{f} let gO be the ideal gO_{f} ∩ E. We have imbedded E in C, and the quotient of C by the lattice gO is an elliptic curve A^{g} on which O acts. Moreover

A_{n}^{g}(ℂ) = \frac{gO}{n} /gO.

If we regard O/nO as O_{f}/nO_{f} we may define λ^{g} as

x → \frac{gx}{n}.

If

K^{n} = {k ∈ I_{f} | k ≡ k^{−1} ≡ 1(mod n)}

then A^{g}, λ^{g} and A^{h}, λ^{h} are isomorphic if and only if

h ∈ E^{×}gk^{n}

so that as a set A_{n}^{O}(ℂ) is just the quotient space E^{×}\I_{f}/K^{n}. The action of I_{f} is the obvious action on the quotient space.

By functoriality the action of 𝔊(Q̄/E) on A_{n}^{O}(Q̄) = A_{n}^{O}(C) commutes with that of I_{f}. Therefore there is a unique homomorphism σ → φ(σ) of 𝔊(Q̄/E) into E^{×}\I_{f}/K^{n} such that the actions of σ and φ(σ) are the same. It follows in particular that 𝔊(Q̄/E) acts through an abelian quotient. To understand the homomorphism σ → φ(σ) we have only to identify φ(σ) when σ is the Frobenius at a prime 𝔭 of E which does not divide n.

Let E_{𝔭} be the completion of E at 𝔭, Ē_{𝔭} an algebraic closure of E_{𝔭}, and Ō_{𝔭} the ring of integers in Ē_{𝔭}. Fix an imbedding Q̄ ↪ Ē_{𝔭}.

A_{n}^{O}(Q̄)\underrightarrow{∼}A_{n}^{O}(Ē_{𝔭}) = Hom_{O}(B_{n}, Ē_{𝔭}) = Hom_{O}(B_{n}, Ō_{𝔭}).

Since B_{n} is unramified at 𝔭, we may use the map Ō_{𝔭} → κ̄_{𝔭}, the algebraic closure of the residue field κ_{𝔭} of O at 𝔭, to obtain

Hom_{O}(B_{n}, Ō_{𝔭}) ≃ Hom_{O}(B_{n}, κ̄_{𝔭}) = A_{n}^{O}(κ̄_{𝔭}).

All these isomorphisms do not affect the action of the Frobenius. Because p is not invertible in κ̄_{𝔭}, the group I_{f} no longer operates, at least not quite as before. However I_{f}^{p}, consisting of those ideles which are 1 at infinity and at p, continues to operate, because for these ideles we can take the auxiliary integer m prime to p, and the difficulties attendant upon the anomalous behavior of p-division points in characteristic p do not appear. Actually because of the simplicity of the present situation, it is rather easy to define an action of the missing part of I_{f}, namely I_{p}, the multiplicative group of O ⊗ Q_{p}. However, we want to avoid all ad hoc techniques. What is needed is an understanding of the finite subgroups, in the scheme-theoretic sense, of an elliptic curve over a field of characteristic p with order a power of p. The general method is the theory of the Dieudonné module. I do not want to give its definition here. It is a module D(A) functorially associated to A. The action of I_{p} is replaced by the action of the O-automorphisms of D(A) ⊗ Q. This group turns out however, because of the special situation with which we are dealing, to be I_{p} so that I_{f} does operate once again. Moreover E^{×} and K^{n} still act trivially. Since I_{f} is generated by E^{×}, I_{f}^{p}, and K^{n} its action is compatible with the isomorphisms of sets introduced above.

If ϖ is a generator of the maximal ideal of O_{𝔭} then ϖ ∈ E_{𝔭}^{×} ⊆ I_{p}. The theory of Dieudonné modules acquired, it is immediate that the action of ϖ on A_{n}^{O}(κ̄_{𝔭}) is the same as that of the Frobenius. It follows that F_{n} is a field and is the abelian extension of E associated to E^{×}I_{∞}K^{n} ⊆ I by class field theory. Moreover the homomorphism

𝔊(Q̄/E) ⟶ 𝔊(F_{n}/E) ≃ I/E^{×}I_{∞}K^{n} ≃ I_{f}/E^{×}K^{n}

given by class-field theory is just σ → φ(σ). So far we have gotten by without any real arithmetic; only the arithmetic of finite fields has played a role. However, it is an essential part of the Jugendtraum that every abelian extension of E is contained in some F_{n}. For this we appeal to class-field theory.

