Definición de una competencia y las situaciones que se le asocian
5.2. Formular una competencia
5.2.2 La formulación técnica de una competencia
We now discuss the promised generalization of the Eichler-Shimura isomor- phism. As before, fix a number field F , with ring of integers OF, and let G
denote the Q-group ResF /Q(GL2). The group cohomology of G can be decom-
posed into two summands - the cuspidal and Eisenstein cohomology of G, the former of which is connected to cuspidal automorphic representations for G, and is our main object of interest. We shall give a more detailed description of this cuspidal cohomology in the following exposition, which follows the spirit of Joachim Schwermer’s treatment in [Sch06].
Before proceeding further, we establish some notation. Let A denote the ring of adeles over the rationals, which we decompose into finite and infinite parts
A = Af × R
as standard. If Sf and S∞ denote the sets of finite and infinite places of F
respectively, then we obtain a similar decomposition G(A) = G(Af) × G(R) ' Y v∈Sf 0 GL2(Fv) × Y v∈S∞ GL2(Fv),
where the product over the finite places Sf is restricted with respect to the
subgroups GL2(Ov).
Noting that
G(R) ' GL2(R)r× GL2(C)s,
where the number field F has signature [r, s], we fix a standard choice of compact open subgroup K∞ of G(R) by setting
K∞= O(2)r× U(2)s.
We shall not yet specify a compact open subgroup Kf of G(Af), instead we
shall simply state that all such subgroups under our consideration will be of the form Kf = Y v∈Sf 0 Kv,
where Kv is a compact subgroup of GL2(Fv), which we take to be GL2(Ov) for
all but finitely many places v ∈ Sf.
Let AG denote the maximal Q-split torus in the centre of G, which can be
identified with the multiplicative Q-group Gm. Moreover, let AG(R) denote
the set of real points of AG, and A0G(R) the connected component of AG(R)
containing the identity, so that A0
G(R) ' R+, embedded diagonally into the
components of G(R) (we refer to A0
Recall from Section 3.1 that we have an identification Γ0(N )\SL2(R) ' GL2(Q)\GL2(A)/A0G(R)K0(N ),
and thus can identify the open modular curve Y0(N ) with the double coset space
GL2(Q)\GL2(A)/A0G(R)K∞K0(N ),
where K∞= SO(2). Motivated by this, we would like to consider spaces of the
form
XKf := G(Q)\G(A)/A
0
G(R)K∞Kf
for various choices of compact open subgroup Kf ⊂ G(Af).
We shall be interested in cohomology with trivial coefficients (in keeping with the classical connection between elliptic curves and modular forms of weight 2). Given a compact subgroup Kf, we define the de Rham complex Ω(XKf, C) to
be the complex of smooth, complex-valued differential forms on XKf, and let
H∗(XKf, C) denote the cohomology of Ω(XKf, C).
While we will be interested in spaces XKf for a specific choice of Kf, it is
useful at first to consider all such subgroups at once by means of a direct limit over the cohomology groups H∗(XKf, C). Explicitly, given a second compact
subgroup Kf0 of G(A), with Kf0 ⊂ Kf, we obtain an inclusion H∗(XKf, C) ,→
H∗(X K0
f, C), thus forming a directed system of cohomology groups. We denote
the direct limit by
H∗(G, C) = lim −→
Kf
H∗(XKf, C).
We note that H∗(G, C) admits a natural G(Af)-module structure, induced
by the natural map g : XKf → Xg−1Kfgfor g ∈ G(Af). Thus, given a particular
compact subgroup Kf⊂ G(Af), one may recover the cohomology of XKf simply
by taking Kf-invariants.
Now, let MG denote the connected component of the intersection of the
kernels of all Q-rational characters of G, and mGthe corresponding Lie algebra of
MG(R). Denoting by g and aGthe Lie algebras of G(R) and A0G(R) respectively,
we have a decomposition
g= aG⊕ mG,
For ease of notation, write A(G) and A0(G) for the spaces A(G(Q)\G(A), 1)
and A0(G(Q)\G(A), 1) of automorphic (respectively cuspidal automorphic) forms
for G with trivial central character (in which we run through all possible com- pact subgroups Kf). Then we have an isomorphism of G(Af)-modules:
H∗(G, C) ' H∗(mG, K∞; A(G)),
where the cohomology on the right-hand side is the relative Lie algebra coho- mology with respect to (mG, K∞) (see, for example, [Sch06], Section 3.2).
There is a decomposition
H∗(G, C) = HEis∗ (G, C) ⊕ Hcusp∗ (G, C) of H∗(G, C) into Eisenstein and cuspidal cohomology, where
Hcusp∗ (G, C) ' H∗(mG, K∞; A0(G)).
We are primarily interested in the cuspidal cohomology, but we briefly men- tion that the Eisenstein cohomology can be thought of as being connected to automorphic forms for parabolic subgroups of G (i.e., it arises from subgroups of G of strictly smaller rank).
