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<a id="X86D5DB887ACB1661" name="X86D5DB887ACB1661"></a>
<div class="ChapSects"><a href="chap13.html#X86D5DB887ACB1661">13 Congruence Subgroups, Cuspidal Cohomology and Hecke Operators</a>
<div class="ContSect"> <a href="chap13.html#X79A1974B7B4987DE">13.1 Eichler-Shimura isomorphism</a>

</div>
<div class="ContSect"> <a href="chap13.html#X7BFA2C91868255D9">13.2 Generators for SL_2( Z) and the cubic tree</a>

</div>
<div class="ContSect"> <a href="chap13.html#X7D1A56967A073A8B">13.3 One-dimensional fundamental domains and
generators for congruence subgroups</a>

</div>
<div class="ContSect"> <a href="chap13.html#X818BFA9A826C0DB3">13.4 Cohomology of congruence subgroups</a>

<div class="ContSSBlock">
 <a href="chap13.html#X7F55F8EA82FE9122">13.4-1 Cohomology with rational coefficients</a>

</div></div>
<div class="ContSect"> <a href="chap13.html#X84D30F1580CD42D1">13.5 Cuspidal cohomology</a>

</div>
<div class="ContSect"> <a href="chap13.html#X80861D3F87C29C43">13.6 Hecke operators on forms of weight 2</a>

</div>
<div class="ContSect"> <a href="chap13.html#X831BB0897B988DA3">13.7 Hecke operators on forms of weight ≥ 2</a>

</div>
<div class="ContSect"> <a href="chap13.html#X84CC51EE8525E0D9">13.8 Reconstructing modular forms from cohomology computations</a>

</div>
<div class="ContSect"> <a href="chap13.html#X8180E53C834301EF">13.9 The Picard group</a>

</div>
<div class="ContSect"> <a href="chap13.html#X858B1B5D8506FE81">13.10 Bianchi groups</a>

</div>
<div class="ContSect"> <a href="chap13.html#X851390E07C3B3BB1">13.11 (Co)homology of Bianchi groups and SL_2(cal O_-d)</a>

</div>
<div class="ContSect"> <a href="chap13.html#X86A6858884B9C05B">13.12 Some other infinite matrix groups</a>

</div>
<div class="ContSect"> <a href="chap13.html#X7EF5D97281EB66DA">13.13 Ideals and finite quotient groups</a>

</div>
<div class="ContSect"> <a href="chap13.html#X7D1F72287F14C5E1">13.14 Congruence subgroups for ideals</a>

</div>
<div class="ContSect"> <a href="chap13.html#X85E912617AFE03F4">13.15 First homology</a>

</div>
</div>

<h3>13 Congruence Subgroups, Cuspidal Cohomology and Hecke Operators</h3>

In this chapter we explain how HAP can be used to make computions about modular forms associated to congruence subgroups Γ of SL_2( Z). Also, in Subsection 10.8 onwards, we demonstrate cohomology computations for the Picard group SL_2( Z[i]), some Bianchi groups PSL_2(cal O_-d) where cal O_d is the ring of integers of Q(sqrt-d) for square free positive integer d, and some other groups of the form SL_m(cal O), GL_m(cal O), PSL_m(cal O), PGL_m(cal O), for m=2,3,4 and certain cal O= Z, cal O_-d.

<a id="X79A1974B7B4987DE" name="X79A1974B7B4987DE"></a>

<h4>13.1 Eichler-Shimura isomorphism</h4>

We begin by recalling the Eichler-Shimura isomorphism <a href="chapBib.html#biBeichler">[Eic57]</a><a href="chapBib.html#biBshimura">[Shi59]</a>

 S_k(\Gamma) \oplus \overline{S_k(\Gamma)} \oplus E_k(\Gamma) \cong_{\sf Hecke} H^1(\Gamma,P_{\mathbb C}(k-2))

which relates the cohomology of groups to the theory of modular forms associated to a finite index subgroup Γ of SL_2( Z). In subsequent sections we explain how to compute with the right-hand side of the isomorphism. But first, for completeness, let us define the terms on the left-hand side.

Let N be a positive integer. A subgroup Γ of SL_2( Z) is said to be a congruence subgroup of level N if it contains the kernel of the canonical homomorphism π_N: SL_2( Z) → SL_2( Z/N Z). So any congruence subgroup is of finite index in SL_2( Z), but the converse is not true.

One congruence subgroup of particular interest is the group Γ_1(N)=ker(π_N), known as the principal congruence subgroup of level N. Another congruence subgroup of particular interest is the group Γ_0(N) of those matrices that project to upper triangular matrices in SL_2( Z/N Z).

A modular form of weight k for a congruence subgroup Γ is a complex valued function on the upper-half plane, f: frakh}={z∈ C : Re(z)>0} → C, satisfying:

<ul>
<li>displaystyle f(fracaz+bcz+d) = (cz+d)^k f(z) for (beginarraylla&b c &d endarray) ∈ Γ,

</li>
<li>f is `holomorphic' on the extended upper-half plane frakh^∗ = frakh ∪ Q ∪ {∞} obtained from the upper-half plane by `adjoining a point at each cusp'.

</li>
</ul>
The collection of all weight k modular forms for Γ form a vector space M_k(Γ) over C.

A modular form f is said to be a cusp form if f(∞)=0. The collection of all weight k cusp forms for Γ form a vector subspace S_k(Γ). There is a decomposition

M_k(\Gamma) \cong S_k(\Gamma) \oplus E_k(\Gamma)

involving a summand E_k(Γ) known as the Eisenstein space. See <a href="chapBib.html#biBstein">[Ste07]</a> for further introductory details on modular forms.

The Eichler-Shimura isomorphism is more than an isomorphism of vector spaces. It is an isomorphism of Hecke modules: both sides admit notions of Hecke operators, and the isomorphism preserves these operators. The bar on the left-hand side of the isomorphism denotes complex conjugation, or anti-holomorphic forms. See <a href="chapBib.html#biBwieser">[Wie78]</a> for a full account of the isomorphism.

On the right-hand side of the isomorphism, the ZΓ-module P_ C(k-2)⊂ C[x,y] denotes the space of homogeneous degree k-2 polynomials with action of Γ given by

\left(\begin{array}{ll}a&b\\ c &d \end{array}\right)\cdot p(x,y) = p(dx-by,-cx+ay)\ .

In particular P_ C(0)= C is the trivial module. Below we shall compute with the integral analogue P_ Z(k-2) ⊂ Z[x,y].

In the following sections we explain how to use the right-hand side of the Eichler-Shimura isomorphism to compute eigenvalues of the Hecke operators restricted to the subspace S_k(Γ) of cusp forms.

<a id="X7BFA2C91868255D9" name="X7BFA2C91868255D9"></a>

<h4>13.2 Generators for SL_2( Z) and the cubic tree</h4>

The matrices S=(beginarrayrr0&-1 1 &0 endarray) and T=(beginarrayrr1&1 0 &1 endarray) generate SL_2( Z) and it is not difficult to devise an algorithm for expressing an arbitrary integer matrix A of determinant 1 as a word in S, T and their inverses. The following illustrates such an algorithm.

<div class="example"><pre>
gap> A:=[[4,9],[7,16]];;
gap> word:=AsWordInSL2Z(A);
[ [ [ 1, 0 ], [ 0, 1 ] ], [ [ 0, 1 ], [ -1, 0 ] ], [ [ 1, -1 ], [ 0, 1 ] ],
 [ [ 0, 1 ], [ -1, 0 ] ], [ [ 1, 1 ], [ 0, 1 ] ], [ [ 0, 1 ], [ -1, 0 ] ],
 [ [ 1, -1 ], [ 0, 1 ] ], [ [ 1, -1 ], [ 0, 1 ] ], [ [ 1, -1 ], [ 0, 1 ] ],
 [ [ 0, 1 ], [ -1, 0 ] ], [ [ 1, 1 ], [ 0, 1 ] ], [ [ 1, 1 ], [ 0, 1 ] ] ]
gap> Product(word);
[ [ 4, 9 ], [ 7, 16 ] ]

</pre></div>

It is convenient to introduce the matrix U=ST = (beginarrayrr0&-1 1 &1 endarray). The matrices S and U also generate SL_2( Z). In fact we have a free presentation SL_2( Z)= ⟨ S,U | S^4=U^6=1, S^2=U^3 ⟩.

The cubic tree cal T is a tree (i.e. a 1-dimensional contractible regular CW-complex) with countably infinitely many edges in which each vertex has degree 3. We can realize the cubic tree cal T by taking the left cosets of cal U=⟨ U⟩ in SL_2( Z) as vertices, and joining cosets xcal U and ycal U by an edge if, and only if, x^-1y ∈ cal U Scal U. Thus the vertex cal U is joined to Scal U, UScal U and U^2Scal U. The vertices of this tree are in one-to-one correspondence with all reduced words in S, U and U^2 that, apart from the identity, end in S.

