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Let F3 = Q(t), where t = ζ8 denotes a primitive eighth root of unity. Let
σ ∈ Gal(F/Q) denote the automorphism of F sending t to t3. There are three equivalence classes of perfect forms under the action of GL2(OF), with rep-
resentatives given by the Hermitian forms [M1, M1σ], [M2, M2σ] and [M3, M3σ],
where M1= 1 16 −2t3+ 2t + 4 −t2− 2t − 2 2t3+ t2− 2 −2t3+ 2t + 4 , M2= 1 8 −t3+ t + 2 −t3− 2t2− 2t − 1 2t3+ 2t2+ t − 1 −4t3+ 4t + 6 and M3= 1 8 −t3+ t + 2 t3+ t2− 1 −t2− t − 1 −t3+ t + 2 ,
whose corresponding perfect pyramids have 12, 12 and 24 vertices respectively. As before, we present details of the decomposition of the Koecher polytope:
Dimension Simplicial Faces Non-Simplicial Faces Boundary Faces Total
1 2 0 1 3 2 8 0 0 8 3 23 1 0 24 4 33 4 0 37 5 27 7 0 34 6 4 10 0 14 7 0 3 0 3 We also note:
• There is a single 3-dimensional non-simplicial face, with 6 vertices. • There are two 4-dimensional non-simplicial faces with 6 vertices, and two
with 7 vertices.
• There are four 5-dimensional non-simplicial faces with 7 vertices, and three with 8 vertices.
• There are four 6-dimensional non-simplicial faces with 8 vertices, four with 9 vertices, one with 10 vertices, and one with 11 vertices.
• There are two 7-dimensional non-simplicial faces with 12 vertices, and one with 24 vertices, as noted previously.
Chapter 5
The Cohomology of
Arithmetic Subgroups
This chapter concerns the practical computation of both the group coho- mology H∗(Γ0(n), C) and the Hecke action on cohomology classes. We begin in
Section 5.1 by presenting a cell complex, known as the sharbly complex, whose homology is dual to the group cohomology we wish to study, and which exhibits a Hecke action.
Section 5.2 provides an in-depth explanation of how we can compute the group cohomology via the homology of the sharbly complex, and the techniques required in order to compute the Hecke action on classes in the sharbly homol- ogy. In Section 5.3 we present details of Hecke eigenclasses in the sharbly homology which correspond to cuspidal automorphic forms, while in Section 5.4 we discuss some of the practical issues regarding our computations.
5.1
The Sharbly Complex
We now move towards our main task of finding modular elliptic curves, beginning by studying the automorphic representations with which we hope to match such curves. As before, let F be a number field, with ring of integers OF, and set G = ResF /Q(GL2). We shall assume throughout that F has trivial
class group, and signature [r, s]. Given an ideal n of F , define an arithmetic subgroup Γ0(n) of G(Q) ' GL2(F ) to be the subgroup
Γ0(n) :=
a b
c d ∈ GL2(OF); c ∈ n .
Let Sf denote the set of finite places of F . Similarly to the classical case in
Section 3.2, define a compact subgroup K0(n) of
G(Af) '
Y
v∈Sf
GL2(Fv)
to be the product of the subgroups Kv(n) for v ∈ Sf, where
Kv(n) =
a b
c d ∈ GL2(Ov); c ≡ 0 (mod n)
if v divides n, and Kv(n) = GL2(Ov) otherwise.
Then, denoting by X the symmetric space G(R)/A0G(R)K∞, we have (since
F is assumed to have trivial class group) an identification Γ0(n)\X ' A0G(R)G(Q)\G(A)/K∞K0(n),
as in Section 3.4. By the results of that section, we can realise automor- phic representations through the cohomology of the locally symmetric space Γ0(n)\X. In turn (as for the classical case in Section 2.6), we can identify this
with the group cohomology H∗(Γ0(n), C).
There are two main approaches for computing this cohomology. The first is perhaps the most obvious; using the Koecher decomposition of the symmetric space X, one can naturally construct a cell complex using the resulting Koecher cells. One can then compute the cohomology of Γ0(n)\X by computing the
Γ0(n)-equivariant cohomology of this complex - since there are only finitely
many Koecher cells under the action of Γ0(n), this computation can indeed be
performed in practice.
We shall take a second approach, which has noticeable advantages over the first. The trouble with working with the Koecher cell complex is that it is fairly restrictive - the Hecke operators which we shall later want to compute do not preserve the Koecher cells, and so we cannot hope to compute their action on the cohomology using this method. The approach which we shall now explain allows us to compute both the cohomology and the Hecke action, by working with a much larger space.
To begin with, we require a few preliminary notions. Let Γ be an arbitrary arithmetic subgroup of G(Q), for G a reductive algebraic group defined over Q. If Γ is torsion-free, we define the cohomological dimension of Γ to be the smallest integer ν such that Hν+1(Γ, M ) = 0 for all coefficient systems M . For an
arbitrary arithmetic subgroup Γ, we define the virtual comological dimension ν of Γ to be the cohomological dimension of any finite-index torsion-free subgroup (this is known to be well-defined).