But no elliptic modular functions have yet appeared. Let V (Z) be the module of column vectors of length two over Z. We can consider the functor which associates to B the isomorphism

λ : V (Z/nZ) ⟶ A_{n}(B).

This functor is also represented by a universal object over a ring J_{n}. The morphism A_{n}^{O} → A_{n} obtained by fixing an isomorphism

O ≃ V (Z)

and then forgetting the action of O yields a homomorphism η : J_{n} ⟶ B_{n}. If we imbed B_{n} ⟶ C over E then of course the image generates a class-field, as described above. Composing the imbedding with η yields a homomorphism of J_{n} or of J_{n} ⊗ C into C.

J_{n} ⊗ C is the ring of rational functions on an algebraic variety S_{n} over C whose points give the homomorphisms of J_{n} into C, that is, the elements of A_{n}(C). In particular, to obtain ψ we have to evaluate the elements of J_{n} at some point of S_{n}(C). There is, at least from the analytic standpoint, a more concrete way of viewing S_{n}(C) and hence J_{n} ⊗ C. Let G be the group GL(2). Let J_{0} be the matrix

( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ).

If g = (g_{∞}, g_{f}) belongs to G(A) with g_{∞} ∈ G(R), g_{f} ∈ G(A_{f}) we set

g_{f}V (Z) = g_{f}V (Z_{f}) ∩ V (Q).

Here Z_{f} is the closure of Z in A_{f} and V (Z_{f}) = V (Z)⊗Z_{f}. Then g_{f}V (Z) is a lattice in V (R). Let J = g_{∞}J_{0}g_{∞}^{−1}. We turn V (R) into a one-dimensional space over C by defining multiplication by \sqrt{−1} to be J. Then

V (R)/g_{f}V (Z)

is an elliptic curve A^{g} over C. Also

A_{n}^{g}(C) = \frac{g_{f}V (Z)}{n}/g_{f}V (Z)

so we may take λ^{g} to be

x → \frac{g_{f}x}{n}.

The isomorphism class of {A^{g}, λ^{g}} is determined solely by the image of g in the double coset space

G(Q)\G(A)/K_{∞}K^{n}

if K_{∞} is the centralizer of J_{0} in G(R) and K^{n} is

{k ∈ G(Z_{f}) | k ≡ 1(mod n)}.

This double coset space has a natural complex structure and may now be identified with S_{n}(C) = A_{n}(C).

Analyzing the double cosets more carefully one sees that S_{n}(C) consists of finitely many connected pieces each of which is the quotient of the Poincaré half-plane by a congruence subgroup. The elements of J_{n} ⊗C, in particular the elements of J_{n}, are functions on these pieces and are in fact just the elliptic modular functions of level n. The points of S_{n}(C) corresponding to the homomorphism ψ introduced above are easily found explicitly. Summing up, we conclude that the class field F_{n} is generated by the values of the elliptic modular functions in J_{n} at a certain easily found point of

G(Q)\G(A)/K_{∞}K^{n}.

As we said, any connected piece of this space is a quotient of the Poincaré half-plane by a discrete group. If we lift the functions in J_{n} to the half-plane they become transcendental.

This aspect, the generation of class fields by the values of transcendental functions, has been emphasized by Hilbert who suggests, in the twelfth problem, that it may be possible to find for an arbitrary number field transcendental functions with a similar property. Whether justly or not, the twelfth problem has received very little attention. Any progress made on it has been an incidental result of research with quite different ends, although it too has its origins in the Jugendtraum. The bulk of this research is due to Shimura.

A characteristic of the number theory of the twentieth century has been the dominant role played by zeta-functions and L-functions, especially at a conjectural level. The analytic properties of the L-functions associated to an algebraic variety over a number field have been particularly difficult, usually impossible, to determine. But Shimura has studied very deeply certain varieties, which, like the varieties defined by elliptic modular functions, are closely related to algebraic groups. For various reasons it is to be expected that the L-functions associated to these Shimura varieties can be expressed in terms of the L-functions associated to automorphic forms on the group defining the variety and on certain related groups. This in itself is not enough to establish the analytic properties but it is a first step. Shimura, inspired by earlier work of Eichler, has been able to confirm the expectation for some of his varieties, basically those which are curves.