From our realization of the cubic tree cal T we see that SL_2( Z) acts on cal T in such a way that each vertex is stabilized by a cyclic subgroup conjugate to cal U=⟨ U⟩ and each edge is stabilized by a cyclic subgroup conjugate to cal S =⟨ S ⟩.

In order to store this action of SL_2( Z) on the cubic tree cal T we just need to record the following finite amount of information.

<img src="images/fdsl2.png" align="center" width="350" alt="Information for the cubic tree"/>

<a id="X7D1A56967A073A8B" name="X7D1A56967A073A8B"></a>

<h4>13.3 One-dimensional fundamental domains and
generators for congruence subgroups</h4>

The modular group cal M=PSL_2( Z) is isomorphic, as an abstract group, to the free product Z_2∗ Z_3. By the Kurosh subgroup theorem, any finite index subgroup M ⊂ cal M is isomorphic to the free product of finitely many copies of Z_2s, Z_3s and Zs. A subset underline x ⊂ M is an independent set of subgroup generators if M is the free product of the cyclic subgroups <x > as x runs over underline x. Let us say that a set of elements in SL_2( Z) is projectively independent if it maps injectively onto an independent set of subgroup generators underline x⊂ cal M. The generating set {S,U} for SL_2( Z) given in the preceding section is projectively independent.

We are interested in constructing a set of generators for a given congruence subgroup Γ. If a small generating set for Γ is required then we should aim to construct one which is close to being projectively independent.

It is useful to invoke the following general result which follows from a perturbation result about free ZG-resolutons in <a href="chapBib.html#biBellisharrisskoldberg">[EHS06, Theorem 2]</a> and an old observation of John Milnor that a free ZG-resolution can be realized as the cellular chain complex of a CW-complex if it can be so realized in low dimensions.

Theorem. Let X be a contractible CW-complex on which a group G acts by permuting cells. The cellular chain complex C_∗ X is a ZG-resolution of Z which typically is not free. Let [e^n] denote the orbit of the n-cell e^n under the action. Let G^e^n ≤ G denote the stabilizer subgroup of e^n, in which group elements are not required to stabilize e^n point-wise. Let Y_e^n denote a contractible CW-complex on which G^e^n acts cellularly and freely. Then there exists a contractible CW-complex W on which G acts cellularly and freely, and in which the orbits of n-cells are labelled by [e^p]⊗ [f^q] where p+q=n and [e^p] ranges over the G-orbits of p-cells in X, [f^q] ranges over the G^e^p-orbits of q-cells in Y_e^p.

Let W be as in the theorem. Then the quotient CW-complex B_G=W/G is a classifying space for G. Let T denote a maximal tree in the 1-skeleton B^1_G. Basic geometric group theory tells us that the 1-cells in B^1_G∖ T correspond to a generating set for G.

Suppose we wish to compute a set of generators for a principal congruence subgroup Γ=Γ_1(N). In the above theorem take X=cal T to be the cubic tree, and note that Γ acts freely on cal T and thus that W=cal T. To determine the 1-cells of B_Γ∖ T we need to determine a cellular subspace D_Γ ⊂ cal T whose images under the action of Γ cover cal T and are pairwise either disjoint or identical. The subspace D_Γ will not be a CW-complex as it won't be closed, but it can be chosen to be connected, and hence contractible. We call D_Γ a fundamental region for Γ. We denote by mathring D_Γ the largest CW-subcomplex of D_Γ. The vertices of mathring D_Γ are the same as the vertices of D_Γ. Thus mathring D_Γ is a subtree of the cubic tree with |Γ|/6 vertices. For each vertex v in the tree mathring D_Γ define η(v)=3 - degree(v). Then the number of generators for Γ will be (1/2)∑_v∈ mathring D_Γ η(v).

The following commands determine projectively independent generators for Γ_1(6) and display mathring D_Γ_1(6). The subgroup Γ_1(6) is free on 13 generators.

<div class="example"><pre>
gap> G:=HAP_PrincipalCongruenceSubgroup(6);;
gap> HAP_SL2TreeDisplay(G);

gap> gens:=GeneratorsOfGroup(G);
[ [ [ -83, -18 ], [ 60, 13 ] ], [ [ -77, -18 ], [ 30, 7 ] ],
 [ [ -65, -12 ], [ 168, 31 ] ], [ [ -53, -12 ], [ 84, 19 ] ],
 [ [ -47, -18 ], [ 222, 85 ] ], [ [ -41, -12 ], [ 24, 7 ] ],
 [ [ -35, -6 ], [ 6, 1 ] ], [ [ -11, -18 ], [ 30, 49 ] ],
 [ [ -11, -6 ], [ 24, 13 ] ], [ [ -5, -18 ], [ 12, 43 ] ],
 [ [ -5, -12 ], [ 18, 43 ] ], [ [ -5, -6 ], [ 6, 7 ] ],
 [ [ 1, 0 ], [ -6, 1 ] ] ]

</pre></div>

<img src="images/pctree6.gif" align="center" width="350" alt="Fundamental region in the cubic tree"/>

An alternative but very related approach to computing generators of congruence subgroups of SL_2( Z) is described in <a href="chapBib.html#biBkulkarni">[Kul91]</a>.

The congruence subgroup Γ_0(N) does not act freely on the vertices of cal T, and so one needs to incorporate a generator for the cyclic stabilizer group according to the above theorem. Alternatively, we can replace the cubic tree by a six-fold cover cal T' on whose vertex set Γ_0(N) acts freely. This alternative approach will produce a redundant set of generators. The following commands display mathring D_Γ_0(39) for a fundamental region in cal T'. They also use the corresponding generating set for Γ_0(39), involving 18 generators, to compute the abelianization Γ_0(39)^ab= Z_2 ⊕ Z_3^2 ⊕ Z^9. The abelianization shows that any generating set has at least 11 generators.

<div class="example"><pre>
gap> G:=HAP_CongruenceSubgroupGamma0(39);;
gap> HAP_SL2TreeDisplay(G);
gap> Length(GeneratorsOfGroup(G));
18
gap> AbelianInvariants(G);
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 3 ]

</pre></div>

<img src="images/g0tree39.gif" align="center" width="350" alt="Fundamental region in the cubic tree"/>

Note that to compute D_Γ one only needs to be able to test whether a given matrix lies in Γ or not. Given an inclusion Γ'⊂ Γ of congruence subgroups, it is straightforward to use the trees mathring D_Γ' and mathring D_Γ to compute a system of coset representative for Γ'∖ Γ.

<a id="X818BFA9A826C0DB3" name="X818BFA9A826C0DB3"></a>

<h4>13.4 Cohomology of congruence subgroups</h4>

To compute the cohomology H^n(Γ,A) of a congruence subgroup Γ with coefficients in a ZΓ-module A we need to construct n+1 terms of a free ZΓ-resolution of Z. We can do this by first using perturbation techniques (as described in <a href="chapBib.html#biBbuiellis">[BE14]</a>) to combine the cubic tree with resolutions for the cyclic groups of order 4 and 6 in order to produce a free ZG-resolution R_∗ for G=SL_2( Z). This resolution is also a free ZΓ-resolution with each term of rank

{\rm rank}_{\mathbb Z\Gamma} R_k = |G:\Gamma|\times {\rm rank}_{\mathbb ZG} R_k\ .

For congruence subgroups of lowish index in G this resolution suffices to make computations.

The following commands compute

H^1(\Gamma_0(39),\mathbb Z) = \mathbb Z^9\ .

<div class="example"><pre>
gap> R:=ResolutionSL2Z_alt(2);
Resolution of length 2 in characteristic 0 for SL(2,Integers) .

gap> gamma:=HAP_CongruenceSubgroupGamma0(39);;
gap> S:=ResolutionFiniteSubgroup(R,gamma);
Resolution of length 2 in characteristic 0 for
CongruenceSubgroupGamma0( 39) .

gap> Cohomology(HomToIntegers(S),1);
[ 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

</pre></div>

This computation establishes that the space M_2(Γ_0(39)) of weight 2 modular forms is of dimension 9.

The following commands show that rank_ ZΓ_0(39) R_1 = 112 but that it is possible to apply `Tietze like' simplifications to R_∗ to obtain a free ZΓ_0(39)-resolution T_∗ with rank_ ZΓ_0(39) T_1 = 11. It is more efficient to work with T_∗ when making cohomology computations with coefficients in a module A of large rank.