A formula for the virtual cohomological dimension is given by the following result (see [BS73], Theorem 11.4.4):
Theorem 5.1.1. Let G be a reductive Q-group, R its radical, and X = G(R)/A0G(R)K
a globally symmetric space, where A0
G(R) denotes the split component of G(R)
and K is a maximal compact subgroup of G(R). Then for any arithmetic sub- group Γ of G(Q), we have
ν = dim(X) − rkQ(G/R).
Returning to the case of G = ResF /Q(GL2), where we recall that F has
signature [r, s], the radical R is the subgroup of diagonal matrices, and subse- quently rkQ(G/R) = 1, while we have seen previously that dim(X) = 3r +4s−1, and so
ν = 3r + 4s − 2 for each arithmetic subgroup Γ of G(Q).
Next, let P1(F ) denote the projective line over our field F , and let Z[P1(F )] denote the free abelian group generated by it. One defines the augmentation map : Z[P1
(F )] → Z by
XnPP
=XnP,
and subsequently we define the Steinberg module St2 for GL2(F ) by the short
exact sequence
0 −→ St2−→ Z[P1(F )]
−→ Z −→ 0.
This clearly admits an action of GL2(F ), induced by the action on P1(F ).
The Steinberg module is a dualizing module for Γ0(n) (in the sense of [BS73],
Section 11.4) and so we have
Hν−k(Γ0(n), C) ' Hk(Γ0(n), St2⊗ZC),
To compute the homology of Γ0(n) with coefficients in the Steinberg module,
we require an appropriate resolution of St2. Such a resolution is provided for us
by the sharbly complex. This is defined as follows: for each k, let Ak denote the
Z-module of Z-linear combinations of (k + 2)-tuples u = [u1, . . . , uk+2], where
ui∈ O2F. In addition, let Rk denote the submodule generated by the relations:
• [u1, . . . , uk+2] − sgn(σ)[uσ(1), . . . , uσ(k+2)], for any permutation σ ∈ Sk+2;
• [u, u2, . . . , uk+2] − [v, u2, . . . , uk+2], for any u, v ∈ O2F with q(u) = λq(v),
for some λ ∈ R+;
• [u1, . . . , uk+2], if the F -span of the vectors u1, . . . , uk+2 is 1-dimensional
(we call such sharblies degenerate).
We then define the Z-module of k-sharblies to be the quotient Sk = Ak/Rk.
Using the relations in Rk, we shall always assume that the vectors ui ∈ O2F
satisfy the property that there is no point of Ξ on the line segment joining q(ui)
with the origin (that is, if vi∈ O2F with q(vi) = λq(ui) for some λ ∈ R+, then
λ ≥ 1).
One can define a boundary map ∂ :Sk→Sk−1 by
∂([u1, . . . , uk+2]) = k+2
X
i=1
(−1)i+1[u1, . . . , ˆui, . . . , uk+2],
where ˆui indicates that we omit ui. The resulting complex S∗ is called the
sharbly complex. The sharbly complex admits an obvious action of GL2(OF),
given by
g · [u1, . . . , uk+2] = [gu1, . . . , guk+2], g ∈ GL2(OF),
and this clearly commutes with the boundary map. In particular, for any sub- group Γ of GL2(OF), we can define the quotient of Γ-coinvariants, (S∗)Γ, by
enforcing the additional relation
• [u1, . . . , uk+2] − γ · [u1, . . . , uk+2] for all γ ∈ Γ.
One can define a map φ : S0 → St2 as follows: given u ∈ OF2, let [u]
denote the line spanned by u, viewed as an element of P1(F ). Then, given
u = [u1, u2] ∈S0, we define
This map is well-defined: indeed, the first and third relations defining the sharbly complex clearly have no effect on φ. For the second, suppose that q(u) = λq(v) for some λ ∈ R+ and u, v ∈ OF2. Without loss of generality (since
the map φ is GL2(F )-equivariant) we may assume that u = (a0) and so v = (0b),
whence u = ba−1v. Since a, b ∈ OF, ba−1∈ F , and so [u] = [v], as required.
Consequently, we can define a sequence . . .−→∂ Sk ∂ −→ . . .−→∂ S1 ∂ −→S0 φ −→ St2 −→ 0.
In fact (see [AGM11], Theorem 5) this sequence is exact, and thus pro- vides an acyclic resolution of the Steinberg module. In particular, we have an isomorphism
Hν−k(Γ0(n), C) ' Hk((S∗)Γ0(n), C),
the latter of which is straightforward to determine computationally.
All the results we have stated apply to an arbitrary number field. Henceforth, with the results of Section 3.5 in mind, we shall specialize to CM fields. In fact, we shall restrict ourselves further to quartic CM fields, for reasons which shall soon become apparent. In this case, F has signature [0, 2], and thus the virtual cohomological dimension of any subgroup Γ0(n) is 6, by Theorem 5.1.1. In
addition, by Corollary 3.4.2, we have
Hcuspi (Γ0(n), C) = 0 if i /∈ [2, 5],
so the smallest degree of the sharbly homology in which we could hope to study cuspidal classes is degree 1. In the next section, we shall describe a method for computing the Hecke action on these particular homology groups.