But many problems remain. I want to discuss one of them, rather casually, in the remainder of the lecture. There are various notions of a reciprocity law, all of them implicit in the laws of class-field theory. For example, one can view a theorem asserting that an L-function defined by diophantine data, that is, by an algebraic variety over a number field, is equal to an L-function defined by analytic data, that is, by an automorphic form, as a reciprocity law. There is good reason for this, for the Artin reciprocity law is such an assertion. The results of Eichler and Shimura are also of this form. There is nonetheless a more concrete notion available.

Suppose one has an algebraic variety S defined over a number field E. Suppose in fact that equations defining S have been chosen whose coefficients are integral outside of some finite set of primes Q. If 𝔭 ∉ Q and κ_{𝔭} is the residue field of E at 𝔭 we can reduce the equations modulo 𝔭 and then speak of the set S(κ̄_{𝔭}) of points of S with coefficients in κ̄_{𝔭}. S(κ̄_{𝔭}) is given together with an action on it, that of the Frobenius Φ_{𝔭}. An explicit description of the sets S(κ̄_{𝔭}) and of the actions of Φ_{𝔭} for all 𝔭 ∉ Q could also be viewed as a reciprocity law. For example, if E = Q and S is defined by the equation

x^{2} + 1 = 0

then S(κ̄_{p}) for p ≠ 2 is a set with two elements and Φ_{p} acts trivially or not according as p ≡ 1 or p ≡ 3 modulo 4. This is the first supplement to the law of quadratic reciprocity.

It is very likely that Shimura varieties admit a reciprocity law in this sense. I want to describe explicitly the form the law will most probably take. The description is speculative, but I have verified its correctness, in so far as my limited command of the necessary techniques allows, for those varieties which arise as solutions of moduli problems for abelian varieties.

To know the zeta-function of a variety, at least in the sense of knowing the factors of its Euler product expansion for almost all 𝔭, one just has to know the number of points in S(κ_{𝔭}^{n}) for all positive n, if κ_{𝔭} is the extension of κ_{𝔭} of degree n. This is just the number of fixed points of Φ_{𝔭}^{n} in S(κ̄_{𝔭}). One might expect that this could be determined from the explicit description of S(κ̄_{𝔭}) and of the action of Φ_{𝔭}; so that from a reciprocity law in the second sense for the Shimura variety S one could obtain one in the first sense, at least for its zeta-function. However, difficult combinatorial problems arise which have not yet been seriously broached. But I have been able to make the transition in a limited number of cases, among which are included varieties of arbitrary large dimension.

The work of Shimura has been expounded in a remarkably clear fashion by Deligne,^{2} who also added improvements of his own. One begins with a reductive algebraic group G over Q and a homomorphism H_{0} : GL(1) → G defined over C. The pair (G, h_{0}) is subject to some simple formal conditions. If R is the torus over R obtained from GL(1) over C by restriction of scalars so that over C

R ≃ GL(1) × GL(1)

then the composition

h : R ⟶^{∼} GL(1) × GL(1) → G,

where the second map is (x, y) → h_{0}(x)^{ρ}h_{0}(y) with ρ the complex conjugation, is to be a homomorphism defined over R. The centralizer of h(R) in G_{der}(R) is to be maximal compact subgroup of G_{der}(R) and if K_{∞} is the centralizer of h(R) in G_{der}(R) then the quotient G(R)/K_{∞} is to carry an invariant complex structure, specified by h_{0}.

It is in fact not h_{0} which is significant but the collection of ad g∘h_{0}, g ∈ G(R). If T is a Cartan subgroup of G defined over Q with T (R) ∩ G_{der}(R) compact we may choose h′_{0} = adg ∘ h_{0} so that it factors through T. We then denote h′_{0} : GL(1) → T by μ̂; it is a coweight of T. If E is defined to be the fixed field of the set of all σ ∈ 𝔊(Q̄/Q) for which σμ̂ = ωμ̂ with ω in the Weyl group of T then E, which is a finite extension of Q in C, plays an important role in the study of Shimura varieties.