<div class="example"><pre>
gap> S!.dimension(1);
112
gap> T:=TietzeReducedResolution(S);
Resolution of length 2 in characteristic 0 for CongruenceSubgroupGamma0(
39) .

gap> T!.dimension(1);
11

</pre></div>

The following commands compute

H^1(\Gamma_0(39),P_{\mathbb Z}(8)) = \mathbb Z_3 \oplus \mathbb Z_6
\oplus \mathbb Z_{168} \oplus \mathbb Z^{84}\ ,

H^1(\Gamma_0(39),P_{\mathbb Z}(9)) = \mathbb Z_2 \oplus \mathbb Z_2 .

<div class="example"><pre>
gap> P:=HomogeneousPolynomials(gamma,8);;
gap> c:=Cohomology(HomToIntegralModule(T,P),1);
[ 3, 6, 168, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
gap> Length(c);
87

gap> P:=HomogeneousPolynomials(gamma,9);;
gap> c:=Cohomology(HomToIntegralModule(T,P),1);
[ 2, 2 ]

</pre></div>

This computation establishes that the space M_10(Γ_0(39)) of weight 10 modular forms is of dimension 84, and M_11(Γ_0(39)) is of dimension 0. (There are never any modular forms of odd weight, and so M_k(Γ)=0 for all odd k and any congruence subgroup Γ.)

<a id="X7F55F8EA82FE9122" name="X7F55F8EA82FE9122"></a>

<h5>13.4-1 Cohomology with rational coefficients</h5>

To calculate cohomology H^n(Γ,A) with coefficients in a QΓ-module A it suffices to construct a resolution of Z by non-free ZΓ-modules where Γ acts with finite stabilizer groups on each module in the resolution. Computing over Q is computationally less expensive than computing over Z. The following commands first compute H^1(Γ_0(39), Q) = H_1(Γ_0(39), Q)= Q^9. As a larger example, they then compute H^1(Γ_0(2^13-1), Q) = Q^1365 where Γ_0(2^13-1) has index 8192 in SL_2( Z).

<div class="example"><pre>
gap> K:=ContractibleGcomplex("SL(2,Z)");
Non-free resolution in characteristic 0 for SL(2,Integers) .

gap> gamma:=HAP_CongruenceSubgroupGamma0(39);;
gap> KK:=NonFreeResolutionFiniteSubgroup(K,gamma);
Non-free resolution in characteristic 0 for <matrix group with
18 generators> .

gap> C:=TensorWithRationals(KK);
gap> Homology(C,1);
9

gap> G:=HAP_CongruenceSubgroupGamma0(2^13-1);;
gap> IndexInSL2Z(G);
8192
gap> KK:=NonFreeResolutionFiniteSubgroup(K,G);;
gap> C:=TensorWithRationals(KK);;
gap> Homology(C,1);
1365

</pre></div>

<a id="X84D30F1580CD42D1" name="X84D30F1580CD42D1"></a>

<h4>13.5 Cuspidal cohomology</h4>

To define and compute cuspidal cohomology we consider the action of SL_2( Z) on the upper-half plane frak h given by

\left(\begin{array}{ll}a&b\\ c &d \end{array}\right) z =
\frac{az +b}{cz+d}\ .

A standard 'fundamental domain' for this action is the region

\begin{array}{ll}
D=&\{z\in {\frak h}\ :\ |z| > 1, |{\rm Re}(z)| < \frac{1}{2}\} \\
&
\cup\ \{z\in {\frak h} \ :\ |z| \ge 1, {\rm Re}(z)=-\frac{1}{2}\}\\
& \cup\ \{z \in {\frak h}\ :\ |z|=1, -\frac{1}{2} \le {\rm Re}(z) \le 0\}
\end{array}


illustrated below.

<img src="images/filename-1.png" align="center" width="450" alt="Fundamental domain in the upper-half plane"/>

The action factors through an action of PSL_2( Z) =SL_2( Z)/⟨ (beginarrayrr-1&0 0 &-1 endarray)⟩. The images of D under the action of PSL_2( Z) cover the upper-half plane, and any two images have at most a single point in common. The possible common points are the bottom left-hand corner point which is stabilized by ⟨ U⟩, and the bottom middle point which is stabilized by ⟨ S⟩.

A congruence subgroup Γ has a `fundamental domain' D_Γ equal to a union of finitely many copies of D, one copy for each coset in Γ∖ SL_2( Z). The quotient space X=Γ∖ frak h is not compact, and can be compactified in several ways. We are interested in the Borel-Serre compactification. This is a space X^BS for which there is an inclusion X↪ X^BS and this inclusion is a homotopy equivalence. One defines the boundary ∂ X^BS = X^BS - X and uses the inclusion ∂ X^BS ↪ X^BS ≃ X to define the cuspidal cohomology group, over the ground ring C, as

 H_{cusp}^n(\Gamma,P_{\mathbb C}(k-2)) = \ker (\ H^n(X,P_{\mathbb C}(k-2)) \rightarrow
H^n(\partial X^{BS},P_{\mathbb C}(k-2)) \ ).

Strictly speaking, this is the definition of interior cohomology H_!^n(Γ,P_ C(k-2)) which in general contains the cuspidal cohomology as a subgroup. However, for congruence subgroups of SL_2( Z) there is equality H_!^n(Γ,P_ C(k-2)) = H_cusp^n(Γ,P_ C(k-2)).

Working over C has the advantage of avoiding the technical issue that Γ does not necessarily act freely on frak h since there are points with finite cyclic stabilizer groups in SL_2( Z). But it has the disadvantage of losing information about torsion in cohomology. So HAP confronts the issue by working with a contractible CW-complex tilde X^BS on which Γ acts freely, and Γ-equivariant inclusion ∂ tilde X^BS ↪ tilde X^BS. The definition of cuspidal cohomology that we use, which coincides with the above definition when working over C, is

 H_{cusp}^n(\Gamma,A) = \ker (\ H^n({\rm Hom}_{\, \mathbb Z\Gamma}(C_\ast(\tilde X^{BS}), A)\, ) \rightarrow
H^n(\ {\rm Hom}_{\, \mathbb Z\Gamma}(C_\ast(\tilde \partial X^{BS}), A)\, \ ).

The following data is recorded and, using perturbation theory, is combined with free resolutions for C_4 and C_6 to constuct tilde X^BS.

<img src="images/filename-2.png" align="center" width="450" alt="Borel-Serre compactified fundamental domain in the upper-half plane"/>

The following commands calculate

H^1_{cusp}(\Gamma_0(39),\mathbb Z) = \mathbb Z^6\ .

<div class="example"><pre>
gap> gamma:=HAP_CongruenceSubgroupGamma0(39);;
gap> k:=2;; deg:=1;; c:=CuspidalCohomologyHomomorphism(gamma,deg,k);
[ g1, g2, g3, g4, g5, g6, g7, g8, g9 ] -> [ g1^-1*g3, g1^-1*g3, g1^-1*g3,
 g1^-1*g3, g1^-1*g2, g1^-1*g3, g1^-1*g4, g1^-1*g4, g1^-1*g4 ]
gap> AbelianInvariants(Kernel(c));
[ 0, 0, 0, 0, 0, 0 ]

</pre></div>

From the Eichler-Shimura isomorphism and the already calculated dimension of M_2(Γ_0(39))≅ C^9, we deduce from this cuspidal cohomology that the space S_2(Γ_0(39)) of cuspidal weight 2 forms is of dimension 3, and the Eisenstein space E_2(Γ_0(39))≅ C^3 is of dimension 3.

The following commands show that the space S_4(Γ_0(39)) of cuspidal weight 4 forms is of dimension 12.

<div class="example"><pre>
gap> gamma:=HAP_CongruenceSubgroupGamma0(39);;
gap> k:=4;; deg:=1;; c:=CuspidalCohomologyHomomorphism(gamma,deg,k);;
gap> AbelianInvariants(Kernel(c));
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

</pre></div>

<a id="X80861D3F87C29C43" name="X80861D3F87C29C43"></a>

<h4>13.6 Hecke operators on forms of weight 2</h4>

A congruence subgroup Γ ≤ SL_2( Z) and element g∈ SL_2( Q) determine the subgroup Γ' = Γ ∩ gΓ g^-1 and homomorphisms

 \Gamma\ \hookleftarrow\ \Gamma'\ \ \stackrel{\gamma \mapsto g^{-1}\gamma g}{\longrightarrow}\ \ g^{-1}\Gamma' g\ \hookrightarrow \Gamma\ . 

