The cusp amplitudes and quasi-level of a
congruence subgroup of SL
2over any Dedekind
domain
BY
A. W. Mason
Department of Mathematics, University of Glasgow Glasgow G12 8QW, Scotland, U.K.
e-mail: awm@maths.gla.ac.uk
AND
Andreas Schweizer
National Center for Theoretical Sciences, Mathematics Division, National Tsing Hua University
Hsinchu 300, Taiwan
e-mail: schweizer@math.cts.nthu.edu.tw Abstract
This is the latest part of an ongoing project aimed at extending algebraic properties of the classical modular group SL2(Z) to equivalent groups in the
theory of Drinfeld modules. We are especially interested in those properties which are important in the classical theory of modular forms. Our results are intended to be applicable to the theory of Drinfeld modular curves and forms. Here we are concerned with the cusp amplitudes and level of a subgroup of such a group (in particular a congruence subgroup). In the process we have discovered that most of the theory of congruence subgroups, including the properties of their cusp amplitudes and level, can be extended to SL2(D),
where D is any Dedekind ring. This means that this theory can be extended to a non-arithmetic setting.
We begin with an ideal theoretic definition of the cusp amplitudes of a subgroup H of SL2(D) and extend the remarkable results of Larcher for the
congruence subgroups of SL2(Z).
We then extend the definition of the cusp amplitude and level of a subgroup H of SL2(D) by introducing the notions of quasi-amplitude and quasi-level.
Quasi-amplitudes and quasi-level encode more information about H since they are not required to be ideals. In general although the level and quasi-level can be very different, we show that for many congruence subgroups they are equal.
As a bonus our results provide several new necessary conditions for a sub-group of SL2(D) to be a congruence subgroup. These include an inequality
Keywords: Dedekind domain, congruence subgroup, cusp amplitude, level, quasi-level, index.
Mathematics Subject Classification (2000): 11F06, 20G30, 20H05
Introduction
This paper is part of an ongoing project which aims to extend algebraic results from the classical modular group, SL2(Z), to equivalent groups occurring in the context
of Drinfeld modular curves [Ge]. We are especially concerned with those results which have proved applicable to the classical theory of modular forms. Our hope is that our results will prove useful to experts in the theory of Drinfeld modules. Here we are primarily concerned with the cusp amplitudes and level of a subgroup of such a group. One of our principal aims is to demonstrate that these concepts, as well as much of the classical theory of congruence subgroups, can be extended to linear groups defined over any Dedekind domain D.
Congruence subgroups, beginning in the 19th century with SL2(Z), are usually
defined for a matrix group with entries in an arithmetic Dedekind domain A (or, more generally, an order in such a domain). The definition involves a non-zero A-ideal. Since every proper quotient of A is finite, every congruence subgroup is necessarily of finite index. This leads naturally to the converse question (for the particular matrix group), namely the celebrated Congruence Subgroup Problem, which has attracted considerable attention for many years. For D which are not arithmetic, i.e. D which might have proper infinite quotients, there is no longer a close connection between the congruence subgroups of SL2(D) and its finite index
subgroups. For example, if k is an infinite field, then SL2(k[t]) has infinitely many
congruence subgroups and no proper finite index subgroups.
The groups SL2(D) include a number of very important special cases, for
exam-ple, the classical modular group SL2(Z), the Bianchi groups, where D is the ring of
integers in an imaginary quadratic number field, and groups occurring in the context of Drinfeld modular curves [Ge]. Other examples are SL2 over a ring of S-integers
of a number field or of a function field in one variable over any constant field, SL2
over any local ring and SL2 over any principal ideal domain.
When generalizing concepts from the classical modular group SL2(Z) to SL2(R)
for more general rings R, one sometimes has to make a choice between loss of structure and loss of information. For example, let H be a subgroup of SL2(R).
Then the set of all a ∈ R such the translation matrix 1 a0 1 is contained in H is an additive subgroup of R. Hence in the special case R = Z this is automatically an ideal. In the general case one can choose between considering this set and losing the ideal structure or considering the biggest ideal contained in this set and thus losing information about H.
This is one reason why it is sometimes difficult to extend results that hold for SL2(Z) to more general groups SL2(R). Either the associated objects do not have
enough structure and hence one cannot prove enough about them, or they have the necessary structure but they don’t encode all the desired information. This also indicates that one should look at both of the possible generalizations and how they are related.
Let us make this precise by introducing the most important instance of this dichotomy structure versus information.
Let R be a commutative ring with identity and let H be a subgroup of SL2(R).
The classical definition of the level l(H) of H opts for structure over information and defines it as the largest R-ideal q such that for every a ∈ q the translation matrix
1 a
0 1 is contained in every conjugate of H in SL2(R). We emphasize information over
structure and call the set of all a ∈ R such the translation matrix 1 a0 1 is contained in every conjugate of H in SL2(R) the quasi-level of H, denoted by ql(H). Implicitly
the quasi-level already occurs for example in the proof of Proposition 1 in [Se1]. Obviously, the level of H is the largest R-ideal contained in the quasi-level of H. Clearly for the special case R = Z we always have ql(H) = l(H). In general however, as we shall see, they can differ by “as much as possible”. By definition the level of a subgroup is not merely an additive subgroup of R and it can be shown to have much stronger properties than the quasi-level. However as a parameter for studying H it is less useful than ql(H), mainly for the following reason. If H is of finite index in SL2(R) then ql(H) is non-zero. On the other hand for some R
(including R = k[t], where k is a finite field) it is known that SL2(R) has infinitely
many finite index subgroups of level zero. Moreover for any finite index subgroup H important information is provided by equations involving the size of the quasi-level of H and its index in SL2(R).
Now let D be any Dedekind domain with quotient field F . The group SL2(D) acts
on the projective line P1(F ) = F ∪ {∞} as a set of linear fractional transformations.
Let H be a subgroup of SL2(D). Clearly H acts on O∞ = {g(∞) : g ∈ SL2(D)},
the SL2(D)-orbit containing ∞. We refer to the orbits of the H-action on O∞ as
the H-cusps.
The classical definition of cusp amplitude again prefers structure over information and defines the cusp amplitude of the H-cusp containing g(∞) as a certain D-ideal c(H, g) associated to the stabilizer of g(∞) in H. See Section 2 for details.
As the cusp amplitude often loses too much information about the stabilizer of the cusp, we also introduce what we call the quasi-amplitude. This represents the other possibility to generalize the notion of cusp-amplitude from SL2(Z) to SL2(D),
going for more information and less structure by dropping the requirement that it is an ideal and considering all translation matrices in Hg.
Again, in general the quasi-amplitude b(H, g) is only an additive subgroup of D. Obviously, the cusp amplitude c(H, g) is the largest ideal contained in b(H, g). In situations where one can control how far the two can be apart, one then obtains
results that generalize theorems for the classical modular group.
Let A(H) denote the set of all the cusp amplitudes of H. It is easy to see that the intersection of all these cusp amplitudes is nothing else than the level of H. In our first principal result we prove that for one important class of subgroups the set A(H) has the following surprising properties.
Theorem A. Let H be a subgroup of SL2(D) and let
cmin = \ q∈A(H) q and cmax = X q∈A(H) q.
If H is a congruence subgroup, then cmin, cmax ∈ A(H).
In particular, if H is a congruence subgroup, there is an H-cusp in the SL2
(D)-orbit of ∞ whose cusp amplitude equals the level of H.
This extends the results of a remarkable paper of Larcher [La] for the particular case D = Z. For this case Stothers [St] has also proved that A(H) has a lower bound by an alternative method. In extending Larcher’s proofs we have simplified his approach in a number of ways. Most importantly we have avoided his use of the Dirichlet theorem on primes in an arithmetic progression (for Z).
Strictly speaking, our definition of the cusp amplitude differs slightly from the one given in [La]. The results in [La] are formulated for P SL2(Z), or equivalently,
for congruence subgroups that contain −I2, and for those both definitions coincide.
Cusp amplitudes were originally introduced for subgroups of the modular group SL2(Z) and play an important role in the theory of modular forms and modular
curves. Theorem A holds (trivially) for every normal subgroup of SL2(D) since
then A(H) reduces to a single ideal. However it is known that it does not hold in general for non-normal non-congruence subgroups. Moreover we show that Theorem A does not hold in general for the quasi-amplitudes of a congruence subgroup.
The intersection of all the quasi-amplitudes (resp. cusp amplitudes) of H is ql(H), its quasi-level (resp. l(H), its level). As previously stated, for the classi-cal case D = Z they clearly coincide. Our second principal result shows, rather surprisingly, that this is also true for “many” congruence subgroups of SL2(D).
We give a name to the precise condition as it will be used in several places in the paper.
Condition L. Let H be a subgroup of SL2(D) of non-zero level l(H) = q. We
say that l(H) (or H) satisfies Condition L if both the following two conditions hold: (i) q + (2) = D;
Note that both conditions are satisfied if, for example, q + (6) = D, so in particular always for char(D) = p ≥ 5. Condition L is also automatically satisfied if D con-tains a field that is a nontrivial extension of F3.
Theorem B. Let H be a congruence subgroup of SL2(D) whose level l(H) = q
satisfies Condition L. Then
ql(H) = l(H).
In other words, then the quasi-level of H is automatically an ideal.