If K is an open compact subgroup of G(A_{f}) then the complex manifold

S_{K}(C) = G(Q)\G(A)/K_{∞}K

is the set of complex points of an algebraic variety of C. It has been conjectured, hesitantly by Shimura, openly by Deligne, that this family of algebraic varieties should have models S_{K} over E. The precise conjecture actually demands certain further properties of the S_{K}, which serve to characterize them uniquely. These properties are patterned on the Jugendtraum; so that implicit in any proof of the existence of these canonical models S_{K} is a partial solution to the twelfth problem. The conjecture, colloquially referred to as the Shimura conjecture, has been solved for many groups but by no means all. My suggestions will only make sense for those groups for which the Shimura conjecture is acquired.

The group G(A_{f}) operates on

lim_{\overleftarrow{K}} S_{K}(C).

It is demanded that this be reflected in an action of G(A_{f}) on

lim_{\overleftarrow{K}} S_{K}

defined over E.

Fix a prime 𝔭 of E and let p be the prime of Q it divides. I shall suppose that the group G is quasi-split over Q_{p} and split over an unramified extension. Recall that if G_{sc} is the simply-connected form of the derived group G_{der} then Bruhat and Tits have associated a building to G_{sc}(Q_{p}) on which G(Q_{p}) acts. A special maximal compact subgroup of G(Q_{p}) is the intersection of the stabilizer in G(Q_{p}) of a special vertex of the Bruhat-Tits building with

{g ∈ G(Q_{p}) | |χ(g)| = 1 for all rational characters of G defined over Q_{p}}.

We shall only be interested in K of the form

K = K^{p}K_{p}

where K^{p} ⊂ G(A_{f}^{p}) and K_{p} is a special maximal compact of G(Q_{p}).

2 Séminaire Bourbaki, 1970/71.

The varieties S_{K} are defined over E and hence over E_{𝔭}. Suppose O_{𝔭} is the ring of integers of E_{𝔭}. To speak of S_{K}(κ̄_{𝔭}) we need models over O_{𝔭}. At the moment I do not know how they should be characterized. Presumably if S_{K}/E is proper and smooth then S_{K}/O_{𝔭} should also be proper and smooth. But if S_{K}/E is not proper, some attention will have to be paid to the behavior at infinity. I simply ignore the difficulty for now and go on to describe the expected structure of S_{K}(κ̄_{𝔭}). It is enough to consider that of

lim_{\overleftarrow{K^{p}}} S_{K}(κ̄_{𝔭}) = S_{K_{p}}(κ̄_{𝔭})

provided that we know how G(A_{f}^{p}) acts on the right-hand side, for

S_{K}(κ̄_{𝔭}) = S_{K_{p}}(κ̄_{𝔭})/K^{p}.

The set S_{K_{p}}(κ̄_{𝔭}) should be the union of certain subsets invariant under G(A_{f}^{p}) and Φ = Φ_{𝔭}. Each of them is constructed from the following data:

(i) a group H over Q and an imbedding H(A_{f}^{p}) ↪ G(A_{f}^{p}); (ii) a group Ḡ over Q_{p} and an imbedding H(Q_{p}) ↪ Ḡ(Q_{p}); (iii) a space X on which Ḡ(Q_{p}) and Φ act, the two actions commuting with each other. The imbeddings H(A_{f}^{p}) ↪ G(A_{f}^{p}), H(Q_{p}) ↪ Ḡ(Q_{p}) when combined with the diagonal imbedding

H(Q) ↪ H(A_{f}) yield an action of H(Q) on G(A_{f}^{p}) × X. The subsets to which I referred have the form

Y = H(Q)\G(A_{f}^{p}) × X.

G(A_{f}^{p}) acts in the obvious way to the right and Φ acts through its action on X.

Before venturing a general prescription for H, G, and X we should orient ourselves with a brief glance at G = GL(2) with h given by

(a + ib, a − ib) ⟶ ( \begin{matrix}a & −b \\ b & a \end{matrix} ), a, b ∈ C, a^{2} + b^{2} ≠ 0.