These homomorphisms give rise to homomorphisms of cohomology groups

H^n(\Gamma,\mathbb Z)\ \ \stackrel{tr}{\leftarrow} \ \ H^n(\Gamma',\mathbb Z) \ \ \stackrel{\alpha}{\leftarrow} \ \ H^n(g^{-1}\Gamma' g,\mathbb Z) \ \ \stackrel{\beta}{\leftarrow} H^n(\Gamma, \mathbb Z) 

with α, β functorial maps, and tr the transfer map. We define the composite T_g=tr ∘ α ∘ β: H^n(Γ, Z) → H^n(Γ, Z) to be the Hecke component determined by g.

For Γ=Γ_0(N), prime integer p coprime to N, and cohomology degree n=1 we define the Hecke operator T_p =T_g where g=(beginarraycc1&00&pendarray). Further details on this description of Hecke operators can be found in <a href="chapBib.html#biBstein">[Ste07, Appendix by P. Gunnells]</a>.

The following commands compute T_2 and T_5 and Γ=Γ_0(39). The commands also compute the eigenvalues of these two Hecke operators. The final command confirms that T_2 and T_5 commute. (It is a fact that T_pT_q=T_qT_p for all p,q.)

<div class="example"><pre>
gap> gamma:=HAP_CongruenceSubgroupGamma0(39);;
gap> p:=2;;k:=2;;T2:=HeckeOperator(gamma,p,k);;
gap> Display(T2);
[ [ -2, -2, 2, 2, 1, 2, 0, 0, 0 ],
 [ -2, 0, 1, 2, -2, 2, 2, 2, -2 ],
 [ -2, -1, 2, 2, -1, 2, 1, 1, -1 ],
 [ -2, -1, 2, 2, 1, 1, 0, 0, 0 ],
 [ -1, 0, 0, 2, -3, 2, 3, 3, -3 ],
 [ 0, 1, 1, 1, -1, 0, 1, 1, -1 ],
 [ -1, 1, 1, -1, 0, 1, 2, -1, 1 ],
 [ -1, -1, 0, 2, -3, 2, 1, 4, -1 ],
 [ 0, 1, 0, -1, -2, 1, 1, 1, 2 ] ]
gap> Eigenvalues(Rationals,T2);
[ 3, 1 ]

gap> p:=5;;k:=2;;h:=HeckeOperator(gamma,p,k);;
gap> Display(T5);
[ [ -1, -1, 3, 4, 0, 0, 1, 1, -1 ],
 [ -5, -1, 5, 4, 0, 0, 3, 3, -3 ],
 [ -2, 0, 4, 4, 1, 0, -1, -1, 1 ],
 [ -2, 0, 3, 2, -3, 2, 4, 4, -4 ],
 [ -4, -2, 4, 4, 3, 0, 1, 1, -1 ],
 [ -6, -4, 5, 6, 1, 2, 2, 2, -2 ],
 [ 1, 5, 0, -4, -3, 2, 5, -1, 1 ],
 [ -2, -2, 2, 4, 0, 0, -2, 4, 2 ],
 [ 1, 3, 0, -4, -4, 2, 2, 2, 4 ] ]
gap> Eigenvalues(Rationals,T5);
[ 6, 2 ]

gap>T2*T5=T5*T2;
true

</pre></div>

<a id="X831BB0897B988DA3" name="X831BB0897B988DA3"></a>

<h4>13.7 Hecke operators on forms of weight ≥ 2</h4>

The above definition of Hecke operator T_p for Γ=Γ_0(N) extends to a Hecke operator T_p: H^1(Γ,P_ Q(k-2) ) → H^1(Γ,P_ Q(k-2) ) for k≥ 2. We work over the rationals since that is a setting of much interest. The following commands compute the matrix of T_2: H^1(Γ,P_ Q(k-2) ) → H^1(Γ,P_ Q(k-2) ) for Γ=SL_2( Z) and k=4;

<div class="example"><pre>
gap> H:=HAP_CongruenceSubgroupGamma0(1);;
gap> h:=HeckeOperator(H,2,12);;Display(h);
[ [ 2049, -7560, 0 ],
 [ 0, -24, 0 ],
 [ 0, 0, -24 ] ]

</pre></div>

<a id="X84CC51EE8525E0D9" name="X84CC51EE8525E0D9"></a>

<h4>13.8 Reconstructing modular forms from cohomology computations</h4>

Given a modular form f: frak h → C associated to a congruence subgroup Γ, and given a compact edge e in the tessellation of frak h (i.e. an edge in the cubic tree cal T) arising from the above fundamental domain for SL_2( Z), we can evaluate

\int_e f(z)\,dz \ .

In this way we obtain a cochain f_1: C_1(cal T) → C in Hom_ ZΓ(C_1(cal T), C) representing a cohomology class c(f) ∈ H^1( Hom_ ZΓ(C_∗(cal T), C) ) = H^1(Γ, C). The correspondence f↦ c(f) underlies the Eichler-Shimura isomorphism. Hecke operators can be used to recover modular forms from cohomology classes.

Let Γ=Γ_0(N). The above defined Hecke operators restrict to operators on cuspidal cohomology. On the left-hand side of the Eichler-Shimura isomorphism Hecke operators restrict to operators T_s: S_2(Γ) → S_2(Γ) for s≥ 1.

Consider the function q=q(z)=e^2π i z which is holomorphic on C. For any modular form f(z) ∈ M_k(Γ) there are numbers a_s such that

f(z) = \sum_{s=0}^\infty a_sq^s 

for all z∈ frak h. The form f is a cusp form if a_0=0.

A non-zero cusp form f∈ S_2(Γ) is a cusp eigenform if it is simultaneously an eigenvector for the Hecke operators T_s for all s =1,2,3,⋯ coprime to the level N. A cusp eigenform is said to be normalized if its coefficient a_1=1. It turns out that if f is normalized then the coefficient a_s is an eigenvalue for T_s (see for instance <a href="chapBib.html#biBstein">[Ste07]</a> for details). It can be shown <a href="chapBib.html#biBatkinlehner">[AL70]</a> that S_2(Γ_0(N)) admits a "basis constructed from eigenforms".

This all implies that, in principle, we can construct an approximation to an explicit basis for the space S_2(Γ_0(N)) of cusp forms by computing eigenvalues for Hecke operators.

Suppose that we would like a basis for S_2(Γ_0(11)). The following commands first show that H^1_cusp(Γ_0(11), Z)= Z⊕ Z from which we deduce that S_2(Γ_0(11)) = C is 1-dimensional and thus admits a basis of eigenforms. Then eigenvalues of Hecke operators are calculated to establish that the modular form

f = q -2q^2 -q^3 +2q^4 +q^5 +2q^6 -2q^7 + -2q^9 -2q^{10} + \cdots 

constitutes a basis for S_2(Γ_0(11)).

<div class="example"><pre>
gap> gamma:=HAP_CongruenceSubgroupGamma0(11);;
gap> AbelianInvariants(Kernel(CuspidalCohomologyHomomorphism(gamma,1,2)));
[ 0, 0 ]

gap> T1:=HeckeOperator(gamma,1,2);; Display(T1);
[ [ 1, 0, 0 ],
 [ 0, 1, 0 ],
 [ 0, 0, 1 ] ]
gap> T2:=HeckeOperator(gamma,2,2);; Display(T2);
[ [ 3, -4, 4 ],
 [ 0, -2, 0 ],
 [ 0, 0, -2 ] ]
gap> T3:=HeckeOperator(gamma,3,2);; Display(T3);
[ [ 4, -4, 4 ],
 [ 0, -1, 0 ],
 [ 0, 0, -1 ] ]
gap> T5:=HeckeOperator(gamma,5,2);; Display(T5);
[ [ 6, -4, 4 ],
 [ 0, 1, 0 ],
 [ 0, 0, 1 ] ]
gap> T7:=HeckeOperator(gamma,7,2);; Display(T7);
[ [ 8, -8, 8 ],
 [ 0, -2, 0 ],
 [ 0, 0, -2 ] ]

</pre></div>

For a normalized eigenform f=1 + ∑_s=2^∞ a_sq^s the coefficients a_s with s a composite integer can be expressed in terms of the coefficients a_p for prime p. If r,s are coprime then T_rs =T_rT_s. If p is a prime that is not a divisor of the level N of Γ then a_p^m =a_p^m-1}a_p - p a_p^m-2}. If the prime p divides N then a_p^m = (a_p)^m. It thus suffices to compute the coefficients a_p for prime integers p only.

The following commands establish that S_12(SL_2( Z)) has a basis consisting of one cusp eigenform

q - 24q^2 + 252q^3 - 1472q^4 + 4830q^5 - 6048q^6 - 16744q^7 + 84480q^8 - 113643q^9

- 115920q^10 + 534612q^11 - 370944q^12 - 577738q^13 + 401856q^14 + 1217160q^15 + 987136q^16

- 6905934q^17 + 2727432q^18 + 10661420q^19 + ...

