We now give the promised example of a Galois representation attached to a cuspidal automorphic representation, as presented by C.P. Mok in [Mok14], which will form the basis for our later study. While Mok’s result is more gen- eral, we shall restrict our attention to automorphic representations with trivial central character, echoing the classical treatment in which elliptic curves corre- spond to modular forms without character.
We begin by stating the result (an adaptation of [Mok14], Theorem 1.1). Recall that we say a cuspidal automorphic representation π is of cohomological type if it corresponds to a non-zero summand in the cohomology Hcusp∗ (G, C). Theorem 3.5.1. Let F be a CM field, and let π be a cuspidal automorphic rep- resentation of ResF /Q(GL2) of cohomological type, with trivial central character,
and fix a prime `. Then there exists an `-adic Galois representation ρπ : Gal(F /F ) → GL2(Q`)
such that, for each place v of F not dividing `, we have the local-to-global com- patibility statement, up to semisimplification:
WD(ρπ,v)ss'Lv(πv⊗ |det| −1
2
v )ss.
Furthermore, if πv is not a twist of Steinberg (e.g., is an unramified princi-
pal series) then we have the full local-to-global compatibility statement, up to Frobenius semisimplification:
WD(ρπ,v)Frob'Lv(πv⊗ |det| −1
2
v ).
We will require a few details regarding the representation ρπ. Note first
that for each place v at which the representation πv is unramified, so too is the
representationLv(πv⊗ |det| −1
2
v ). Since Frobenius semisimplification preserves
the action on inertia groups, this implies that ρ is similarly unramified at these places.
Next, for each unramified place v, we have Tr(Lv(πv⊗ |det| −1 2 v )(Frobv)) = q 1 2Tr(tπ v),
where q is the cardinality of the residue field of Fv, and tπvdenotes the Langlands
In particular, suppose that ϕ is an automorphic form of level K0(n), for some
ideal n of F , which is an eigenform for the Hecke operators Tp, and let π be
its associated automorphic representation. If v is a place of F not dividing n, then the subgroup Kv(n) of K0(n) is, by definition, the group GL2(Ov). Since
ϕ is invariant under K0(n), it follows that there is a non-trivial vector fixed
by the action of GL2(Ov) under πv (which in fact spans the one-dimensional
space of GL2(Ov)-fixed vectors, by the results of Section 3.2), and thus πv is
unramified. In this case, q12Tr(tπ
v) is equal to θ(Tv) as in Section 3.2 (where
we extend the notion of a Hecke operator to arbitrary fields in the obvious manner).
Finally, by Theorem 3.3.1, the determinant of the local Galois representa- tion ρv is equivalent to |det|
−1 2
v . Denoting by $ a uniformiser of Fv, we have
|det($)|−12
v = q,
where q is the cardinality of the residue field of Fv. Under the aforementioned
correspondence, we observe that
det(ρv(Frobv)) = q,
and thus the determinant of ρvis given by the local cyclotomic character, which
we recall is the same as the determinant of the Galois representation attached to a rational elliptic curve.
Chapter 4
Koecher Theory
Motivated by the results of Section 3.4, we aim to study automorphic repre- sentations through the corresponding cohomology of certain symmetric spaces for the group ResF /Q(GL2), with the aim of being able to compute the data
attached to the Galois representations constructed in Section 3.5 in the case where F is a CM quartic field.
We begin by establishing the structure of the global symmetric spaces we will study in Section 4.1, and show that they can be realised as cones of binary Hermitian forms over the field F . Such cones are examples of positivity domains, and in Section 4.2 we recall the theory of Koecher (generalising work of Vorono¨ı) which provides us with a decomposition of such domains. In Section 4.3 we continue our exploration of this theory by studying the Koecher polytope, an infinite polytope which captures the information of this decomposition in a manner that will lend itself more readily to our future calculations.
Finally, in Sections 4.4 and 4.5 we return, armed with the knowledge of the previous sections, to our case of interest, and provide some details of the Koecher polytope specific to our global symmetric space.