For mnemonic reasons, I adhere to a slightly different convention than Deligne, so that my h is the inverse of his. For this pair G, h there is one subset for each imaginary quadratic extension F of Q. H is the group F^{∗} over Q associated in the usual way to F so that H(Q) = F^{×}, Ḡ is also H, and X is the quotient of H(Q_{𝔭}) = (E ⊗ Q_{p})^{×} ≃ Q_{p}^{×} × Q_{p}^{×} by H(Z_{p}), the group of units, Z_{p}^{×} × Z_{p}^{×}. If 𝔤 is one of the prime divisors of p in E and ϖ the corresponding uniformizing parameter then Φ is multiplication by ϖ ∈ (E ⊗ Q_{p})^{×}. There is one additional subset. For it, H is the multiplicative group of the quaternion algebra over Q split everywhere except at infinity and p and Ḡ is H. X is the quotient Ḡ(Q_{p})/Ḡ(Z_{p}), if Ḡ(Z_{p}) is the multiplicative group of the maximal order in the completion of the algebra at p. Φ is multiplication by any ϖ in this order which generates the maximal ideal.

There is an alternative description of the X for the final subset which yields more insight into the general situation. Let 𝔨 be the completion of the maximal unramified extension of Q_{p} and 𝔬 its ring of integers. Denote by a → ^{σ}a the action of the Frobenius. Let H be the set of 𝔬-lattices in the space of column vectors of length 2 over 𝔨. H is the set of vertices in the Bruhat-Tits building of G(𝔨). Set

b = ( \begin{matrix}0 & 1 \\ p & 0 \end{matrix} ).

Define an action of Φ on H by

Φ𝔨 = b^{σ}𝔨.

Then

Ḡ(Q_{p}) = {g ∈ G(𝔨) | b^{σ}gb^{−1} = g}

and X is the set of all 𝔵 in H for which

p𝔵 ⊂_{≠} Φ𝔵 ⊂_{≠} 𝔵.

Geometrically this means that the images of 𝔵 and Φ𝔵 in the Bruhat-Tits building of G_{sc}(𝔨) = SL(2, 𝔨) are joined by an edge. To verify that the two descriptions of X are not essentially different, one uses the fact that the Bruhat-Tits building is a tree. It is an amusing exercise.

To define H, Ḡ, and X in general we fix Q̄ ↪ C and then choose, once and for all, an imbedding Q̄ ↪ Q̄_{p} so that the prime of E it defines is 𝔭. Suppose γ belongs to G(Q) and is semi-simple. Suppose moreover that all the eigenvalues of γ have absolute value 1 away from infinity and p. Let

H^{o} = {g ∈ G | gγ^{m} = γ^{m}g for some m ≠ 0 in Z}.

H^{o} is connected and of course defined over Q. Suppose h^{o} : R → H^{o} and the composition

R ⟶^{h^{o}} H^{O} ↪ G

is conjugate under G(R) to h. If T is a Cartan subgroup of H^{o} defined over Q with T (R)∩G_{der}(R) compact, then, as before, replacing h^{o} by ad g ∘ h^{o}, g ∈ H^{o}(R) if necessary, we may suppose h^{o} factors through T. The associated

h_{0}^{o} : GL(1) → T

is a coweight μ̂ of T. μ̂ is not uniquely determined by h^{o}, but its orbit under the Weyl group of T in H^{o} is; and that suffices for the following.

If L(T) is the Z-module

Hom(T, GL(1))

and

L̂(T) = Hom(GL(1), T)

then L̂(T) is also

Hom(L(T), Z).

Define λ̂(γ) ∈ L̂(R) by

|λ(γ)|_{p} = p^{−⟨λ,λ̂(γ)⟩}, λ ∈ L(T).

Let M be the lattice of rational characters of H^{o} defined over Q_{p}. We say that the pair (γ, h^{o}) is of Frobenius type if there is an r > 0 in Q such that λ̂(γ) − rμ̂ is orthogonal to M.

Later an equivalence relation will be introduced on pairs of Frobenius type. To each equivalence class will be associated H, Ḡ, and X, as well as

Y = H(Q)\G(A_{f}^{p}) × X.

For each equivalence class we will also define a multiplicity d. If dY is the disjoint union of d copies of Y then, as a set on which Φ and G(A_{f}^{p}) act, S_{K_{p}}(κ̄_{𝔭}) should be isomorphic to the disjoint union over equivalence classes of pairs of Frobenius type of the sets dY.