<div class="example"><pre>
gap> R:=ResolutionSL2Z_alt(2);;
gap> G:=R!.group;;
gap> P:=HomogeneousPolynomials(G,14);
MappingByFunction( SL(2,Integers), <matrix group with
2 generators>, function( x ) ... end )
gap> Cohomology(HomToIntegralModule(R,P),1);
[ 2, 2, 156, 0, 0, 0 ]
gap> #Thus the space S_12 of cusp forms is of dimension 1

gap> G:=HAP_CongruenceSubgroupGamma0(1);;
gap> for p in [2,3,5,7,11,13,17,19] do
> T:=HeckeOperator(G,p,12);;
> Print("eigenvalues= ",Eigenvalues(Rationals,T), " and eigenvectors = ", Eigenvectors(Rationals,T)," for p= ",p,"\n");
> od;
eigenvalues= [ 2049, -24 ] and eigenvectors = [ [ 1, -2520/691, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] for p= 2
eigenvalues= [ 177148, 252 ] and eigenvectors = [ [ 1, -2520/691, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] for p= 3
eigenvalues= [ 48828126, 4830 ] and eigenvectors = [ [ 1, -2520/691, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] for p= 5
eigenvalues= [ 1977326744, -16744 ] and eigenvectors = [ [ 1, -2520/691, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] for p= 7
eigenvalues= [ 285311670612, 534612 ] and eigenvectors = [ [ 1, -2520/691, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] for p= 11
eigenvalues= [ 1792160394038, -577738 ] and eigenvectors = [ [ 1, -2520/691, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] for p= 13
eigenvalues= [ 34271896307634, -6905934 ] and eigenvectors = [ [ 1, -2520/691, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] for p= 17
eigenvalues= [ 116490258898220, 10661420 ] and eigenvectors = [ [ 1, -2520/691, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] for p= 19

</pre></div>

<a id="X8180E53C834301EF" name="X8180E53C834301EF"></a>

<h4>13.9 The Picard group</h4>

Let us now consider the Picard group G=SL_2( Z[ i]) and its action on upper-half space

{\frak h}^3 =\{(z,t) \in \mathbb C\times \mathbb R\ |\ t > 0\} \ . 

To describe the action we introduce the symbol j satisfying j^2=-1, ij=-ji and write z+tj instead of (z,t). The action is given by

\left(\begin{array}{ll}a&b\\ c &d \end{array}\right)\cdot (z+tj) \ = \ \left(a(z+tj)+b\right)\left(c(z+tj)+d\right)^{-1}\ .

Alternatively, and more explicitly, the action is given by

\left(\begin{array}{ll}a&b\\ c &d \end{array}\right)\cdot (z+tj) \ = \
\frac{(az+b)\overline{(cz+d) } + a\overline c t^2}{|cz +d|^2 + |c|^2t^2} \ +\
\frac{t}{|cz+d|^2+|c|^2t^2}\, j
 \ .

A standard 'fundamental domain' D for this action is the following region (with some of the boundary points removed).


\{z+tj\in {\frak h}^3\ |\ 0 \le |{\rm Re}(z)| \le \frac{1}{2}, 0\le {\rm Im}(z) \le \frac{1}{2}, z\overline z +t^2 \ge 1\}


<img src="images/picarddomain.png" align="center" width="350" alt="Fundamental domain for the Picard group"/>

The four bottom vertices of D are a = -frac12 +frac12i +fracsqrt2}2j, b = -frac12 +fracsqrt3}2j, c = frac12 +fracsqrt3}2j, d = frac12 +frac12i +fracsqrt2}2j.

The upper-half space frak h^3 can be retracted onto a 2-dimensional subspace cal T ⊂ frak h^3. The space cal T is a contractible 2-dimensional regular CW-complex, and the action of the Picard group G restricts to a cellular action of G on cal T.

Using perturbation techniques, the 2-complex cal T can be combined with free resolutions for the cell stabilizer groups to contruct a regular CW-complex X on which the Picard group G acts freely. The following commands compute the first few terms of the free ZG-resolution R_∗ =C_∗ X. Then R_∗ is used to compute

H^1(G,\mathbb Z) =0\ ,

H^2(G,\mathbb Z) =\mathbb Z_2\oplus \mathbb Z_2\ ,

H^3(G,\mathbb Z) =\mathbb Z_6\ ,

H^4(G,\mathbb Z) =\mathbb Z_4\oplus \mathbb Z_{24}\ ,

and compute a free presentation for G involving four generators and seven relators.

<div class="example"><pre>
gap> K:=ContractibleGcomplex("SL(2,O-1)");;
gap> R:=FreeGResolution(K,5);;
gap> Cohomology(HomToIntegers(R),1);
[ ]
gap> Cohomology(HomToIntegers(R),2);
[ 2, 2 ]
gap> Cohomology(HomToIntegers(R),3);
[ 6 ]
gap> Cohomology(HomToIntegers(R),4);
[ 4, 24 ]
gap> P:=PresentationOfResolution(R);
rec( freeGroup := <free group on the generators [ f1, f2, f3, f4 ]>,
 gens := [ 184, 185, 186, 187 ],
 relators := [ f1^2*f2^-1*f1^-1*f2^-1, f1*f2*f1*f2^-2,
 f3*f2^2*f1*(f2*f1^-1)^2*f3^-1*f1^2*f2^-2,
 f1*(f2*f1^-1)^2*f3^-1*f1^2*f2^-1*f3^-1,
 f4*f2*f1*(f2*f1^-1)^2*f4^-1*f1*f2^-1, f1*f4^-1*f1^-2*f4^-1,
 f3*f2*f1*(f2*f1^-1)^2*f4^-1*f1*f2^-1*f3^-1*f4*f2 ] )

</pre></div>

We can also compute the cohomology of G=SL_2( Z[i]) with coefficients in a module such as the module P_ Z[i](k) of degree k homogeneous polynomials with coefficients in Z[i] and with the action described above. For instance, the following commands compute

H^1(G,P_{\mathbb Z[i]}(24)) = (\mathbb Z_2)^4 \oplus \mathbb Z_4
\oplus \mathbb Z_8 \oplus \mathbb Z_{40} \oplus \mathbb Z_{80}\, ,

H^2(G,P_{\mathbb Z[i]}(24)) = (\mathbb Z_2)^{24} \oplus \mathbb Z_{520030}\oplus \mathbb Z_{1040060} \oplus \mathbb Z^2\, ,

H^3(G,P_{\mathbb Z[i]}(24)) = (\mathbb Z_2)^{22} \oplus \mathbb Z_{4}\oplus (\mathbb Z_{12})^2 \, .

<div class="example"><pre>
gap> G:=R!.group;;
gap> M:=HomogeneousPolynomials(G,24);;
gap> C:=HomToIntegralModule(R,M);;
gap> Cohomology(C,1);
[ 2, 2, 2, 2, 4, 8, 40, 80 ]
gap> Cohomology(C,2);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
 520030, 1040060, 0, 0 ]
gap> Cohomology(C,3);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 12, 12
]

</pre></div>

<a id="X858B1B5D8506FE81" name="X858B1B5D8506FE81"></a>

<h4>13.10 Bianchi groups</h4>

The Bianchi groups are the groups G=PSL_2(cal O_-d) where d is a square free positive integer and cal O_-d is the ring of integers of the imaginary quadratic field Q(sqrt-d). More explicitly,

{\cal O}_{-d} = \mathbb Z\left[\sqrt{-d}\right]~~~~~~~~ {\rm if~} d \equiv 1,2 {\rm ~mod~} 4\, ,

{\cal O}_{-d} = \mathbb Z\left[\frac{1+\sqrt{-d}}{2}\right]~~~~~ {\rm if~} d \equiv 3 {\rm ~mod~} 4\, .