There are examples of (normal) congruence subgroups which show that both re-strictions on q from Condition L are necessary in Theorem B. For non-congruence subgroups H Theorem B breaks down completely, i.e. ql(H) and l(H) can “differ by as much as possible”. More precisely we prove that SL2(k[t]), where k is a field,
contains normal, non-congruence subgroups whose quasi-level has k-codimension 1 in k[t] and whose level can take any possible value (including zero). (There are no proper subgroups of SL2(k[t]) whose quasi-level is k[t].)
Presumably one of the most interesting and in practice one of the most useful results on congruence subgroups would be to bound the level in terms of the index of the subgroup, i.e. to generalize the classical result that the level of a congru-ence subgroup of SL2(Z) can not be bigger than its index. In Section 5 we discuss
different generalizations of this. The proofs use many of the previous results. We highlight a version that holds for general D.
Theorem C. Let D be any Dedekind domain and let H be a congruence subgroup of index n in SL2(D) whose level l(H) satisfies Condition L. Then
|D/l(H)| divides n!, and if H is a normal subgroup then even
|D/l(H)| divides n.
The structure of the paper is as follows:
In Section 1 we clarify by means of theorems and counterexamples how the properties ‘Every subgroup of finite index has non-zero level’, ‘Every subgroup of non-zero level is a congruence subgroup’ and ‘Every subgroup of finite index is a congruence subgroup’ are related. In Section 2 we present the quite lengthy proof of Theorem A. In Section 3 we introduce quasi-amplitudes. In Section 4 we define the quasi-level and prove Theorem B. Using the previous results, we then can in Section 5 prove Theorem C and similar relations between the level and the index of a congruence subgroup.
On the way we also prove several other interesting results. Whenever appropriate we show by means of examples the necessity of conditions in theorems and the limitations of certain definitions.
Of course, every result that holds for congruence subgroups can also be refor-mulated as a necessary condition that a subgroup must satisfy in order to stand a chance of being a congruence subgroup. For the classical modular group SL2(Z)
many such criteria are known, and some of them are very useful in practice (compare Example 2.12). In order to get an overview, we now summarize the criteria that our results furnish. Several of them seem to be new or at least new in such a general setting.
Theorem D. Let D be any Dedekind domain and q a non-zero ideal of D. Let H be a subgroup of G = SL2(D) with level l(H) = q and quasi-level ql(H). In order
for H to be a congruence subgroup, the following conditions are necessary: a) ( Corollary 1.4) G(q) ⊆ H.
b) (Corollary 2.8) The ideal-theoretic sum of all cusp amplitudes of H is again a cusp amplitude of H.
c) (Theorem 2.10) The intersection of all cusp amplitudes of H is again a cusp amplitude of H.
d) (Corollary 2.11) The level q of H is a cusp amplitude of H.
e) (Lemma 4.2) For any α ∈ D that is invertible modulo q we have α2ql(H) ⊆ ql(H).
Now suppose moreover that the level q of H satisfies Condition L. Then the following conditions are necessary for H to be a congruence subgroup:
f ) (Theorem 4.6) ql(H) = l(H).
g) (Corollary 5.7) |D/l(H)| divides (|G : H|)!.
Finally suppose that in addition H is a normal subgroup of G (and that l(H) satisfies Condition L). Then the following conditions are necessary for H to be a congruence subgroup:
h) (Corollary 4.7) Every quasi-amplitude of H is actually a cusp amplitude of H.
With the exception of a) in general none of the conditions is sufficient for H to be a congruence subgroup. Conditions e), f), and h) hold automatically if D = Z and Remark 2.12 points out non-congruence subgroups of SL2(Z) that satisfy conditions
b), c), d), g) and i).
On the other hand, none of the conditions is trivial in the sense of automatically true for every D. Remark 2.12 provides examples of non-congruence subgroups of SL2(Z) that violate conditions b), c) and d), and Theorem 4.12 allows the
construc-tion of normal non-congruence subgroups that satisfy Condiconstruc-tion L, but violate the conditions e), f), g), h) and i).
If H is a congruence subgroup such that l(H) does not satisfy Condition L, then Examples 4.10 and 5.4 show that none of the conditions f), g), h) and i) need hold, even if H is normal. Compare also Example 4.9 for conditions f) and h).
Finally, the condition that H is normal cannot simply be dropped for parts h) and i). Examples 3.2 and 3.5 exhibit non-normal congruence subgroups H that satisfy Condition L, but neither h) nor i).
1. Congruence subgroups
Let R be a (commutative) ring and let R∗ be its group of units. For each r ∈ R, α ∈ R∗, we put T (α, r) := α r 0 α−1 and T (r) := T (1, r) = 1 r 0 1 . In addition we define S(r) := 1 0 r 1 and R(r) := 1 + r r −r 1 − r .
Let E2(R) be the subgroup of SL2(R) generated by all T (r) and S(r) with r ∈ R.
For each R-ideal q we denote by E2(R, q) the normal subgroup of E2(R) generated
by all T (q) with q ∈ q. By definition we note that E2(R, R) = E2(R). Let N E2(R, q)
be the normal subgroup of SL2(R) generated by E2(R, q). We put N E2(R, R) =
N E2(R). We also define
SL2(R, q) = {X ∈ SL2(R) : X ≡ I2 (mod q)} .
where I2 denotes the 2-dimensional unit matrix. Obviously
E2(R, q) ≤ N E2(R, q) E SL2(R, q).
Occasionally we will also need the Borel group, consisting of the upper triangular matrices,
If H is a subgroup of G, we denote the core of H in G, i.e. the biggest normal subgroup of G contained in H by NH = \ g∈G Hg.
Definition. The level of a subgroup H of SL2(R), l(H), is the largest ideal q0, say,
for which N E2(R, q0) ≤ H. The level is well-defined since N E2(R, q1)N E2(R, q2) =
N E2(R, q1+ q2).
Since N E2(R, q0) is the normal subgroup of SL2(R) generated by all T (r) with
r ∈ q0, we can equivalently say that the level of H is the largest ideal q0 such that the core NH of H contains all translation matrices T (r) with r ∈ q0.
Definition. A subgroup C of SL2(R) is called a congruence subgroup if
SL2(R, q0) ≤ C
for some q0 6= {0}.
It is clear that if R/q0 is finite then C is of finite index in SL2(R).
Let q, q0 be ideals for which q ⊇ q0 ⊇ q2. We will require some properties of
the quotient group SL2(R, q)/SL2(R, q0). We put r = q/q0 and we denote the image
of any q ∈ q in r by q. Let X ∈ SL2(R, q). Then
X = 1 + x y z 1 + t
, for some x, y, z, t ∈ q. We define a map
θ : SL2(R, q) −→ r3
by
θ(X) = (x, y, z). (Note that x + t ≡ 0 (mod q0).)
Lemma 1.1. With the above notation, the map θ induces the following isomor-phism
SL2(R, q)/SL2(R, q0) ∼= (r+)3,
where r+ is the additive group of r.
Moreover the quotient group SL2(R, q)/SL2(R, q0) is generated by the images of
Proof. See the proof of [MSt, Theorem 4.1]. To obtain more precise properties of congruence subgroups we require some re-strictions on R. We recall that R is said to be an SR2-ring if, for all a, b ∈ R such
that aR + bR = R, there exists c ∈ R for which (a + cb) ∈ R∗. Every semi-local ring, for example, is an SR2−ring ([B, Theorem 3.5, p.239]).
Theorem 1.2. Let q, q0 be R-ideals such that R/q0 is an SR2-ring. Then
E2(R, q) SL2(R, q0) = SL2(R, q + q0).
Proof. See the proof of [B, (9.3) Corollary, p. 267].
Since every proper quotient of a Dedekind domain is semi-local the following is an immediate consequence of Theorem 1.2.
Corollary 1.3. Let D be a Dedekind domain and let q, q0 be D-ideals. (i) If q0 6= {0} then
E2(D, q) SL2(D, q0) = SL2(D, q + q0).
(ii)
SL2(D, q) SL2(D, q0) = SL2(D, q + q0).
The classical version of our next result (for the special case D = Z) is due to Fricke. (See [W].)
Corollary 1.4. Let D be a Dedekind domain and let H be a subgroup of SL2(D)
with l(H) = q. Then H is a congruence subgroup if and only if H ≥ SL2(D, q).
Remarks 1.5.
a) By virtue of Theorem 1.2, Corollaries 1.3 and 1.4 also apply to the case where R is a Noetherian domain of Krull dimension one. For example all orders in algebraic number fields are of this type. One “standard” example of a Noetherian domain of Krull dimension one which is not Dedekind is the ring Z[
√
−3]. A Dedekind domain D is an integrally closed Noetherian domain of Krull dimension one.
b) But Corollaries 1.3 and 1.4 do in general not hold for an integrally closed Noetherian domain of Krull dimension bigger than one. Take for example D = Fq[x, y], the polynomial ring in two variables over a finite field. Then
SL2(D)/SL2(D, yD) ∼= SL2(Fq[x]).
Let f (x) ∈ Fq[x] be irreducible of degree at least 2. By Theorem 4.12 below
or by [MSch2, Theorem 5.5] then SL2(Fq[x]) has a normal non-congruence
subgroup N of finite index and level (f (x)). The inverse image of N under reduction modulo y is a normal subgroup of finite index in SL2(D) of level
yD + f (x)D that contains SL2(D, yD) but not SL2(D, yD + f (x)D).