For the moment fix γ and h^{o}. H will be obtained from H^{o} by an inner twisting. Since the Hasse principle is valid for the adjoint group H_{ad}^{o}, it is enough to specify the twisting locally! Of course it has also to be verified that there is a global twisting with the specified local behavior; but this turns out to be a matter of standard techniques. The twisting is trivial except at infinity and p. At infinity it is so arranged that H_{der}(R) is compact. Before describing the twisting at p, we introduce a subgroup Ḡ^{o} of G defined over Q_{p}. It is the connected subgroup whose Lie algebra is spanned by those elements V in the Lie algebra of G satisfying

Adγ(V) = ϵV

with ϵ ∈ Q̄_{p} and |ϵ|_{p} = 1. Ḡ will be a twisted form of Ḡ^{o}.

We shall in fact twist Ḡ^{o} and H^{o} simultaneously. If T is as above, let T_{ad} be its image in H_{ad}^{o} and T̄_{ad} its image in Ḡ_{ad}^{o}. We choose T so that T_{ad} is anisotropic over Q_{p}. Choose a finite Galois extension k of Q in Q̄ over which T splits. Suppose a_{σ,τ} is a fundamental 2-cocycle for k_{p}/Q_{p}. Since

T (k_{p}) = L̂(T) ⊗ k_{p}^{×}

we may introduce the 1-cochain

σ → a_{σ} = ∑_{τ∈𝔊(k_{p}/Q_{p})} στμ̂ ⊗ a_{σ,τ} .

It takes values in T (k_{p}) but is not a 1-cocycle. However its image in T_{ad}(k_{p}) or T̄_{ad}(k_{p}) is. Composing with the maps

H^{1}(𝔊(k_{p}/Q_{p}), T_{ad}(k_{p})) → H^{1}(Q̄_{p}, H^{o})
H^{1}(𝔊(k_{p}/Q_{p}), T̄_{ad}(k_{p})) → H^{1}(Q̄_{p}, Ḡ^{o})

we obtain the twisting cocycles for H^{o} and Ḡ^{o} at p. One must of course verify that the twistings are independent of all auxiliary data.

The homomorphism H^{o} → G_{der}\G yields H → G_{der}\G. The multiplicity d is the number of elements in H^{1}(Q̄, H) which are trivial at every place except p, including infinity, and which lie in the kernel of

H^{1}(Q̄, H) → H^{1}(Q̄, G_{der}\G).

It may be, however, a little rash to predict d on the basis of the examples studied, for the groups involved have special cohomological properties.

The set X is the object must complicated to define. Set

ν̂ = ∑_{τ∈𝔊(k_{p}/Q_{p})} τμ̂

and denote

ν̂ ⊗ x ∈ L̂(T) ⊗ k_{p}^{×} = T (k_{p})

by xν̂. We define the Weil group, W_{k_{p}/Q_{p}}, by means of the cocycle a_{σ,τ}. If w = (x, σ) ∈ W_{k_{p}/Q_{p}}, with x ∈ k_{p}^{×}, σ ∈ 𝔊(k_{p}/Q_{p}), set

b_{w} = xν̂a_{σ}.

Then w → b_{w} is a 1-cocycle. Let D be the maximal torus in the centre of H^{o} split over Q_{p}. Let 𝔨 be the maximal unramified extension of Q_{p}. It turns out that if we enlarge k_{p} to some k′_{p} and inflate b_{w} to W_{k′_{p}/Q_{p}}

then we may represent its class by a cocycle {b̄_{w}} such that

b̄_{w} = b̄′_{w}b̄′′_{w}

where b̄′_{w} ∈ T (𝔨), b̄′′_{w} ∈ D(Q̄_{p}) and

|λ(b̄′′_{w})|_{p} = 1

for all rational characters of D. Moreover if W^{o} is the kernel of W_{k_{p}}/Q_{p} → 𝔊(F̄_{p}/F_{p}) then we may take b̄′_{w} = 1 for w ∈ W^{o}. If w is any element of W_{k′_{p}}/Q_{p} which maps to the Frobenius in 𝔊(κ̄_{𝔭}/κ_{𝔭}) set

b = b̄′_{w}.

We regard b as an element of G(𝔨). Any other choice of the auxiliary data replaces b by cb^{σ}c^{−1} if σ is the Frobenius on 𝔨. Such a change is irrelevant for our purposes. Observe that we may realize Ḡ(Q_{p}) as

{g ∈ G(𝔨) | b^{σ}gb^{−1} = g}.