These groups act on upper-half space frak h^3 in the same way as the Picard group. Upper-half space can be tessellated by a 'fundamental domain' for this action. Moreover, as with the Picard group, this tessellation contains a 2-dimensional cellular subspace cal T⊂ frak h^3 where cal T is a contractible CW-complex on which G acts cellularly. It should be mentioned that the fundamental domain and the contractible 2-complex cal T are not uniquely determined by G. Various algorithms exist for computing cal T and its cell stabilizers. One algorithm due to Swan <a href="chapBib.html#biBswan">[Swa71a]</a> has been implemented by Alexander Rahm <a href="chapBib.html#biBrahmthesis">[Rah10]</a> and the output for various values of d are stored in HAP. Another approach is to use Voronoi's theory of perfect forms. This approach has been implemented by Sebastian Schoennenbeck <a href="chapBib.html#biBschoennenbeck">[BCNS15]</a> and, again, its output for various values of d are stored in HAP. The following commands combine data from Schoennenbeck's algorithm with free resolutions for cell stabiliers to compute

H^1(PSL_2({\cal O}_{-6}),P_{{\cal O}_{-6}}(24)) =
(\mathbb Z_2)^4
\oplus \mathbb Z_{12}
\oplus \mathbb Z_{24}
\oplus \mathbb Z_{9240}
\oplus \mathbb Z_{55440}
\oplus \mathbb Z^4\,,


H^2(PSL_2({\cal O}_{-6}),P_{{\cal O}_{-6}}(24)) =
\begin{array}{l}
(\mathbb Z_2)^{26}
\oplus \mathbb (Z_{6})^8
\oplus \mathbb (Z_{12})^{9}
\oplus \mathbb Z_{24}
\oplus (\mathbb Z_{120})^2
\oplus (\mathbb Z_{840})^3\\
\oplus \mathbb Z_{2520}
\oplus (\mathbb Z_{27720})^2
\oplus (\mathbb Z_{24227280})^2
\oplus (\mathbb Z_{411863760})^2\\
\oplus \mathbb Z_{2454438243748928651877425142836664498129840}\\
\oplus \mathbb Z_{14726629462493571911264550857019986988779040}\\
\oplus \mathbb Z^4\end{array}\ ,


H^3(PSL_2({\cal O}_{-6}),P_{{\cal O}_{-6}}(24)) =
(\mathbb Z_2)^{23}
\oplus \mathbb Z_{4}
\oplus (\mathbb Z_{12})^2\ .


Note that the action of SL_2(cal O_-d) on P_{cal O_-d}(k) induces an action of PSL_2(cal O_-d) provided k is even.

<div class="example"><pre>
gap> R:=ResolutionPSL2QuadraticIntegers(-6,4);
Resolution of length 4 in characteristic 0 for PSL(2,O-6) .
No contracting homotopy available.

gap> G:=R!.group;;
gap> M:=HomogeneousPolynomials(G,24);;
gap> C:=HomToIntegralModule(R,M);;
gap> Cohomology(C,1);
[ 2, 2, 2, 2, 12, 24, 9240, 55440, 0, 0, 0, 0 ]
gap> Cohomology(C,2);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
 2, 6, 6, 6, 6, 6, 6, 6, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 24, 120, 120,
 840, 840, 840, 2520, 27720, 27720, 24227280, 24227280, 411863760, 411863760,
 2454438243748928651877425142836664498129840,
 14726629462493571911264550857019986988779040, 0, 0, 0, 0 ]
gap> Cohomology(C,3);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 12,
 12 ]

</pre></div>

We can also consider the coefficient module

 P_{{\cal O}_{-d}}(k,\ell) = P_{{\cal O}_{-d}}(k) \otimes_{{\cal O}_{-d}} \overline{P_{{\cal O}_{-d}}(\ell)} 

where the bar denotes a twist in the action obtained from complex conjugation. For an action of the projective linear group we must insist that k+ℓ is even. The following commands compute

H^2(PSL_2({\cal O}_{-11}),P_{{\cal O}_{-11}}(5,5)) =
(\mathbb Z_2)^8 \oplus \mathbb Z_{60} \oplus (\mathbb Z_{660})^3 \oplus \mathbb Z^6\,, 

a computation which was first made, along with many other cohomology computationsfor Bianchi groups, by Mehmet Haluk Sengun <a href="chapBib.html#biBsengun">[Sen11]</a>.

<div class="example"><pre>
gap> R:=ResolutionPSL2QuadraticIntegers(-11,3);;
gap> M:=HomogeneousPolynomials(R!.group,5,5);;
gap> C:=HomToIntegralModule(R,M);;
gap> Cohomology(C,2);
[ 2, 2, 2, 2, 2, 2, 2, 2, 60, 660, 660, 660, 0, 0, 0, 0, 0, 0 ]

</pre></div>

The function <code class="code">ResolutionPSL2QuadraticIntegers(-d,n)</code> relies on a limited data base produced by the algorithms implemented by Schoennenbeck and Rahm. The function also covers some cases covered by entering a sring "-d+I" as first variable. These cases correspond to projective special groups of module automorphisms of lattices of rank 2 over the integers of the imaginary quadratic number field Q(sqrt-d) with non-trivial Steinitz-class. In the case of a larger class group there are cases labelled "-d+I2",...,"-d+Ik" and the Ij together with O-d form a system of representatives of elements of the class group modulo squares and Galois action. For instance, the following commands compute

H_2(PSL({\cal O}_{-21+I2}),\mathbb Z) = \mathbb Z_2\oplus \mathbb Z^6\, .

<div class="example"><pre>
gap> R:=ResolutionPSL2QuadraticIntegers("-21+I2",3);
Resolution of length 3 in characteristic 0 for PSL(2,O-21+I2)) .
No contracting homotopy available.

gap> Homology(TensorWithIntegers(R),2);
[ 2, 0, 0, 0, 0, 0, 0 ]

</pre></div>

<a id="X851390E07C3B3BB1" name="X851390E07C3B3BB1"></a>

<h4>13.11 (Co)homology of Bianchi groups and SL_2(cal O_-d)</h4>

The (co)homology of Bianchi groups has been studied in papers such as <a href="chapBib.html#biBSchwermer">[SV83]</a> <a href="chapBib.html#biBVogtmann">[Vog85]</a> <a href="chapBib.html#biBBerkove00">[Ber00]</a> <a href="chapBib.html#biBBerkove06">[Ber06]</a> <a href="chapBib.html#biBRahm11">[RF13]</a> <a href="chapBib.html#biBRahm13">[Rah13b]</a> <a href="chapBib.html#biBRahm13a">[Rah13a]</a> <a href="chapBib.html#biBRahm20">[BLR20]</a>. Calculations in these papers can often be verified by computer. For instance, the calculation

H_q(PSL_2({\cal O}_{-15}),\mathbb Z) =
\left\{\begin{array}{ll}
\mathbb Z^2 \oplus \mathbb Z_6 & q=1,\\
\mathbb Z \oplus \mathbb Z_6 & q=2,\\
\mathbb Z_6 & q\ge 3\\
\end{array}\right.


obtained in <a href="chapBib.html#biBRahm11">[RF13]</a> can be verified as follows, once we note that Bianchi groups have virtual cohomological dimension 2 and, if all stabilizer groups are periodic with period dividing m, then the homology has period dividing m in degree ≥ 3.

<div class="example"><pre>
gap> K:=ContractibleGcomplex("SL(2,O-15)");;
gap> PK:=QuotientOfContractibleGcomplex(K,Group(-One(K!.group)));;
gap> for n in [0..2] do
> for k in [1..K!.dimension(n)] do
> Print( CohomologicalPeriod(K!.stabilizer(n,k))," ");
> od;od;
2 2 2 2 2 2 2 2 2 2 2 2 2 2
gap> R:=FreeGResolution(PK,5);;
gap> for n in [0..4] do
> Print("H_",n," = ", Homology(TensorWithIntegers(R),n),"\n");
> od;
H_0 = [ 0 ]
H_1 = [ 6, 0, 0 ]
H_2 = [ 6, 0 ]
H_3 = [ 6 ]
H_4 = [ 6 ]

</pre></div>

All finite subgroups of SL_2(cal O_-d) are periodic. Thus the above example can be adapted from PSL to SL for any square=free d≥ 1. For example, the calculation

H^q(SL_2({\cal O}_{-2}),\mathbb Z) =
 \left\{\begin{array}{ll}
 \mathbb Z & q=1,\\
 \mathbb Z_6 & q=2 {\rm \ mod\ } 4,\\
\mathbb Z_2 \oplus \mathbb Z_{12} & q= 3 {\rm \ mod\ } 4\\
\mathbb Z_2 \oplus \mathbb Z_{24} & q= 0 {\rm \ mod\ } 4 (q>0)\\
\mathbb Z_{12} & q= 1 {\rm \ mod\ } 4 (q > 1)\\
\end{array}\right.


obtained in <a href="chapBib.html#biBSchwermer">[SV83]</a> can be verified as follows.

