3.5 Ibukiyama’s correspondence
3.5.2 The level U 2 and the theory of O-lattices
In order to construct the relevant open compact subgroup U2 of GU2(DAf) we
must think about how to generalise the one used in Eichler’s correspondence to higher dimensions. The “level 1” open compact subgroup U =Q
qO×q ⊂ D×Af
can be viewed as StabD× Af
(O) under an action defined by right multiplication. Since D×
Af = GU1(DAf) an obvious generalisation would be to consider the
open compact subgroup
U1= StabGU2(DAf)(O
2) ⊂ GU 2(DAf)
or indeed the GU2(DAf)-stabilizer of any free O-module of rank 2 with action
given by matrix multiplication. With this in mind we investigate the general theory of O-lattices in D2 and return to defining U
3.5. Ibukiyama’s correspondence
Viewing Dn as a left D-module, we have a special Hermitian form given by
hx, yi =
n
X
i=1
xiyj
(where x is the standard involution on D). Under this interpretation we find that GUn(D) is the similitude group of this form.
Definition 3.5.6. Let L be a Z-module in Dn. Then L is a left O-lattice of Dn if it is a free left O-module of Dn
of rank n and also a Z-lattice of Dn of
rank n.
A similar definition can be made for right O-lattices, the distinction being made due to non-commutativity of multiplication in D.
Given a left O-lattice L ⊆ Dn we can consider its localizations Lq := L ⊗ Zq,
one for each prime q. By extension of scalars these are left Oq-lattices in Dqn
(the definition being the same as above after tensoring everything with Zq).
We are mainly interested in equivalence classes of these lattices, both locally and globally.
Definition 3.5.7. Let M, M0 be two left Oq-lattices in Dqn. We say that M
and M0 are equivalent if there exists gq ∈ GU2(Dq) such that M = M0gq.
Let L, L0 be two left O-lattices in Dn. We say that L and L0 are locally
equivalent at q if Lq and L0q are equivalent.
We also say that L and L0are globally equivalent if there exists g ∈ GU2(D)
such that L0= Lg.
Clearly global equivalence implies local equivalence everywhere. Naturally we care about the converse since it tells us about local-global behaviour. Un- fortunately the converse fails.
Definition 3.5.8. A genus of O-lattices is a full set of O-lattices that are locally equivalent everywhere.
We can now speak of two lattices lying in the same genus, this being a weaker notion than global equivalence.
We will only concern ourselves with maximal lattices in this thesis. In order to learn something about the number of genera amongst maximal left O-lattices we should first study equivalence classes of maximal left Oq-lattices.
Shimura tells us the following (see [59]):
Theorem 3.5.9. If D is split at q, i.e Dq ∼= M2(Qq), then all maximal left
If Dq is a division algebra then there are exactly two equivalence classes of
maximal left Oq-lattices in Dnq, one of which is represented by Onq.
Corollary 3.5.10. Let D be definite and ramified at m distinct finite primes. Then there are 2m genera of maximal left O-lattices in Dn.
Proof. Let L be a maximal left O-lattice in Dn. Then for any unramified place
q of D we know that Lq is equivalent to Oqn.
At any ramified prime p we know that Lpis equivalent to one of two lattices,
due to the Theorem. Since there are m ramified places of D and local equivalence is independent between choices of ramified primes we have at most 2mgenera.
It is shown in [59] that in fact all of these possibilities can occur. Thus we are done.
In particular the above shows that in our case, where D has exactly one ramified finite prime, there should be exactly two genera. From now on assume we are in this case.
Definition 3.5.11. Let D be ramified at p, ∞ for some prime p:
• If a maximal left O-lattice L is locally equivalent to On
q for all q then we
say that L lies in the principal genus.
• If at the ramified prime p we have that Lp is locally inequivalent to Onp
then we say that L lies in the non-principal genus.
For example the standard lattice Onlies in the principal genus. The following
explicit description of O-lattices is due to Ibukiyama [40].