From now on D will always be a Dedekind domain. We will sometimes use the abbreviations
G = SL2(D)
and
G(q) = SL2(D, q)
for each D-ideal q.
Lemma 1.6. If D has characteristic p > 0 and every unit of D has finite or-der, then D∗ ∪ {0} is a field k. Moreover, k is the biggest algebraic extension of Fp
contained in the field of fractions of D.
Proof. Obviously K, the quotient field of D, contains the prime field Fp. Let
k be the biggest algebraic extension of Fp contained in K.
Let u be a unit of D that has finite order. Then u is a root of unity and hence contained in an algebraic extension of Fp. Thus u ∈ k. Conversely, every element of
k∗ is a root of unity, so it is in D since D is integrally closed, and obviously it is a
unit of D.
Lemma 1.7. If D contains an infinite field, then every subgroup of finite index in SL2(D) has level D.
Proof. The following proof is inspired by the proof of [Se1, Proposition 1, p.491]. If H is a subgroup of finite index in SL2(D), then its core is a normal subgroup
of SL2(D) and still has finite index. So it suffices to prove the lemma for a normal
subgroup N of finite index in SL2(D). We define
ql(N ) := {r ∈ D : T (r) ∈ N }.
This set will be investigated in more detail in Section 4. Here we only need that ql(N ) is a subgroup of (D, +) and that u2ql(N ) ⊆ ql(N ) for every u ∈ D∗. The last
claim can be seen by conjugating with the diagonal matrix with entries u and u−1, as N is normal.
Let k be the infinite field in D. Then obviously k∗ ⊆ D∗. So ql(N ) is stable
under multiplication with x2 and even with x2− y2 for all x, y ∈ k.
If the characteristic of k is different from 2, then it is easy to see that every element of k is of the form x2− y2 with x, y ∈ k. Thus ql(N ) is a subspace of the
k-vector space D. Since the quotient space D/ql(N ) is finite because [D : ql(N )] ≤ [SL2(D) : N ] ≤ ∞,
but k is infinite, we must have ql(N ) = D.
If the characteristic is 2, the squares in k still form an infinite field k2, and we
can apply the same proof using k2-vector spaces.
Proposition 1.8. If D has characteristic 0 or if D∗ is infinite, then every sub-group of finite index in SL2(D) has non-zero level.
Proof. As in the previous proof we can assume that the subgroup N is normal. Again we show that ql(N ) contains a non-zero ideal. The first two cases were already treated in the proof [Se1, Proposition 1, p.491], namely:
If char(D) = 0 and [SL2(D) : N ] = n, then T (nr) = (T (r))n lies in N for every
r ∈ D, so ql(N ) contains the ideal nD.
If char(D) = p > 0 and D∗ contains a unit u of infinite order, then D and ql(N ) are Fp[u2]-modules. So the quotient group D/ql(N ) is also an Fp[u2]-module, and
since it is finite (whereas Fp[u2] is isomorphic to a polynomial ring), it is annulated
by some nonzero element a of Fp[u2]. Thus aD ⊆ ql(N ).
Finally, if char(D) = p > 0 and there are no units of infinite order, then Lemma 1.6 implies D∗ = k∗ where k is an infinite algebraic extension of Fp. So in this case
every subgroup of finite index has level D by Lemma 1.7. We recall from the introduction that if D is of arithmetic type then every con-gruence subgroup of SL2(D) is of finite index. It is known that for some such D
(including the classical case R = Z) the converse does not hold. At this point we record a class of arithmetic Dedekind domains for which the converse does hold.
Let K be a global field. In other words, K is either an algebraic number field, i.e. a finite extension of Q, or K is an algebraic function field of one variable with finite constant field, i.e., K is a finite extension of a rational function field Fq(t).
Let S be a proper subset of all the places of K which contains all archimedean places. Let OS be the set of all elements of K that are integral outside S. Then OS
is an arithmetic Dedekind domain whose prime ideals correspond to the places of K outside S.
Note that we do not assume that S is finite (as in the standard definition of the ring of S-integers of K). If S is infinite, then O∗S is not finitely generated. If S
contains all places of K except one, then OS is a discrete valuation ring. The next
theorem is a minor extention of a famous result of Serre [Se1].
Theorem 1.9. Let K be a global field and OS the ring of S-integers where S
is an infinite set. Then every subgroup of finite index in SL2(OS) has non-zero
level, and every subgroup of non-zero level is a congruence subgroup.
Proof. The first claim follows from Proposition 1.8. The second claim is equiv-alent to showing
SL2(OS, q) = N E2(OS, q)
for every non-zero ideal q of OS. We use a “local” argument to reduce this to a
suitable case where S is finite. Take any g ∈ SL2(OS, q). Then the entries of g,
being elements of K, lie in OS0 for some finite subset S0 of S. We may assume that
|S0| > 1 and that S0 contains at least one non-archimedean place. Let q0 be the ideal
q∩ OS0 of OS0. Then
SL2(OS0, q0) = N E2(OS0, q0)
by [Se1, Th´eor`eme 2 (b), p.498]. Since obviously g ∈ SL2(OS0, q0) and N E2(OS0, q0) ⊆
N E2(OS, q), we have shown SL2(OS, q) ⊆ N E2(OS, q). The converse inclusion
al-ways holds.
We present another class of Dedekind domains that have the same congruence sub-group property.
Theorem 1.10. If D has only finitely many maximal ideals, then every subgroup of finite index in SL2(D) has nonzero-level, and every subgroup (finite index or not)
of non-zero level is a congruence subgroup.
Proof. Let a be the product of the maximal ideals of D. Since the ideals a, a2, a3, . . . are all different, a must be an infinite set. Now every element of the form 1 + a with a ∈ a obviously does not lie in any of the maximal ideals; so it must be a unit. Thus D∗ is infinite. So if H is a subgroup of finite index in SL2(D), it has
non-zero level by Proposition 1.8.
Now let H be a subgroup, not necessarily of finite index, and assume that H has non-zero level, say l(H) = q. As D is semi-local and hence an SR2-ring by [B,
Theorem 3.5, p.239], we can apply Theorem 1.2 with q0 being the zero ideal, and
obtain SL2(D, q) ≤ H.
In the previous two results the reason for the congruence subgroup property was that N E2(D, q) = SL2(D, q) for every ideal q of D. We now construct examples
Lemma 1.11. If N E2(D) 6= SL2(D), then N E2(D, q) 6= SL2(D, q) for every
non-zero ideal q of D.
Proof. Let N E2(D) 6= SL2(D). Then D has infinitely many maximal ideals
by Theorem 1.10. Asssume N E2(D, q) = SL2(D, q). Choosing an ideal q0 with
q+ q0 = D and using Corollary 1.3 we would obtain
N E2(D) = N E2(D, q + q0) = N E2(D, q) N E2(D, q0)
= SL2(D, q) N E2(D, q0) = SL2(D, q + q0) = SL2(D),
a contradiction.
Example 1.12. Let k be an algebraically closed field of characteristic 0. Let D = k[x, y] with y2 = x3+Ax+B such that the cubic polynomial x3+Ax+B ∈ k[x]
has no multiple roots.
Then N E2(D, q) 6= SL2(D, q) for every non-zero ideal q of D. So SL2(D) has
non-congruence subgroups of every level. But every subgroup of finite index is a congruence subgroup for the trivial reason that SL2(D) has no subgroups of finite
index.
To see all this, we use Takahashi’s description [Ta] of the action of GL2(D) on
the Bruhat-Tits tree of GL2(k((t))) where t = xy. Note that since every element of
k∗ is a square, there exists a natural isomorphism P SL2(D) ∼= P GL2(D).
Since the fundamental domain of this action is a connected graph without loops [Ta], by the theory of groups acting on trees [Se2], the group is generated by the stabilizers of the vertices. By [Ta, Theorem 5] these stabilizers are built from sub-groups isomorphic to k, k∗ and P SL2(k). As the groups k and k∗ are infinitely
divisible, they have no subgroups of finite index; and since k is infinite, P SL2(k)
also has no finite index subgroups. Hence a subgroup of finite index in P GL2(D)
would have to contain all stabilizers and so be equal to P GL2(D).
Moreover, the fact that the fundamental domain has a vertex with trivial sta-bilizer and the description of its neighbours [Ta] shows that P GL2(D) is a free
product
P GL2(D) ∼=
∗
`∈P1(k)∆(`)
where ∆(∞) is GL2(k)
∗
B2(k)B2(D) modulo the center. This shows that moduloscalar matrices SL2(D)/N E2(D) is isomorphic to the infinite group
∗
`∈k∆(`).To appreciate this example we point out that even if a Dedekind domain D has no finite quotients, this does not imply that the group SL2(D) has no subgroups of
Example 1.13. Let k = [
n∈N
F25n and D = k[x, y] with y2+ y = x3.
Then D∗ = k∗, and since every element of k∗ is a square we again have P SL2(D) ∼=
P GL2(D). By Takahashi’s results [Ta] again we have a free product
P SL2(D) ∼=
∗
`∈P1(k)∆(`)
with ∆(∞) equal to SL2(k)
∗
B2(k)B2(D) modulo scalar matrices. However, sincethe equation y2+ y = 1 has no solution in k, by [Ta, Theorem 5] this time
∆(1) ∼= L∗/k∗ where L = k(ω) with ω2 + ω = 1.