<div class="example"><pre>
gap> K:=ContractibleGcomplex("SL(2,O-2)");;
gap> for n in [0..2] do
> for k in [1..K!.dimension(n)] do
> Print(CohomologicalPeriod(K!.stabilizer(n,k))," ");
> od;od;
2 4 2 4 2 2 2 2 2 2 2 2
gap> R:=FreeGResolution(K,11);;
gap> for n in [0..10] do
> Print("H^",n," = ", Cohomology(HomToIntegers(R),n),"\n");
> od;
H^0 = [ 0 ]
H^1 = [ 0 ]
H^2 = [ 6 ]
H^3 = [ 2, 12 ]
H^4 = [ 2, 24 ]
H^5 = [ 12 ]
H^6 = [ 6 ]
H^7 = [ 2, 12 ]
H^8 = [ 2, 24 ]
H^9 = [ 12 ]
H^10 = [ 6 ]

</pre></div>

A quotient of a periodic group by a central subgroup of order 2 need not be periodic. For this reason the (co)homology of PSL can be a bit more tricky than SL. For example, the calculation

H^q(PSL_2({\cal O}_{-13}),\mathbb Z) =
 \left\{\begin{array}{ll}
 \mathbb Z^3 \oplus (\mathbb Z_2)^2 & q=1,\\
 \mathbb Z^2 \oplus \mathbb Z_4 \oplus (\mathbb Z_3)^2 \oplus \mathbb Z_2 & q=2,\\
 (\mathbb Z_2)^q \oplus (\mathbb Z_{3})^2 & q= 3 {\rm \ mod\ } 4\\
 (\mathbb Z_2)^q & q= 0 {\rm \ mod\ } 4 (q>0)\\
 (\mathbb Z_2)^q & q= 1 {\rm \ mod\ } 4 (q>1)\\
 (\mathbb Z_2)^q \oplus (\mathbb Z_{3})^2 & q= 2 {\rm \ mod\ } 4 (q > 2)\\
\end{array}\right.


was obtained in <a href="chapBib.html#biBRahm11">[RF13]</a>. The following commands verify the calculation in the first 34 degrees, but for a proof valid for all degrees one needs to analyse the computation to spot that there is a certain "periodicity of period 2" in the computations for q≥ 3. This analysis is done in <a href="chapBib.html#biBRahm11">[RF13]</a>.

<div class="example"><pre>
gap> K:=ContractibleGcomplex("SL(2,O-13)");;
gap> PK:=QuotientOfContractibleGcomplex(K,Group(-One(K!.group)));;
gap> for n in [0..2] do
> for k in [1..PK!.dimension(n)] do
> S:=SmallGroup(IdGroup(PK!.stabilizer( n, k )));
> Print( [n,k]," is periodic ",IsPeriodic(S),"\n ");
> od;od;
[ 0, 1 ] is periodic true
[ 0, 2 ] is periodic true
[ 0, 3 ] is periodic true
[ 0, 4 ] is periodic true
[ 0, 5 ] is periodic true
[ 0, 6 ] is periodic true
[ 0, 7 ] is periodic false
[ 0, 8 ] is periodic false
[ 1, 1 ] is periodic true
[ 1, 2 ] is periodic true
[ 1, 3 ] is periodic true
[ 1, 4 ] is periodic true
[ 1, 5 ] is periodic true
[ 1, 6 ] is periodic true
[ 1, 7 ] is periodic true
[ 1, 8 ] is periodic true
[ 1, 9 ] is periodic true
[ 1, 10 ] is periodic true
[ 1, 11 ] is periodic true
[ 1, 12 ] is periodic true
[ 1, 13 ] is periodic true
[ 2, 1 ] is periodic true
[ 2, 2 ] is periodic true
[ 2, 3 ] is periodic true
[ 2, 4 ] is periodic true
[ 2, 5 ] is periodic true
[ 2, 6 ] is periodic true
[ 2, 7 ] is periodic true
[ 2, 8 ] is periodic true
[ 2, 9 ] is periodic true
[ 2, 10 ] is periodic true
[ 2, 11 ] is periodic true

gap> R:=ResolutionPSL2QuadraticIntegers(-13,35);;
gap> for n in [0..34] do
> Print("H_",n," = ", Homology(TensorWithIntegers(R),n),"\n");
> od;
H_0 = [ 0 ]
H_1 = [ 2, 2, 0, 0, 0 ]
H_2 = [ 6, 12, 0, 0 ]
H_3 = [ 2, 6, 6 ]
H_4 = [ 2, 2, 2, 2 ]
H_5 = [ 2, 2, 2, 2, 2 ]
H_6 = [ 2, 2, 2, 2, 6, 6 ]
H_7 = [ 2, 2, 2, 2, 2, 6, 6 ]
H_8 = [ 2, 2, 2, 2, 2, 2, 2, 2 ]
H_9 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
H_10 = [ 2, 2, 2, 2, 2, 2, 2, 2, 6, 6 ]
H_11 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6 ]
H_12 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
H_13 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
H_14 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6 ]
H_15 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6 ]
H_16 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
H_17 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
H_18 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6 ]
H_19 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6 ]
H_20 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
H_21 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
H_22 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6 ]
H_23 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6 ]
H_24 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
H_25 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
H_26 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6 ]
H_27 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6 ]
H_28 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
H_29 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
H_30 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6 ]
H_31 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6 ]
H_32 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
H_33 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
H_34 = [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6 ]

</pre></div>

The Lyndon-Hochschild-Serre spectral sequence H_p(G/N,H_q(N,A)) ⇒ H_p+q(G,A) for the groups G=SL_2(mathcal O_-d) and N≅ C_2 the central subgroup with G/N≅ PSL_2(mathcal O_-d), and the trivial module A = Z_ℓ, implies that for primes ℓ>2 we have a natural isomorphism H_n(PSL_2(mathcal O_-d), Z_ℓ) ≅ H_n(SL_2(mathcal O_-d), Z_ℓ). It follows that we have an isomorphism of ℓ-primary parts H_n(PSL_2(mathcal O_-d), Z)_(ℓ) ≅ H_n(SL_2(mathcal O_-d), Z)_(ℓ). Since H_n(SL_2(mathcal O_-d), Z)_(ℓ) is periodic in degrees ≥ 3 we can recover the 3-primary part of H_n(PSL_2(mathcal O_-13), Z) in all degrees q≥1 from the following computation by ignoring all 2-power factors in the output.

<div class="example"><pre>
gap> R:=ResolutionSL2QuadraticIntegers(-13,13);;
gap> for n in [3..12] do
> Print("H_",n," at prime p=3 is: ", Filtered(Homology(TensorWithIntegers(R),n), m->IsInt(m/3)),"\n");
> od;
H_3 at prime p=3 is: [ 6, 24 ]
H_4 at prime p=3 is: [ ]
H_5 at prime p=3 is: [ ]
H_6 at prime p=3 is: [ 6, 12 ]
H_7 at prime p=3 is: [ 6, 24 ]
H_8 at prime p=3 is: [ ]
H_9 at prime p=3 is: [ ]
H_10 at prime p=3 is: [ 6, 12 ]
H_11 at prime p=3 is: [ 6, 24 ]
H_12 at prime p=3 is: [ ]
gap> #Ignore the 2-power factors in the output

</pre></div>

The ring mathcal O_-163 is an example of a principal ideal domain that is not a Euclidean domain. It seems that no complete calculation of H_n(PSL_2(mathcal O_-163), Z) is yet available in the literature. The following comands compute this homology in the first 31 degrees. The computation suggests a general formula in higher degrees. All but two of the stabilizer groups for the action of PSL_2(mathcal O_-163) are periodic. The non-periodic group A_4 occurs twice in degree 0.

<div class="example"><pre>
gap> R:=ResolutionPSL2QuadraticIntegers(-163,32);;
gap> for n in [1..31] do
> Print("H_",n,"= ",Homology(TensorWithIntegers(R),n),"\n");
> od;
H_1= [ 0, 0, 0, 0, 0, 0, 0 ]
H_2= [ 2, 12, 0, 0, 0, 0, 0, 0 ]
H_3= [ 6 ]
H_4= [ ]
H_5= [ 2, 2, 2 ]
H_6= [ 2, 6 ]
H_7= [ 6 ]
H_8= [ 2, 2, 2, 2 ]
H_9= [ 2, 2, 2 ]
H_10= [ 2, 6 ]
H_11= [ 2, 2, 2, 2, 6 ]
H_12= [ 2, 2, 2, 2 ]
H_13= [ 2, 2, 2 ]
H_14= [ 2, 2, 2, 2, 2, 6 ]
H_15= [ 2, 2, 2, 2, 6 ]
H_16= [ 2, 2, 2, 2 ]
H_17= [ 2, 2, 2, 2, 2, 2, 2 ]
H_18= [ 2, 2, 2, 2, 2, 6 ]
H_19= [ 2, 2, 2, 2, 6 ]
H_20= [ 2, 2, 2, 2, 2, 2, 2, 2 ]
H_21= [ 2, 2, 2, 2, 2, 2, 2 ]
H_22= [ 2, 2, 2, 2, 2, 6 ]
H_23= [ 2, 2, 2, 2, 2, 2, 2, 2, 6 ]
H_24= [ 2, 2, 2, 2, 2, 2, 2, 2 ]
H_25= [ 2, 2, 2, 2, 2, 2, 2 ]
H_26= [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 6 ]
H_27= [ 2, 2, 2, 2, 2, 2, 2, 2, 6 ]
H_28= [ 2, 2, 2, 2, 2, 2, 2, 2 ]
H_29= [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
H_30= [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 6 ]
H_31= [ 2, 2, 2, 2, 2, 2, 2, 2, 6 ]