Since we are concerned with the cuspidal cohomology Hcusp∗ (G, C), we would like to understand more about its structure. Given a cuspidal automorphic representation π, let Vπ = Vπ∞ ⊗ Vπf denote the (g, K∞) × G(Af)-module
associated with the representation π. Then we have a decomposition of G(Af)-
modules
Hcusp∗ (G, C) =M
π
H∗(mG, K∞; Vπ∞) ⊗ Vπf,
where the sum ranges over those cuspidal automorphic subrepresentations of the space A(G) (see [Sch06], Theorem 4.1).
In particular, fixing a compact open subgroup Kf and taking Kf-invariants,
we find that Hcusp∗ (XKf, C) = M π H∗(mG, K∞; Vπ∞) ⊗ V Kf πf ,
where now the sum is restricted to those cuspidal automorphic representations of level Kf of the space A0(G).
For our purposes, we will say that a cuspidal automorphic representation π is of cohomological type and weight two if the summand H∗(mG, K∞; Vπ∞)
is non-zero. Higher weight representations correspond to non-trivial coefficient systems in the cohomology.
As in the classical case, the cohomology H∗(XKf, C) admits a Hecke action.
Indeed, let K = Kf, and choose g ∈ G(Af) such that each of the subgroups
K1 := K ∩ g−1Kg and K2 := K ∩ gKg−1 have finite index in K. Then the
Hecke correspondence K1 ι1 αgf // K2 ι2 K K
from Section 3.1 induces a correspondence on the cohomology groups
H∗(XK1, C) αgf ,∗ // H∗(X K2, C) ι2,∗ H∗(X K, C) ι∗ 1 OO H∗(X K, C)
via the action on the corresponding de Rham complexes. We shall not give an explicit computation here, rather we shall wait until later, when we have shaped the cohomology into a more computationally accessible form.
We will make one final observation regarding the action of Hecke operators on cohomology. Suppose first that we restrict to cohomology with rational co- efficients. According to [Har06], Chapter 2, Proposition 2.2, H∗(G, Q) is a G(Af)-module, and by taking Kf-coinvariants we obtain the rational cohomol-
ogy groups H∗(XKf, Q), on which the Hecke operators act as defined previously.
Crucially, it can be shown that the rational cuspidal cohomology in fact gener- ates the complex vector space Hcusp∗ (XKf, C), and so in particular the action of
the Hecke operators on cuspidal cohomology groups can be defined rationally. We will bear this in mind for future reference.
While we do not have an explicit description of the (mG, K∞)-cohomology
appearing in the decomposition of Hcusp∗ (XKf, C), we can at least state a result
concerning the degrees in which we can have non-vanishing cuspidal cohomology. Indeed, let X denote the symmetric space
X = G(R)/A0G(R)K∞,
and let `0(G) = rk(g) − rk(k) − 1, where g and k denote the Lie algebras of the
(real) Lie groups G(R) and K∞respectively (here the rank of a real Lie algebra
is given by the dimension of a Cartan subalgebra). Then we have the following result: Proposition 3.4.1. Hcuspi (G, C) = 0 if i /∈ 1 2(dim(X) − `0(G)), 1 2(dim(X) + `0(G)) .
This is similar to, but differs slightly from, [Sch06], Theorem 6.2, in that the result there allows arbitrary coefficient systems, and makes no mention of the split component A0
G(R). Justification for this result can be found on page
34 of [Gun11].
For our example, in which G = ResF /Q(GL2), we can give a simple formula
for the degrees in which the cuspidal cohomology is non-vanishing. As before, let F have signature [r, s], so that
G(R) ' GL2(R)r× GL2(C)s,
and
K∞' O(2)r× U(2)s.
In the next chapter, we shall see that the dimension of the symmetric space X is given by
dim(X) = 3r + 4s − 1, which makes use of the identifications
SL2(R)/SO(2) ' h2 and SL2(C)/SU(2) ' h3,
where h2 and h3 denote hyperbolic 2- and 3-space respectively.
To work out the value `0(G), we note that the Lie algebras gl2(R) and gl2(C)
comprise all 2 × 2 real (respectively complex) matrices, while o(2) and u(2) comprise all 2 × 2 real skew-symmetric (respectively complex skew-hermitian) matrices. For each of the above, with the exception of o(2), the subalgebra of all diagonal matrices is a Cartan subalgebra, while o(2) is a 1-dimensional (and thus abelian) Lie algebra. Thus
rk(g) = 2; if g = gl2(R), 4; if g = gl2(C), 1; if g = o(2), 2; if g = u(2), and so `0(G) = r + 2s − 1.
Combining this information, we have the following result:
Corollary 3.4.2. Let G = ResF /Q(GL2), where F is a number field with sig-
nature [r, s]. Then Hi
(G, C) is non-zero only if i ∈ [r + s, 2r + 3s − 1].