So L = F4k is the unique quadratic extension of k. Every element of k∗ is a third
power of an element of k∗. Moreover, F8 * L, so F64* L. Thus L contains the 3-rd
roots of unity, but not the 9-th roots of unity. Hence the third powers in L∗ form a subgroup of index 3 that contains k∗. So L∗/k∗ has a subgroup of index 3. Because of the free product there exists a surjective homomorphism from P SL2(D) to ∆(1)
whose kernel is the normal subgroup generated by all other ∆(`). Thus SL2(D)
contains a normal non-congruence subgroup of index 3 and level D.
We mention two types of Dedekind domains for which SL2(D) does not have the
congruence subgroup property.
Every subgroup of finite index in SL2(Z) has non-zero level, but SL2(Z) has
finite index non-congruence subgroups.
Let OS be the ring of S-integers in a global function field K where |S| = 1.
Then SL2(OS) has uncountably many finite index subgroups of level zero [MSch2,
Corollary 3.6] and SL2(OS) has finite index non-congruence subgroups of almost
every level [MSch2, Theorem 5.5].
Remark 1.14. This leaves open the question whether there exist examples SL2(D)
that contain finite index subgroups of level zero but in which every finite index subgroup of non-zero level is a congruence subgroup.
By Lemmas 1.6 and 1.7 and Proposition 1.8 such a D must be an Fq-algebra
with unit group F∗q. One might be tempted to think that the rings OS in a global
function field K with |S| = 1 are the only instances of such rings. This is true for finitely generated Fq-algebras D, because then the Krull dimension of D equals the
transcendence degree of its quotient field.
However, O. Goldman [Go] has constructed Dedekind domains D with D∗ = F∗q such that D/m is a finite field for every maximal ideal m of D, and the quotient field K of D has transcendence degree n over Fq where n can be any given natural
2. Cusp amplitudes
Throughout D denotes a Dedekind domain with quotient field F (6= D). For each x ∈ D we denote the principal D-ideal xD by (x). We write (x, y) = z, if (x) + (y) = (z). All the results of Section 1 apply to SL2(D) which from now on we denote by
G.
We recall the action of G on ˆF = P1(F ). Let g = a b
c d be an element of G. Then g(z) = (az + b)/(cz + d)−1 , z ∈ F, (cz + d) 6= 0 ∞ , c 6= 0, z = −dc−1 ac−1 , c 6= 0, z = ∞ ∞ , c = 0, z = ∞
In particular G acts transitively on ˆF if and only if D is a principal ideal domain. In general the G-orbit in ˆF containing ∞ is
O∞= G(∞) = {a/b : a, b ∈ D, b 6= 0, (a, b) = 1} ∪ {∞}.
Let H be a subgroup of G. For each g ∈ G, let Hg denote the conjugate subgroup
g−1Hg. For each z ∈ ˆF , we denote the stabilizer of z in H by Hz = {h ∈ H : h(z) = z}.
It is clear that
Hg(z)= ((Hg)z)g
−1
, for all g ∈ G and z ∈ ˆF , and that
G∞= B2(D) = {T (α, r) : α ∈ D∗, r ∈ D}.
The following correspondence is obvious. Definition. Let z1, z2 ∈ ˆF . We write
z1 ≡ z2 (mod H) ⇐⇒ z2 = h(z1),
for some h ∈ H.
Lemma 2.1. Let r, s ∈ G. Then
r(∞) ≡ s(∞) (mod H) ⇐⇒ HrG∞ = HsG∞.
Lemma 2.1 provides the following bijection:
between the orbits of the H-action on O∞ and the (H, G∞) double cosets in G. We
refer to the elements of H\O∞ as the H-cusps.
It is clear that the double coset HgG∞ is a union of a set S of (right) H-cosets and
that
S ←→ G∞/G∞∩ Hg ←→ Gz/Hz,
where z = g(∞). From now on let {gλ : λ ∈ Λ} be a complete set of representatives
for the double coset space H\G/G∞. Then {zλ = gλ(∞) : λ ∈ Λ} is a complete
set of representatives for the H-cusps in O∞, by Lemma 2.1. Our next lemma is an
immediate consequence of the above.
Lemma 2.2. Let H have finite index in G. Then, with the above notation, |G : H| =X λ∈Λ |Gzλ : Hzλ| = X λ∈Λ |G∞ : (Hgλ)∞|.
It follows from the above that, if g ∈ G and z = g(∞), then every unipotent matrix in Gz is of the form gT (x)g−1, for some x ∈ D. It is obvious that {x ∈ D : gT (x)g−1 ∈
H} is a subgroup of (the additive group of) D. Now let gi ∈ G, (i = 1, 2). Suppose
now that Hg1G∞ = Hg2G∞. Then g1 = hg2s, for some h ∈ H and s ∈ G∞. It is
clear that
g1T (x)g1−1 ∈ H ⇐⇒ g2T (µx)g2−1 ∈ H,
where µ ∈ D∗ is completely determined by s. We are now able to make the following definition.
Definition. For each g ∈ G, let c(H, g) denote the largest D-ideal q with the property that
gT (q)g−1 ∈ H, for all q ∈ q.
It is clear from the above that c(H, g) = c(H, g0), whenever g(∞) ≡ g0(∞) (mod H). We call c(H, g) the cusp amplitude of the H-cusp containing g(∞).
It is clear that, for each g ∈ G, there exists a unique λ0 ∈ Λ such that c(H, g) = c(H, gλ0).
Definition. We denote a complete set of cusp amplitudes for H by A(H) = {c(H, gλ) : λ ∈ Λ}.
The following is an immediate consequence of the above. Lemma 2.3. For each subgroup H of G,
l(H) = \ g∈G c(H, g) = \ λ∈Λ c(H, gλ). Remarks 2.4.
(i) If N is a normal subgroup then c(N, g1) = c(N, g2), for all g1, g2 ∈ G. In this
case A(N ) involves a single D-ideal.
(ii) If l(H) is non-zero (for example when H is a congruence subgroup), then A(H) involves only finitely many ideals.
(iii) To avoid any possible confusion we emphasize that if D is not a principal ideal domain the cusps (and the cusp amplitudes) of H we consider are only a part of the H-orbits on P1(F ), namely those contained in the G-orbit of ∞.
It is easily verified that cusp amplitudes are invariant under conjugation as given below.
Lemma 2.5. Let k ∈ G. Then
c(Hk, k−1g) = c(H, g), for all g ∈ G.
We require a more detailed description of the unipotent matrices in each Gz, where
z ∈ O∞. If z = g(∞) then gT (−x)g−1 ∈ Gz is of the form
U (a, b; x) = 1 + xab −xa
2
xb2 1 − xab
,
where (a, b) = 1. In this case z = a/b, when b 6= 0, and z = ∞, when b = 0. Note that U (1, 0; x) = T (x) and U (0, 1; x) = S(x).
Before our principal results we record this well-known useful property of Dedekind domains, which follows from the Chinese Remainder Theorem (CRT).
Lemma 2.6. Let p1, · · · , pt be distinct prime D-ideals and let α1, · · · , αt be
non-negative integers, where t ≥ 1. Then there exists d ∈ D such that d ∈ pαi
i \p αi+1
where 1 ≤ i ≤ t.
For the first principal result our approach is more direct than that of Larcher [La]. In particular we avoid his use of the Dirichlet theorem on primes in an arithmetic progression (for Z.)
Theorem 2.7. Let H be a congruence subgroup of G and let qi be a non-zero
D-ideal contained in c(H, gi), where i = 1, 2. Then there exists g0 ∈ G such that
q1+ q2 ⊆ c(H, g0).
Proof. Now H ≥ G(q), for some q 6= {0}. It is clear that, if X, Y ∈ G and X ≡ Y (mod q), then X ∈ H if and only if Y ∈ H.
By Lemma 2.5 it is sufficient to prove the theorem for the case g1 = I2. We may
also assume that q1 + q2 6= qi and (by Corollary 1.4 and Lemma 2.3) that qi ⊇ q,
where i = 1, 2.
Now U (1, 0; q) = T (q) ∈ H, for all q ∈ q1 and U (c, d; q) ∈ H, for all q ∈ q2, for
some c, d ∈ D, with (c, d) = 1. Let q1 + q2 = q0. For our purposes it is sufficient
to find a, b, with (a, b) = 1 and ideals qi, contained in qi, where i = 1, 2, such that
q1+ q2 = q0, with the following properties. For all λ ∈ q1 and all µ ∈ q2, there exist
λ0 ∈ q1 and µ0 ∈ q2 such that:
(a) U (a, b; λ) ≡ T (λ0) (mod q);
(b) U (a, b; µ) ≡ U (c, d; µ0) (mod q).
We may then take g0 to be any matrix of the form
a ∗ b ∗
∈ G.
By definition qi = q0iq0, where i = 1, 2. Then q01+ q02 = D. Let q00i be the “smallest”
divisor of q−10 q all of whose prime divisors also divide q0i, where i = 1, 2. (Clearly q00i ⊆ q0 i.) Then q 00 1+ q 00 2 = D and q= q001q002q0r, say, where r + q001q002 = D.