4.1
A Model for the Symmetric Space of GL
2 Let F be a number field, with signature [r, s] and ring of integers OF, and letG denote the reductive Q-group ResF /Q(GL2), where ResF /Qdenotes the Weil
restriction of scalars. By the results of Section 3.4, we can study automorphic forms for G by instead looking at the cohomology of certain symmetric spaces
XKf = G(Q)\G(A)/A
0
G(R)K∞Kf,
where A0
G(R) is the split component of G, and Kf is a compact open subgroup of
G(Af). These spaces can, in turn, be realized as quotient spaces of the globally
symmetric space
X = G(R)/A0G(R)K∞.
As touched upon earlier, there is a geometric realization of this space, echoing the role of the complex upper half-plane in the theory of classical modular forms. Indeed, note that
G(R) ' GL2(R)r× GL2(C)s,
while
K∞' O(2)r× U(2)s.
In addition, we have identifications
GL2(R)/O(2) ' h2× R+
and
GL2(C)/U(2) ' h3× R+,
where h2 and h3 denote hyperbolic 2- and 3-space respectively. Recalling that
A0
G(R) ' R+, we obtain the final identification
X ' hr2× hs 3× R
r+s−1
+ .
We will now present an alternative description of the symmetric space X, in terms of a cone of binary Hermitian forms over F , which will have the benefit of being more amenable to computations. The field F has r real embeddings and s conjugate pairs of complex embeddings; for each conjugate pair, fix a particular embedding F ,→ C. For each infinite place v, define
Vv =
Sym2(R); if v is real; Herm2(C); if v is complex,
where Sym2(R) and Herm2(C) denote the real vector spaces of real symmetric
Define the space of Hermitian forms over F to be
V =Y
v
Vv.
This is a real vector space, with
dimR(V) = 3r + 4s.
We can equip V with an inner product h , i by setting
hX, Y i =X
v
cvTr(XvYv),
where cv= 1 if v is a real place of F , and cv= 2 if v is a complex place of F .
The vector space V admits an action of the group G(R). Indeed, identifying G(R) ' GL2(R)r× GL2(C)s via the embeddings corresponding to the infinite
places of F , we have, for an arbitrary element g = (gv) ∈ G(R):
g · X = (gvXvgv∗) for all X ∈ V,
where g∗v denotes the transpose of gv if v is a real embedding, and the complex
conjugate transpose of gv if v is a complex embedding.
We can define a cone C contained in V by setting Cvto be the cone of positive
definite matrices in Vv for each v, and then defining
C =Y
v
Cv.
With respect to the inner product h , i previously defined on V, C is self- adjoint, meaning that we have a characterization
C = {X ∈ V; hX, Y i > 0 for all Y ∈ C \ {0}},
where the closure C of C consists of all positive semi-definite forms (in the sense that we allow each component to be positive semi-definite).
The group action of G(R) on V restricts to an action on C, and in fact any linear automorphism of C arises in this way. Moreover, we have the following result:
Proposition 4.1.1. The action of G(R) on the cone C described above is tran- sitive.
Proof. Since both C and G(R) decompose into products indexed by the infinite places of F , it suffices to prove transitivity of the action componentwise. Let X1, X2∈ Cvfor some place v. Since both are positive definite real symmetric
(respectively Hermitian) matrices, there exist orthogonal (respectively unitary) matrices M1 and M2such that the matrices MiXiMi∗are diagonal, say
MiXiMi∗= αi 0 0 βi , where the αiand βi are positive real numbers.
If we define g = M2∗ ( α2 α1) 1 2 0 0 (β2 β1) 1 2 ! M1,
then g ∈ G(R), and we have g · X1= X2, as required.
Now, consider the point I = (Iv) ∈ C, where each Ivis the 2 × 2 identity ma-
trix in the factor Sym2(R) or Herm2(R). It is clear to see that under the action
of G(R), each Iv is fixed by the orthogonal subgroup O(2) or unitary subgroup
U(2) of GL2(Fv), depending on whether v is real or complex. It therefore follows
that C ' G(R)/K∞, where K∞ is the standard maximal compact subgroup of
G(R) defined previously. Furthermore, if we quotient out C by positive real homotheties, we obtain an isomorphism
C/R+' G(R)/A0G(R)K∞,
thus confirming our earlier statement that we can realise our symmetric space X as a cone of Hermitian forms over F .