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, I2), 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(NH) = c(NH, g) for all g ∈ G.
(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 triv-ial centre.
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(NH ∩ 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 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.
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
M = hT (1), N i.
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].)