Since (a, b) = 1 the congruence (a) is equivalent to
bλ ≡ λ0+ a2λ ≡ 0 (mod q). (∗)
Since (c, d) = 1 the congruence (b) is equivalent to:
for some x ∈ D such that µ0 ≡ µx (mod q). A consequence of (∗∗) is that, for all
µ ∈ q2,
(ad − bc)µ ≡ 0 (mod q). We begin by finding a, b ∈ D for which
(i) ad ≡ bc (mod q001), (ii) (a, b) = 1,
(iii) b ∈ q002r.
By Lemma 2.6 we may choose d0 ∈ (c) + q001 and b ∈ q 00 2rsuch that (i) (dd0) + q001 = (bc) + q 00 1, (ii) (d0) + (b) = D.
By applying the CRT to the factors in the prime decomposition of q001 we can find y ∈ D such that
(i) ydd0 ≡ bc (mod q001),
(ii) (y) + q001 = D. (If pα||q00
1, where p is prime with α > 0, and dd0 ≡ bc ≡ 0 (mod q001), take y = 1.)
Now for each prime divisor p of (b) we again use the CRT to find z ∈ D for which (i) z ≡ y (mod pα), where pα||q00
1,
(ii) z ≡ 1 (mod p), when p - q001.
The elements a = zd0 and b (as above) satisfy the requirements.
We can use the CRT to find x ∈ D such that ab ≡ cdx (mod q001) , a2 ≡ c2x (mod q00
1) , b
2 ≡ d2x (mod q00 1).
(Suppose that pα||q00
1. If c /∈ p, take x0 ≡ c−2a2 (mod pα). If d /∈ p, take x0 ≡
d−2b2 (mod pα).) Then, for all µ ∈ q 2 = q
00
2q0r, the congruences (∗∗) are satisfied.
In addition, for all λ ∈ q1 = q001q0, the congruences (∗) are satisfied (with λ0 = −a2λ).
This completes the proof.
Corollary 2.8. Let H be a congruence subgroup of G and let cmax= X q∈A(H) q=X g∈G c(H, g). Then cmax∈ A(H).
Proof. As previously noted A(H) involves only finitely many distinct ideals q1· · · qt,
say. By repeated applications of Theorem 2.7 it follows that there exists g0 ∈ G such that
q1+ · · · + qt⊆ c(H, g0).
But c(H, g0) ∈ A(H) and so c(H, g0) = q1+ · · · + qt = cmax.
We now come to our second principal result. Here our approach is similar to that of Larcher.
Lemma 2.9. Let H be a congruence subgroup of G of level q and let q∞= c(H, I2)
and q0 = c(H, g0), where g0(∞) = 0. Then
q⊇ q0q∞.
Proof. Suppose to the contrary that q0q∞+ q 6= q. We will prove that there exists
an ideal q0 such that
G(q0) ≤ H and q0 ) q, which contradicts Corollary 1.4.
Recall that q∞⊇ q and q0 ⊇ q, by Lemma 2.3. We assume for now that
q∞q0+ q = q0,
where pq0 = q, for some prime ideal p. Then T (x), S(x) ∈ H, for all x ∈ q0. There are two possibilities.
Case 1: p + q0 = D. By Corollary 1.3
G(q0)/G(q) ∼= G/G(p) ∼= SL2(D/p).
Now D/p is a field and so SL2(D/p) is generated by elementary matrices. It follows
that
hT (x), S(x) (x ∈ q0), G(q)i = G(q0), and hence that G(q0) ≤ H.
Case 2: p + q0 = p.
Note that here Lemma 1.1 applies to the quotient group G(q0)/G(q), since (q0)2 ⊆ q.
For all x ∈ q∞ and all y ∈ q0, the element
Now
V ≡ 1 + xy ∗ ∗ 1 − xy
(mod q). It follows that H ∩ G(q0) contains elements
W ≡ 1 + z ∗ ∗ 1 − z
(mod q),
for all z ∈ q0 and also S(z) and T (z) (as mentioned before Case 1). This implies that G(q0) ≤ H, by Lemma 1.1.
If q∞q0+ q 6= q0 we replace q0, say, with q∗, divisible by q0, where q∞q∗+ q = q0,
and repeat the above argument.
An alternative proof of our second principal result (for the case D = Z) can be found in [St].
Theorem 2.10. Let H be a congruence subgroup of G and let cmin = \ q∈A(H) q= \ g∈G c(H, g). Then cmin ∈ A(H).
Proof. Recall from Lemma 2.3 that c(H, g) ⊇ q, for all g ∈ G, where q = l(H). We will assume that
c(H, g) 6= q, for all g ∈ G, and obtain a contradiction.
By Lemma 2.5 it is sufficient to prove the theorem for the case where cmax= c(H, I2).
(See Corollary 2.8.) Let c(H, I2) = q∞ and let c(H, g0) = q0, where g0 ∈ G is any
element for which g0(∞) = 0. ( For example, g0 = T (−1)S(1).) By Corollary 2.8 it
follows that q∞ ⊇ q0. Now q = q∞q∞0 = q0q00, say. Let r = q∞+ q 0
∞. Then q00 ⊇ r,
by Lemma 2.9.
By Lemma 2.6 we can choose z ∈ D for prime ideals p such that (i) z ∈ p, when p|r, p - q00,
(ii) z /∈ p, when p|q0 0.
(Possibly z = 1.) Consider the H-cusp, c(H, gz), where gz ∈ G is given by gz = z ∗ 1 ∗ .
Then gz(∞) = z and c(H, gz) = qz is determined by the matrices
U (z, 1; x) = Q(x),
where x ∈ D. Let q = qzq0z. If t = T (z), then, by Lemma 2.5,
c(H, I2) = c(Ht, t−1) and c(H, gz) = c(Ht, t−1gz).
Now t−1(∞) = ∞ and t−1gz(∞) = 0. Applying Lemma 2.9 therefore to Ht, it
follows that q0z ⊇ r. (By definition l(H) = l(Ht) = q.) Note that by our initial
hypothesis q00, q0z 6= D. There are two possibilities. Case 1: q00+ q0z 6= D.
Choose a prime divisor p of q00 + q0z. Let q = pq0. Then (q0)2 ⊆ q, since p divides q∞ and q0∞. It is clear that q0 is divisible by q∞, q0, qz. It follows that
T (x), S(x), Q(x) ∈ H, for all x ∈ q0.
Now by definition p + (z) = D and so q + (z)q0 = q0. Hence, for all q ∈ q0, there exist q0 ∈ q0 such that zq0 ≡ q (mod q). As in the proof of Lemma 2.9 (Case
2) it follows from Lemma 1.3 that G(q0) ≤ H, which contradicts Corollary 1.4. Case 2: q00+ q0z = D.
Choose a prime p dividing q0z. Since q0z ⊇ r and p + q0
0 = D, it follows that z ∈ p.
As before let q = pq0. Then q0 ⊆ qz, so Q(x) ∈ H and
Q(x) ≡ S(x) (mod q),
for all x ∈ q0. It follows that S(x) ∈ H, for all x ∈ q0+ q0. But q0+ q0 6= q0, since
q0 + q0, which contradicts the maximality of c(H, g0). This completes the proof.
From Theorem 2.10 we obtain in particular the following necessary condition for a subgroup to be a congruence subgroup.
Corollary 2.11. If H is a congruence subgroup of G, then there exists some g ∈ G with l(H) = c(H, g).
group. It is appropriate therefore that we include a detailed account of how they apply to this important special case.
Example 2.12. Cusp amplitudes were first introduced for finite index subgroups of the modular group SL2(Z) (motivated by the theory of modular forms). In this
case the cusp ∞ is P1(Q), since Z is a PID. Let H be a subgroup of finite index in
G containing ±I2. For each g ∈ G, the Z-ideal c(H, g) is non-zero, generated by a
(unique) positive integer n(H, g), say. Since Z∗ = {±1} it follows that in this case |G∞: G∞∩ Hg| = |Gz : Hz| = n(H, g),
where z = g(∞). The double coset space H\G/G∞is finite, represented by elements
g1, · · · , gt, say, of G. Let ni = n(H, gi), where 1 ≤ i ≤ t. We assume that n1 ≤
· · · ≤ nt. The sequence (n1, · · · , nt) is called the cusp-split of H and satisfies the
important cusp-split equation
n1+ · · · + nt= |SL2(Z) : H|.
If n0(≥ 1) is a generator of the (Z-)ideal l(H), then n0 = lcm{n1, · · · , nt}, by Lemma
2.3.
Suppose now that H is a congruence subgroup of index µ in SL2(Z). Then, by
Corollary 2.8 and Theorem 2.10,
n1 = gcd{n1, · · · , nt} and nt = n0.
There are two immediate consequences, namely, µ ≥ n0, (∗)
and
µ divides |SL2(Z) : SL2(Z, n0Z)|. (∗∗)
(See Corollary 1.4.) The inequality (∗) can be used [St] to obtain an upper bound for the number of congruence subgroups of SL2(Z) of bounded index.
Among the index 7 subgroups of SL2(Z) are those with cusp-splits (3, 4) and (2, 5).
(See [St].) They are non-congruence by Corollary 2.8 or Theorem 2.10 (or inequal-ity (∗)). On the other hand these results provide only necessary conditions. For example, it is known [AS] that there exist subgroups with cusp-splits (1, 6) and (1, 1, 7) which, despite being consistent with Corollary 2.8 and Theorem 2.10, are non-congruence since neither satisfies (∗∗). However non-congruence subgroups do exist for which Corollary 2.8, Theorem 2.10 and (∗∗) all hold. It is known [AS] there exists a non-congruence subgroup with cusp-split (8).