The group K_{p} determines a special vertex in the Bruhat-Tits building of G_{sc}(Q_{p}) and hence of G_{sc}(𝔨) which in turn determines a parahoric subgroup of K_{p}(𝔨) of G(𝔨). Set

H = G(𝔨)/K_{p}(𝔨).

Let F be the map H → H which takes the point represented by g to the point represented by b^{σ}g.

There is a bijection between conjugacy classes of parabolic subgroups of G and conjugacy classes of parahoric subgroups with a representative in K_{p}(𝔨). Let ℑ be the class determined by the parabolic subgroup generated by T and the family of one-parameter subgroups corresponding to roots α with ⟨α, μ̂⟩ ≤ 0. Any point 𝔵 of H determines a special spoint 𝔵_{i} in the Bruhat-Tits building of each simple factor G_{i}(𝔨) of G_{sc}(𝔨). We consider only those 𝔵 such that if 𝔶 = F 𝔵 then, for each i, 𝔵_{i} and 𝔶_{i} are either the same or are joined by an edge. Then 𝔵_{i} and 𝔶_{i} determine a parahoric subgroup of G_{i}(𝔨) and, as a consequence, 𝔵 and 𝔶 determine a parahoric subgroup of G(𝔨). X consists of those 𝔵 for which this parahoric subgroup lies in the class ℑ. G(Q_{p}) acts on X. If r = [E_{𝔨} : Q_{p}] we define the action of Φ_{𝔨} to be F^{r}.

The correct conditions defining the equivalence of two pairs (γ_{1}, h_{1}^{o}), (γ_{2}, h_{2}^{o}) seem to be local, one condition at each finite place, but none at the infinite place. There should be positive integers m and n and a δ in the centre of G(Q) with every eigenvalue a unit such that, first of all, γ_{1}^{m} and δγ_{2}^{n} are conjugate in G(Q_{ℓ}) for each ℓ ≠ p. They should also be conjugate in G(𝔨). Let

δγ_{2}^{n} = gγ_{1}^{m}g^{−1}, g ∈ G(𝔨).

Suppose b_{1} and b_{2} in H_{1}^{o}(ℜ) and H_{2}^{o}(ℜ) are associated to (γ_{1}, h_{1}^{o}) and (γ_{2}, h_{2}^{o}) as above. Then gb_{1}^{σ}g^{−1} ∈ H_{2}^{o}(𝔨). The final condition for equivalence is that there be a c in H_{2}^{o}(𝔨) such that

cgb_{1}^{σ}g^{−1} ^{σ}c^{−1} = b_{2}.

In order to define the Γ-factors that should appear in the functional equation of the zeta-function of a Shimura variety one must also know something about their behavior at the infinite places of E. Two problems arise. If τ is an automorphism of Q̄ over Q we may apply τ to the family S_{K} over E to obtain a family of varieties ^{τ}S_{K} over ^{τ}E. This new family should be again just the canonical models for the Shimura varieties defined by some new pair (^{τ}G,^{τ}h_{0}). There is an obvious guess. ^{τ} G should be obtained from G by an inner twisting which is trivial at every finite place. If ρ is the complex conjugation and T is chosen as above then the twisting cocycle at infinity should be ρ → t_{ρ} with

λ(t_{ρ}) = (−1)^{⟨λ,τμ̂−μ̂⟩}, λ ∈ L(T).

Then T may also be regarded as a Cartan subgroup of ^{τ} G over R. The homomorphism ^{τ} h_{0} should just be the composition

GL(1) ⟶^{τμ̂} T^{⊂⟶}^{I}τG.

If the field E is real then the complex involution acts on S_{K}(C) which as a complex manifold is isomorphic to

G(Q)\G(A)/K_{∞}K.

It should be possible to define the resulting involution on the double coset space explicitly. E can be real only if ρμ̂ = ωμ̂ with ω in the Weyl group of T in G. If this is so then ω can be realized by an element w in the normalizer of T in G(R). The element w will normalize K_{∞} so that the map g → gw may be transferred to the quotient. This should give the involution.

Shimura and Shih are working on these two problems, which are deeper than they appear at first glance.