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
|OS/a| ≤ |OS : L|d.
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
peiiOS. 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,
G/G(q0) ∼= P1× · · · × Pd,
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 m1m2d2 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 Example 4.10 for a special case.)
Thus it looks like Theorem 5.2 is optimal. Also, it is not possible to prove a function field analogue of Theorem 5.2 along the same lines, as there cannot be a function field analogue of the inequality in Lemma 5.1. Even in Fq[t] one can construct addi-tive subgroups L of index q such that the biggest ideal a contained in L is the zero ideal or any prescribed nontrivial ideal. (This is actually the key point for many constructions in [MSch2]. See also Theorem 4.12.)
However, somewhat surprisingly, under the condition that the level is prime to certain ideals one can give a relation between the level and the index (if finite) of a congruence subgroup of SL2(D) that is valid for any Dedekind domain D.
Theorem 5.6. Let D be any Dedekind domain and N a normal congruence subgroup of SL2(D). If the level of N satisfies Condition L, then
|D/l(N )| divides |SL2(D) : N |.
Proof. If N is normal, the level is equal to any cusp amplitude. Under Condition L this cusp amplitude is equal to the quasi-amplitude by Corollary 4.7. So the result
follows from Theorem 3.1.
Examples 3.2 and 3.5 show that even for “relatively simple” rings like F9[t] or Z[
√−13] we cannot expect
|D/l(H)| ≤ |SL2(D) : H|
for non-normal congruence subgroups H that satisfy Condition L. But of course we have
Corollary 5.7. Let D be any Dedekind domain and H a congruence subgroup of SL2(D) of index n. If the level of H satisfies Condition L, then
|D/l(H)| divides n!.
Proof. H has the same level as its core in SL2(D), and the index of this core
divides n!.
Corollary 5.7 is stronger than Theorem 5.2 if n is sufficiently small compared to d.
Examples 4.10 and 5.4 show that Condition L cannot simply be dropped in The-orem 5.6 and Corollary 5.7. But for arithmetic Dedekind domains D one can prove somewhat weaker results than Theorem 5.6 and Corollary 5.7 for congruence sub-groups that do not satisfy Condition L. The key is that by results of [M2] one can still controll the relation between l(N ) and ql(N ). See our Remark 4.11 b). We content
ourselves with the function field case. Since the case of characteristic p > 3 is al-ready fully covered by Theorem 5.6, we only have to deal with characteristic 2 and 3.
Theorem 5.8. Let K be a global function field with constant field Fq of and let D = OS where |S| is finite. Let N be a normal congruence subgroup of SL2(D).
a) If char(Fq) = 3, then
|D/l(N )| divides 3r|SL2(D) : N | where r is the number of prime ideals p of D with |D/p| = 3.
b) If char(Fq) = 2, then
|D/l(N )| divides 4s|SL2(D) : N |2 where s is the number of prime ideals p of D with |D/p| = 2.
Proof. We only give the proof for characteristic 2 as they are almost the same.
By [M2, Theorem 3.14] (compare also the end of Section 3 in [M2]) we have s2(o(N ))2 ≤ l(N ) where s is the product of all prime ideals p in D with |D/p| = 2.
Hence |D/l(N )| divides 4s|D/o(N )|2, which divides 4s|D/ql(N )|2 since ql(N ) is an Fq-subspace of o(N ). As N is normal, we have ql(N ) = b(N, g), and the claim
follows from Theorem 3.1.
Remarks 5.9.
a) If K is a number field and |S| is finite, then using [M2, Theorem 3.6] and the end of Section 3 of [M2], for a normal congruence subgroup N of SL2(OS) we obtain in the worst case
|OS/l(N )| divides 12d|SL2(OS) : N | where d = |K : Q|.
b) Theorems 5.6 and 5.8 can be used to improve [MSch2, Proposition 4.5 b)] in the sense that the condition that m is a maximal ideal is not really needed there. This condition was used to show that the constructed groups of level m are non-congruence subgroups. But if deg(m) is big enough, then Theorem 5.6 (resp. Theorem 5.8) guarantees the non-congruence property.
Acknowledgement. We thank Peter V´amos for pointing out the reference [Go] to us.
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