As previously emphasised we wish to demonstrate explicitly that our principal re-sults, Corollary 2.8 and Theorem 2.10, do extend to D which are not of arithmetic type.
Example 2.13. Let D = k[t], the polynomial ring over a field k. Then D is of arithmetic type if and only if k is finite. Let p be a prime k[t]-ideal.
We define
G0(p) = hT (α, β) (α ∈ k∗, β ∈ k), SL2(k[t], p)i.
Now T (x) ∈ G0(p) if and only if x ∈ k + p. Let g ∈ G. Then z = g(∞) = a/b,
where a, b ∈ k[t], with (a, b) = 1. If U (a, b; x) ∈ G0(p), then xb2 ≡ 0 (mod p). There
are two possibilities. (i): b /∈ p.
In this case x ∈ p and hence c(G0(p), g) = p.
(ii): b ∈ p.
In this case a /∈ p and U (a, b; x) ≡ T (−xa2) (mod p). Thus U (a, b; x) ∈ G 0(p)
if and only if x ∈ c2k + p, where ac ≡ 1 (mod p). We are reduced to two possible
outcomes.
If dimk(k[t]/p) > 1, then
c(G0(p), g) = p,
for all g ∈ G. We note that G0(p) is an example of a non-normal subgroup all of
whose cusp-amplitudes are equal.
Suppose then that dimk(k[t]/p) = 1 (equivalently, k[t]/p ∼= k). (This always
hap-pens when k is algebraically closed.) It is easily verified that there are precisely two (G0(p), S∞) double cosets in G = SL2(k[t]) and that {T (1), S(1)} is a complete set
of representatives for the G0(p)-cusps, G0(p)\ dk(t)). It follows that
A(G0(p)) = {k[t], p}.
3. Quasi-amplitudes
Definition. As usual let H be a subgroup of G. We define the quasi-amplitude b(H, g) = {b ∈ D : gT (b)g−1 ∈ H} = {b ∈ D : T (b) ∈ Hg}.
Clearly the cusp amplitude c(H, g) is the biggest D-ideal contained in b(H, g). Note that, in contrast to c(H, g), the quasi-amplitude b(H, g) does not just depend on the H-cusp containing g(∞). More precisely it is easily verified that
if g(∞) ≡ g0(∞) (mod H), then b(H, g) = u2b(H, g0),
for some u ∈ D∗. It is clear that if H is normal in G, then b(H, g1) = b(H, g2), for
all g1, g2 ∈ G.
We put
U = {T (r) : r ∈ D}.
It is clear that, for all g ∈ G, the subgroup (Hg)∞ normalizes U , so that U · (Hg)∞
is a subgroup of G. We then have the following one-one correspondences D/b(H, g) ↔ U/U ∩ (Hg)∞ ↔ U · (Hg)∞/(Hg)∞.
Definition. If H is of finite index in G we define m(H, g) = |G∞ : U · (Hg)∞|.
Our next result is a generalization of the cusp split formula for subgroups of SL2(Z).
Theorem 3.1. Let H be a subgroup of finite index in G. If {gλ(∞) : λ ∈ Λ}
is a complete set of representatives for the H-cusps in O∞, then
|G : H| =X
λ∈Λ
m(H, gλ)|D : b(H, gλ)|.
Proof. We note that, for all g ∈ G,
|G∞: (Hg)∞| = |G∞: U · (Hg)∞| · |U · (Hg)∞: (Hg)∞| = m(H, g)|D : b(H, g)|.
The proof follows from Lemma 2.2.
Example 3.2. Let C = F9[t]. Then
SL2(C)/ ± SL2(C, (t)) ∼= P SL2(F9) ∼= A6.
So there exists a (non-normal) congruence subgroup H of SL2(F9[t]) of index 6 and
level (t).
Obviously (t) ⊆ b(H, I2) ⊆ F9[t]. Actually, both inclusions are proper: b(H, I2)
cannot equal F9[t] since H/ ± SL2(C, (t)) ∼= A5 does not contain a subgroup of order
9; on the other hand
so b(H, I2) cannot equal (t). In particular, this shows that b(H, I2) is not an F9
-subspace of F9[t], although, being an additive group, it is clearly an F3-subspace.
All in all we see that
b(H, I2) = F3β + (t)
where β is an element from F∗9.
Let ζ ∈ F9be a primitive 8-th root of unity. Then H cannot contain any matrices
of the form
ζ ∗ 0 ζ−1
since H/ ± SL2(C, (t)) has no elements of order 4. Hence |G∞ : H∞| is divisible by
2. Together with |C : b(H, I2)| = 3 we see that |G∞ : H∞| = 6 and hence that ∞ is
the only cusp of H.
Here we see the problem discussed earlier. The matrix γ = ζ 0
0 ζ−1
represents the same cusp as I2, namely ∞, but
b(H, γ) = ζ2b(H, I2) = F3iβ + (t) 6= b(H, I2)
where i is a primitive 4-th root of unity in F9.
Proposition 3.3.
a) If q > 3, then SL2(Fq[t]) has no proper, normal subgroup of finite index with
only one cusp.
b) If q ≤ 3, then for every positive integer e there are uncountably many normal subgroups of index qe in SL
2(Fq[t]) that have only one cusp.
Proof. a) If N is a normal subgroup of finite index in G with only one cusp, then Theorem 3.1 shows that the Borel group G∞ contains a system of coset
represen-tatives for N in G. Thus G/N is a semidirect product of an elementary abelian p-group where p = char(Fq) and a cyclic group of order m where m|(q − 1). If
m > 1, then G/N contains a normal subgroup of index m, in contradiction to the fact that the minimal index of a normal subgroup in G is |P SL2(Fq)|. Compare
[MSch1, Theorem 6.2]. If m = 1, then G/N contains a normal subgroup of index p, again contradicting the same minimal index.
b) Recall Nagao’s Theorem
SL2(Fq[t]) = SL2(Fq)
∗
B2(Fq)Compare for example [Se2, exercise 2, p.88]. We define V = {T (r) : r ∈ (t)}.
Since for q ∈ {2, 3} all diagonal matrices are in the center of SL2(Fq[t]), this
amal-gamated product shows that the identity on V extends to a surjective group homo-morphism φ : SL2(Fq[t]) → V whose kernel is the normal subgroup generated by
SL2(Fq). Now for each of the uncountably many Fq-subspaces W of codimension qe
in V the inverse image φ−1(W ) is a normal subgroup of SL2(Fq[t]) of index qe, and
Theorem 3.1 shows that it has only one cusp.
The following lemma will be required later on.
Lemma 3.4. Let H be a congruence subgroup of G of level q. Assume that x ∈ b(H, I2). Then for every α ∈ D that is invertible modulo q there exists an
element g ∈ G with α2x ∈ b(H, g).
Proof. Since α is invertible mod q, there exists δ ∈ D such that αδ − 1 =: γ ∈ q. Let g−1 = α 1γ δ ∈ G. Then g−1Hg contains
g−1T (x)g = 1 − αγx α 2x −γ2x 1 + αγx . Moreover, g−1T (x)gT (−α2x) = 1 − αγx α 3γx −γ2x 1 + α2γ2x2 + αγx
is an element of G(q), which is a subgroup of g−1Hg. Hence T (α2x) ∈ g−1Hg. We conclude this section with two examples which show that in general neither Corollary 2.8 nor Theorem 2.10 hold for the quasi-amplitudes of a congruence sub-group. If H is any subgroup of G, we denote the subgroup hH, −I2i by ±H.
Example 3.5. Here we use G to denote the Bianchi group SL2(Od), where Od
is the ring of integers in the imaginary quadratic number field Q(√−d). We will assume that d ≡ 1(mod 3), which ensures that q0 = (3) is a prime ideal in D = Od.
Then D/q0 ∼= F9 and
G/ ± G(q0) ∼= P SL2(F9) ∼= A6.
Hence there exists a (non-normal) subgroup H of G for which H/ ± G(q0) ∼= A5.
Let g ∈ G. Then q0 ⊆ b(H, g) ⊆ D. Suppose that b(H, g) = D. Then T (x) ∈ Hg,
for all x ∈ D, which implies that A5 has a subgroup of order 9. Hence b(H, g) 6= D.
On the other hand, by Theorem 3.1,
|D : b(H, g)| ≤ |G : H| = 6. It follows that |D : b(H, g)| = 3.
Now suppose moreover that d 6= 1; then D∗ = {±1}. With the above notation therefore, there exists λ ∈ Λ such that b(H, g) = ±b(H, gλ). By Theorem 3.1 we
deduce that
|G : H| =X
λ∈Λ
|D : b(H, gλ)| = 6.
Hence there are precisely two (H, G∞)-double cosets in G. Then
b(H, I2) = {0, b, −b} + q0,
for some b ∈ D \ q0. For a representative of the other double coset we fix α ∈ D
which maps onto a primitive root of F9 ∼= D/q0. By Lemma 3.4 there exists g0 ∈ G
for which
b(H, g0) = {0, α2b, −α2b} + q0 6= b(H, I2).
Then g0 can represent the other double coset. We conclude that
{b(H, g) : g ∈ G} = {±{0, b, −b} + q0, ±{0, α2b, −α2b} + q0}.
Clearly this set has neither a minimum nor maximum member under set-theoretic containment.