Theorem 3.5.12. Let n ≥ 2. Every maximal left O-lattice in Dncan be written in the form Ong for some g ∈ GLn(D).
Further Ibukiyama was able to find a criterion on the matrix g that deter- mines which genus the lattice belongs to.
Theorem 3.5.13. Let n ≥ 2 and suppose L = Ong is a maximal left O-lattice.
• L lies in the principal genus if and only if ggT = mx for some positive
m ∈ Q and some x ∈ GLn(O) such that x = xT and such that x is positive
3.5. Ibukiyama’s correspondence
• L lies in the non-principal genus if and only if ggT = m
ps r
r pt
where m ∈ Q is positive, s, t ∈ N, r ∈ O lies in the two sided ideal of O above p and is such that p2st − N (r) = p (so that the matrix on the right has
determinant p).
Using the above it is easy to produce O-lattices that are in either genus. Example 3.5.14. The choice g = I satisfies the properties in part 1 of the theorem and so On is a lattice in the principal genus, as expected.
In practice one may almost always take m = s = t = 1 to produce lattices in the non-principal genus. We will make these choices from now on.
Given a maximal O-lattice L = Ong one can consider StabGUn(DAf)(L)
(where we view L as the collection of its localizations). Such stabilizer subgroups are open compact subgroups of GUn(DAf) and so serve well as level structures
for algebraic modular forms on GUn(D).
Let n = 2 now. Earlier we defined U1= StabGU2(DAf)(O
2). We now have an
interpretation of this as the stabilizer of a lattice lying in the principal genus. In a similar vein we fix a choice of g ∈ GL2(D) as in part 2 of Theorem
3.5.13 (taking m = s = t = 1). Then the corresponding lattice lies in the non- principal genus and so we get a genuinely different open compact subgroup U2=
StabGU2(DAf)(O
2g). This is the open compact subgroup used in Ibukiyama’s
correspondence.
As a final remark the adelic modular curves GU2(D)\GU2(DAf)/Ui for i =
1, 2 have interpretations in this setting as global equivalence classes of lattices within the genus determined by Ui. Thus the class number has an arithmetic
significance here.
3.5.3
Hecke operators
Before discussing the new subspace of A(GU2(D), U2, Vj,k−3) it remains to ex-
plain the transfer of the Hecke operators. The story is similar to the Eichler correspondence but has subtle differences. Again, we will only see this for q 6= p. The Hecke action of Tq at q 6= p for level p Siegel forms is defined by the
double coset operator for the matrix
Mq= 1 0 0 0 0 1 0 0 0 0 q 0 0 0 0 q ∈ GSp4(Qq).
Now since D splits at q we have that GU2(Dq) ∼= GSp4(Qq). Fixing such an
isomorphism as in Corollary 3.5.4 we may choose vq ∈ GU2(Dq) such that
vq 7→ Mq.
Let gq∈ GL2(Dq) be the q component of the matrix g as chosen in Theorem
3.5.13 (as embedded diagonally into GL2(DAf)). We know by definition of the
non-principal genus that O2g is locally equivalent to O2 at q 6= p. Thus there exists hq ∈ GU2(Dq) such that Oq2gq = Oq2hq. We then have a corresponding
uq∈ GU2(Dq) given by uq = hqvqh−1q .
This may seem like a convoluted way to construct uq in comparison to previ-
ous choices but the key point here is we cannot assume that gq lies in GU2(Dq).
We will see why this is important later.
Let u ∈ GU2(DAf) have uq as the component at q and have identity com-
ponent elsewhere.
Definition 3.5.15. For the above choice of u, the corresponding Hecke operator on Anew(GU2(D), U2, Vj,k−3) will be called Tu,q.
Under Ibukiyama’s (conjectural) correspondence it is predicted that: Tq ←→ Tu,q.
Recall that we are really only interested in eigenvalues for the Tq operator
in Harder’s conjecture so we will only be interested in decomposing U2uU2
into left cosets. This will be done later too in a similar fashion to the Eichler correspondence.