Example 3.6. For our second example of this type we return to Example 2.13. Let p be a prime ideal in k[t], where k is a field. We recall that
G0(p) = hT (α, β) (α ∈ k∗, β ∈ k), SL2(k[t], p)i.
From Example 2.13 it follows that, for all g ∈ G0,
b(G0(p), g) = a2k + p = b0(a), say,
for some a ∈ k[t]. It is clear that b0(a) = b0(b), if a ≡ b (mod p), and that b0(a) = p
if and only if a ∈ p. We now restrict our attention to those b0(a) for which a /∈ p.
Let K = k[t]/p. For each pair x, y ∈ K∗ we write
Then ρ is an equivalence relation on K∗. Let {aω : ω ∈ Ω} be a subset of k[t] which
maps bijectively onto a complete set of representatives for the ρ-classes. It follows that
{b(G0(p), g) : g ∈ SL2(k[t])} = {bo(aω) : ω ∈ Ω} ∪ {p}.
This set always has a minimal member, i.e. p. However it has a maximal member only when it reduces to {k + p, p}. This happens only when x2 ∈ k∗, for all x ∈ K∗.
It is easy to find examples of K without this property when k is not algebraically closed.
4. Quasi-level
Definition. Let H be a subgroup of G. We define the quasi-level of H as ql(H) = \ g∈G b(H, g). Since b(H, g) = b(Hg, I 2), we see that ql(H) = \ g∈G b(Hg, I2) = b(NH, I2)
where NH is the core of H in G.
However, Example 3.2 shows that in contrast to Lemma 2.3 the intersection over g ∈ G can in general not be replaced by the intersection over a system of representatives of the cusps (even when H is a congruence subgroup).
The second equality shows that our definition of quasi-level coincides with the one we gave in [MSch2]. Actually ql(H) = b(NH, g) for any g ∈ G. But Example
3.5 shows that in contrast to Corollary 2.11 there is in general no g ∈ G for which ql(H) = b(H, g) (even when H is a congruence subgroup).
We summarize the basic properties: Lemma 4.1.
(i) ql(H) = {d ∈ D : T (d) ∈ NH}.
(ii) ql(H) is an additive subgroup of D with the property that d ∈ ql(H), u ∈ D∗ =⇒ u2d ∈ ql(H). (iii) ql(H) ⊇ l(H).
(iv) ql(H) = ql(Hg) = ql(NH) = b(NH, g) for all g ∈ G.
(v) l(H) = l(Hg) = l(N
(vi) l(H) is the largest D-ideal contained in ql(H). (vii) |D : ql(H)| ≤ |SL2(D) : NH|.
For congruence subgroups we can combine Lemmas 3.4 and 4.1 to obtain the fol-lowing extension of Lemma 4.1(ii).
Lemma 4.2. Let H be a congruence subgroup and let α ∈ D be invertible mod-ulo l(H). Then
α2ql(H) ⊆ ql(H).
We will also show that when H is a congruence subgroup the inequality in Lemma 4.1(iii) becomes an equality (in “most” cases). For this purpose we require a number of preliminaries.
Lemma 4.3. Let H be a congruence subgroup of G and let l(H) = q = q1q2,
where q1+ q2 = D. Then
l((H ∩ G(q1)) · G(q2)) = q2.
Proof. Let the required level be q02. Then q02 ⊇ q2. Now
G(q02) ≤ (H ∩ G(q1)) · G(q2)
and so
G(q1q02) = G(q1) ∩ G(q02) ≤ (H ∩ G(q1)) · G(q) ⊆ H.
Hence q ⊇ q1q02 by Corollary 1.4. The result follows.
Lemma 4.4. Let N be a normal congruence subgroup of level q = q1q2, where
q1+ q2 = D. Let N0 = (N ∩ G(q1)) · G(q2) and N = N · G(q2). Then
N /N0 is a central subgroup of G/N0.
Proof. Now G = G(q1) · G(q2), by Corollary 1.3. It follows that
[G, N ] = [G(q1) · G(q2), N · G(q2)] ≤ [G(q1), N ] · G(q2) ≤ N0.
The result follows.
Our next lemma is almost certainly well-known. In the absence of a reference we provide a proof.
Lemma 4.5. Let L be a local ring for which 2 ∈ L∗. Then P SL2(L) has
Proof. Note that, since 2 ∈ L∗, the only involutions in L∗ are ±1. Let g = α β
γ δ
∈ SL2(L).
Then, if g maps into the centre of P SL2(L), it follows that, for all x ∈ L,
gT (x) = λT (x)g and gS(x) = µS(x)g,
where λ2 = µ2 = 1. If γ ∈ L∗, then λ = 1 and so γx = 0. Thus γ /∈ L∗ and similarly
β /∈ L∗. We deduce that α, δ ∈ L∗. From the first of the above equations it follows
that α = λ(α + xγ) and hence that 2xαγ + x2γ2 = 0. The latter equation holds for
x = ±1 and so 4αγ = 0. From the above γ = 0 and similarly β = 0. It then follows
from the above that α = δ, i.e. g = ±I2.
Before coming to our next principal result we make another definition.
Definition. For a subgroup H of G = SL2(D) we define o(H) as the ideal of
D generated by all elements a − d, b, c with a bc d ∈ H. Somewhat unfortunately, o(H) is sometimes called the order of H. Every matrix in H is congruent modulo o(H) to a scalar matrix xI2 for some x ∈ D, and o(H) is the smallest ideal of D
with this property. Obviously
ql(H) ⊆ o(H). Conversely, for each D-ideal q we define
Z(q) = {X ∈ G : X ≡ xI2 (mod q) for some x ∈ D}.
Then H ≤ Z(q) ⇔ o(H) ≤ q. So it is clear that if G(q) ≤ H ≤ Z(q), then ql(H) = l(H) = q.
Theorem 4.6. Let H be a congruence subgroup of SL2(D) such that l(H) = q
satisfies Condition L (from the Introduction). Then ql(H) = l(H), equivalently, the quasi-level is actually an ideal.
Proof. By Lemma 4.1(iv), (v) we may assume that H = NH (i.e. H E G). From
the above it is sufficient to prove that NH ≤ Z(q), and hence that NH ≤ Z(pα),
where p is any prime ideal for which α = ordp(q) > 0.
Let L denote the local ring D/pα and let
π : G −→ SL2(L),
denote the natural map. Now SL2(L) is generated by elementary matrices by [K,
Theorem 1] and so (again by [K, Theorem 1])
π(G(r)) = E2(L, r) = SL2(L, r),
for all D-ideals r, where r is the image of r in L. (In particular π is an epimorphism.) Let q = q0pα. Now suppose that π(N
H ∩ G(q0)) is not central in SL2(L). Then by
[K, Theorem 3] the hypotheses on q and the above ensure that G(pβ) ≤ (NH ∩ G(q0)) · G(pα).
for some β < α which contradicts Lemma 4.3 (with q2 = pα and q1 = q0). It follows
that
π(NH ∩ G(q0)) ≤ {±I2}.
The map π extends to an epimorphism
π : G −→ P SL2(L).
By Lemma 4.4 π(NH) is central in P SL2(L). We now apply Lemma 4.5 to conclude
that
π(NH) ≤ {±I2}.
For normal subgroups we can reformulate Theorem 4.6 as follows.
Corollary 4.7. Let N be a normal congruence subgroup of G. If the level of N satisfies Condition L, then
b(N, g) = c(N, g),
that is, the quasi-amplitudes of N are actually the cusp amplitudes.
Proof. Since N is normal any quasi-amplitude is equal to the quasi-level and
any cusp amplitude is equal to the level.
Remark 4.8. A word of warning is in order here. If ql(H) = l(H) for a (non-normal) congruence subgroup H and H has only one cusp, this does not imply
b(H, g) = c(H, g), not even if l(H) satisfies Condition L. See Example 3.2. Ulti-mately the problem is caused by diagonal matrices that are not central.
McQuillan [Mc, Theorem 1] has proved, for the special case D = Z, that, if N is a normal congruence subgroup of G of level q, then N ≤ Z(q), using a similar approach. We now provide a pair of examples to show that both restrictions in Theorem 4.6 are necessary.
Example 4.9. Our first example [M1, Example 2.3] shows that Theorem 4.6 can fail when q is not prime to 2. Let p be a prime D-ideal for which 2 ∈ p2. We recall from Lemma 1.1 that
G(p2)/G(p4) ∼= {(a, b, c) : a, b, c ∈ p2/p4},
where the latter group is additive. Let Λ = {t2+ p4 : t ∈ p}. We define a subgroup
K, where G(p4) ≤ K ≤ G(p2), by
K/G(p4) = {(a, b, c) : b, c ∈ Λ}.
Since 2p2 ⊆ p4, it is easily verified that K is a subgroup of G, normalized by
S(x), T (x), for all x ∈ D. Since K is also normalized by G(p4), it follows from
Theorem 1.2 that K is normal in G.
Clearly ql(K) = {t2 + q : t ∈ p, q ∈ p4}. Now suppose that l(K) 6= p4. Then
G(p3) ≤ K. Let h be a generator of p3 (mod p4). It follows that, there exists k ∈ D,
such that h ≡ k2 (mod p4). We conclude that l(K) = p4.
Explicit examples that satisfy the requirement 2 ∈ p2 are, among others, D =
Z[ √
−2] with p = (√−2), or to take a local example, D = Z2[
√
2] with p = (√2). More generally, this example actually works for any Dedekind domain of character-istic 2.
Example 4.10. Suppose that q = m1m2, where m1 + m2 = D and |D/mi| =
3 (i = 1, 2). (Consider, for example, D = Z[√−2] with m1 = (1 +
√
−2) and m2 = (1 −
√
−2) or D = F3[t] with m1 = (t) and m2 = (t + 1).) Then, by Corollary
1.3(ii),
G/G(q) ∼= SL2(F3) × SL2(F3).
From the well-known structure of SL2(F3) it follows that there exists a normal
subgroup, N , of G, containing G(q), such that
|G : N | = 9 and |N : G(q)| = 64. Now let
Since 9 = 32, M E G and |G : M | = 3. Obviously, ql(M ) contains 1. If 1 ∈ l(M ), then l(M ) = D, in which case M = G(D) = G. Thus ql(M ) 6= l(M ).
In particular, there exists a normal congruence subgroup of index 3 in SL2(F3[t])
that has level t(t + 1). Remarks 4.11.
a) If the level of a congruence subgroup H is a prime ideal p, then ql(H) = l(H). This follows immediately from the simplicity of the group P SL2(D/p), when
|D/p| > 3. The cases for which |D/p| ≤ 3 can be checked directly.
b) If D is any arithmetic Dedekind domain, the quasi-level of a congruence sub-group is not “ too far from” its level. For a normal congruence subsub-group N the relation between l(N ) and o(N ) is described in [M2, Theorems 3.6, 3.10 and 3.14]. See also the end of Section 3 of [M2].
For example, for a finite index subgroup H of SL2(Z[
√
11]) we obtain 4ql(NH) ⊆ 4o(NH) ⊆ l(NH) ⊆ ql(NH) ⊆ o(NH)
from [M2, Theorem 3.6] since 2 is ramified and 3 is inert in Z[√11]. Actually even 4ql(H) ( l(H) since ql(NH) = o(NH) would mean that ql(H) is an ideal
and hence equal to l(H).
Our final result demonstrates that for a non-congruence subgroup there is in general almost no connection between its quasi-level and level (in contrast with Theorem 4.6). We note that there is no proper normal subgroup of SL2(k[t]) whose quasi-level
is k[t]. (Since k[t] is a Euclidean ring, SL2(k[t]) is generated by T (r), S(r), where
r ∈ k[t].)
Theorem 4.12. Let k be any field and let f ∈ k[t] with deg(f ) ≥ 2. Suppose that f (0) 6= 0 and, further, that f0(0) 6= 0, when deg(f ) = 2. Then there exists a normal non-congruence subgroup N of SL2(k[t]) of level (f ) with the following
properties.
(i) N · SL2(k) = SL2(k[t]).
(ii) l(N ) = (f ).
(iii) ql(N ) has k-codimension 1 in k[t].
Proof. We note that, by hypothesis, t - f . We define the k-subspace Q = (f ) ⊕ kt ⊕ kt2⊕ · · · ⊕ ktd−1
where d = deg(f ). Let N = ∆(Q) be the normal subgroup of SL2(k[t]) generated by
all T (q), where q ∈ Q. Since SL2(k[t]) is generated by all T (r), S(r), where r ∈ k[t],
part (i) follows. In addition
ql(∆(Q)) = Q,
by [M3, Theorem 3.8]. Part (iii) follows. Suppose that l(∆(Q)) 6= (f ). Then (f ) ⊆ (h) ⊆ Q,
for some polynomial divisor h of f , with deg(h) < deg(f ). Then by the definition of Q, h, and hence f , must be divisible by t. Part (ii) follows.
Finally, suppose that N is a congruence subgroup. Since, by hypothesis t is prime to l(N ) = (f ),
t2Q ⊆ Q,
by Lemma 4.2. If deg(f ) > 2, then td∈ Q. If deg(f ) = 2, then t3 ∈ Q. Now for this
case tf ∈ Q and so t2 ∈ Q, by the extra hypothesis. In either case (t) ⊆ Q, which
implies that Q = k[t], a contradiction. The proof is complete. Remarks 4.13.
a) Obviously the group N in Theorem 4.12 shows that Lemmas 3.4 and 4.2 do not hold in general for non-congruence subgroups, even if they are normal. b) The restriction on the degree of f in Theorem 4.12 is necessary. It is
well-known that, if deg(f ) ≤ 1, then every subgroup of SL2(k[t]) of level (f ) is a
congruence subgroup.
c) Several versions of Theorem 4.12 are already known for subgroups of level zero. (See Section 4 of [MSch2].)
5. Level and index
We now make use of quasi-amplitudes to extend the index/level inequality (∗) for finite index subgroups of SL2(Z) (in Example 2.12) to other arithmetic Dedekind
domains.
First, let K be an algebraic number field of degree d over Q. We recall the def-inition of OS from Section 1, where S is a suitable, not necessarily finite, set of
places of K.
Lemma 5.1. Let L be a subgroup of OS such that OS/L has exponent e. Then
L contains the OS-ideal generated by e.
In particular, every finite index subgroup L of OS contains an OS-ideal a such
that
Proof. The first claim is obvious since ea ∈ L for every a ∈ OS.
For the second claim let |OS : L| = n. Then L contains nOS. Denote the ring of
integers of K by OK. If nOK =Q peii is the decomposition of nOK into a product
of prime ideals of OK, then
nOS = Y pi6∈S pei i OS. So |OS/nOS| = Y pi6∈S |OS/piOS|ei = Y pi6∈S |OK/piOK|ei ≤Y|OK/piOK|ei = |OK/nOK| = nd. Theorem 5.2. Let K be an algebraic number field with [K : Q] = d and let OS be
the ring of S-integers of K (with |S| not necessarily finite). If H is a congruence subgroup of SL2(OS), then
|OS/l(H)| ≤ |SL2(OS) : H|d.
Proof. By Corollary 2.11 there exists g ∈ SL2(OS) with l(H) = c(H, g). Let L
be the quasi-amplitude b(H, g). Then |OS : L| ≤ |SL2(OS) : H|, and the Theorem
follows from Lemma 5.1.
Remarks 5.3.
a) Theorem 5.2 also holds for normal non-congruence subgroups of SL2(OS) since
then every cusp amplitude is equal to the level.
b) If |S| > 1 and S contains at least one real or non-archimedean place, or if |S| = ∞, then by [Se1] resp. Theorem 1.9 the inequality in Theorem 5.2 holds for all subgroups of finite index in SL2(OS).
c) Lubotzky [Lu, (1.6) Lemma] has given a version of Theorem 5.2 for more general algebraic groups using different methods.
Next we exhibit some examples for which the inequality in Theorem 5.2 is sharp. Example 5.4. Let K be a number field of degree d over Q. Let D = OK
de-note the ring of integers of K (i.e. the ring of S-integers of K, where S consists precisely of the archimedean places of K). We suppose that the ideal q0 = (2) = 2D
splits into the product of d distinct prime ideals in D. Let G = SL2(D). Then,
using Corollary 1.3,
where for 1 ≤ i ≤ d,
Pi ∼= SL2(F2) ∼= S3,
Let p be any prime ideal dividing q0 and let q0 = pp0. Then, again by Corollary 1.3,
under the first isomorphism above
G(p0)/G(q0) ∼= Pj,
for some j. For each i, let Ni be the normal subgroup of Pi of order 3. There exists
an epimorphism
θ : P1× · · · × Pd−→ S2,
such that N = ker θ contains N1× · · · × Nd but not any Pi. Let M/G(q0) be the
inverse image of N in G/G(q0). If l(M ) 6= q0, then, by Corollary 1.4, M contains
G(p0), for some prime divisor p of q0, which contradicts the above. Hence l(M ) = q0.
We conclude that M is a normal congruence subgroup with |D/l(M )| = |G : M |d= 2d.
By the way, |G : M | = 2 also shows
[D : ql(M )| = |D : b(M, g)| = 2.
Remark 5.5. Actually, for each d there are infinitely many number fields K of degree d that satisfy the condition in Example 5.4. This can be seen as follows: By Dirichlet’s Theorem on primes in arithmetic progressions there are infinitely many odd primes that are congruent to 1 modulo d. Pick two of these, say p1 and
p2 with pi = mid + 1. Then 2mid ≡ 1 mod pi and hence 2m1m2d≡ 1 mod p1p2. So
by the decomposition law in cyclotomic fields the inertia degree of (2) in the p1p2-th
cyclotomic field Q(ζp1p2) divides m1m2d. Since Q(ζp1p2) has degree m1m2d
2
over Q, the decomposition field of (2) in Q(ζp1p2) is an abelian extension of Q whose degree
is divisible by d. Thus it contains a subfield K of degree d over Q with the desired property.
Alternatively we could argue as follows: Gal(Q(ζp1p2)/Q) ∼= Z/m1dZ ⊕ Z/m2dZ
and the Frobenius at 2 generates a cyclic subgroup; so its fixed field is an abelian extension of Q whose degree is divisible by d.
Taking two other primes p1, p2 one obtains a different field K since Q(ζp1p2) and
hence K is unramified outside p1p2.
By an analogous proof for each d there are infinitely many number fields K of de-gree d such that (3) splits completely in OK. For these one can similarly construct
a normal congruence subgroup of index 3 in SL2(OK) with level 3OK. (Compare