</pre></div>

<a id="X86A6858884B9C05B" name="X86A6858884B9C05B"></a>

<h4>13.12 Some other infinite matrix groups</h4>

Analogous to the functions for Bianchi groups, HAP has functions

<ul>
<li><code class="code">ResolutionSL2QuadraticIntegers(-d,n)</code>

</li>
<li><code class="code">ResolutionSL2ZInvertedInteger(m,n)</code>

</li>
<li><code class="code">ResolutionGL2QuadraticIntegers(-d,n)</code>

</li>
<li><code class="code">ResolutionPGL2QuadraticIntegers(-d,n)</code>

</li>
<li><code class="code">ResolutionGL3QuadraticIntegers(-d,n)</code>

</li>
<li><code class="code">ResolutionPGL3QuadraticIntegers(-d,n)</code>

</li>
</ul>
for computing free resolutions for certain values of SL_2(cal O_-d), SL_2( Z[frac1m]), GL_2(cal O_-d) and PGL_2(cal O_-d). Additionally, the function

<ul>
<li><code class="code">ResolutionArithmeticGroup("string",n)</code>

</li>
</ul>
can be used to compute resolutions for groups whose data (provided by Sebastian Schoennenbeck, Alexander Rahm and Mathieu Dutour) is stored in the directory <code class="code">gap/pkg/Hap/lib/Perturbations/Gcomplexes</code> .

For instance, the following commands compute

H^1(SL_2({\cal O}_{-6}),P_{{\cal O}_{-6}}(24)) =
(\mathbb Z_2)^4
\oplus \mathbb Z_{12}
\oplus \mathbb Z_{24}
\oplus \mathbb Z_{9240}
\oplus \mathbb Z_{55440}
\oplus \mathbb Z^4\,,


H^2(SL_2({\cal O}_{-6}),P_{{\cal O}_{-6}}(24)) =
\begin{array}{l}
(\mathbb Z_2)^{26}
\oplus \mathbb (Z_{6})^7
\oplus \mathbb (Z_{12})^{10}
\oplus \mathbb Z_{24}
\oplus (\mathbb Z_{120})^2
\oplus (\mathbb Z_{840})^3\\
\oplus \mathbb Z_{2520}
\oplus (\mathbb Z_{27720})^2
\oplus (\mathbb Z_{24227280})^2
\oplus (\mathbb Z_{411863760})^2\\
\oplus \mathbb Z_{2454438243748928651877425142836664498129840}\\
\oplus \mathbb Z_{14726629462493571911264550857019986988779040}\\
\oplus \mathbb Z^4\end{array}\ ,


H^3(SL_2({\cal O}_{-6}),P_{{\cal O}_{-6}}(24)) =
(\mathbb Z_2)^{58}
\oplus (\mathbb Z_{4})^4
\oplus (\mathbb Z_{12})\ .


<div class="example"><pre>
gap> R:=ResolutionSL2QuadraticIntegers(-6,4);
Resolution of length 4 in characteristic 0 for PSL(2,O-6) .
No contracting homotopy available.

gap> G:=R!.group;;
gap> M:=HomogeneousPolynomials(G,24);;
gap> C:=HomToIntegralModule(R,M);;
gap> Cohomology(C,1);
[ 2, 2, 2, 2, 12, 24, 9240, 55440, 0, 0, 0, 0 ]
gap> Cohomology(C,2);
gap> Cohomology(C,2);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
 2, 6, 6, 6, 6, 6, 6, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 24, 120,
 120, 840, 840, 840, 2520, 27720, 27720, 24227280, 24227280, 411863760,
 411863760, 2454438243748928651877425142836664498129840,
 14726629462493571911264550857019986988779040, 0, 0, 0, 0 ]
gap> Cohomology(C,3);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 12, 12 ]

</pre></div>

The following commands construct free resolutions up to degree 5 for the groups SL_2( Z[frac12]), GL_2(cal O_-2), GL_2(cal O_2), PGL_2(cal O_2), GL_3(cal O_-2), PGL_3(cal O_-2). The final command constructs a free resolution up to degree 3 for PSL_4( Z).

<div class="example"><pre>
gap> R1:=ResolutionSL2ZInvertedInteger(2,5);
Resolution of length 5 in characteristic 0 for SL(2,Z[1/2]) .

gap> R2:=ResolutionGL2QuadraticIntegers(-2,5);
Resolution of length 5 in characteristic 0 for GL(2,O-2) .
No contracting homotopy available.

gap> R3:=ResolutionGL2QuadraticIntegers(2,5);
Resolution of length 5 in characteristic 0 for GL(2,O2) .
No contracting homotopy available.

gap> R4:=ResolutionPGL2QuadraticIntegers(2,5);
Resolution of length 5 in characteristic 0 for PGL(2,O2) .
No contracting homotopy available.

gap> R5:=ResolutionGL3QuadraticIntegers(-2,5);
Resolution of length 5 in characteristic 0 for GL(3,O-2) .
No contracting homotopy available.

gap> R6:=ResolutionPGL3QuadraticIntegers(-2,5);
Resolution of length 5 in characteristic 0 for PGL(3,O-2) .
No contracting homotopy available.

gap> R7:=ResolutionArithmeticGroup("PSL(4,Z)",3);
Resolution of length 3 in characteristic 0 for <matrix group with 655 generators> .
No contracting homotopy available.

</pre></div>

<a id="X7EF5D97281EB66DA" name="X7EF5D97281EB66DA"></a>

<h4>13.13 Ideals and finite quotient groups</h4>

The following commands first construct the number field Q(sqrt-7), its ring of integers cal O_-7=cal O( Q(sqrt-7)), and the principal ideal I=⟨ 5 + 2sqrt-7⟩ ◃ cal O( Q(sqrt-7)) of norm cal N(I)=53. The ring I is prime since its norm is a prime number. The primality of I is also demonstrated by observing that the quotient ring R=cal O_-7/I is an integral domain and hence isomorphic to the unique finite field of order 53, R≅ Z/53 Z . (In a ring of quadratic integers prime ideal is the same as maximal ideal).

The finite group G=SL_2(cal O_-7/I) is then constructed and confirmed to be isomorphic to SL_2( Z/53 Z). The group G is shown to admit a periodic ZG-resolution of Z of period dividing 52.

Finally the integral homology

H_n(G,\mathbb Z) = \left\{\begin{array}{ll}
0 & n\ne 3,7, {\rm~for~} 0\le n \le 8,\\
\mathbb Z_{2808} & n=3,7,
\end{array}\right.

is computed.

<div class="example"><pre>
gap> Q:=QuadraticNumberField(-7);
Q(Sqrt(-7))

gap> OQ:=RingOfIntegers(Q);
O(Q(Sqrt(-7)))

gap> I:=QuadraticIdeal(OQ,5+2*Sqrt(-7));
ideal of norm 53 in O(Q(Sqrt(-7)))

gap> R:=OQ mod I;
ring mod ideal of norm 53

gap> IsIntegralRing(R);
true

gap> gens:=GeneratorsOfGroup( SL2QuadraticIntegers(-7) );;
gap> G:=Group(gens*One(R));;G:=Image(IsomorphismPermGroup(G));;
gap> StructureDescription(G);
"SL(2,53)"

gap> IsPeriodic(G);
true
gap> CohomologicalPeriod(G);
52

gap> GroupHomology(G,1);
[ ]
gap> GroupHomology(G,2);
[ ]
gap> GroupHomology(G,3);
[ 8, 27, 13 ]
gap> GroupHomology(G,4);
[ ]
gap> GroupHomology(G,5);
[ ]
gap> GroupHomology(G,6);
[ ]
gap> GroupHomology(G,7);
[ 8, 27, 13 ]
gap> GroupHomology(G,8);
[ ]

</pre></div>

The following commands show that the rational prime 7 is not prime in cal O_-5=cal O( Q(sqrt-5)). Moreover, 7 totally splits in cal O_-5 since the final command shows that only the rational primes 2 and 5 ramify in cal O_-5.

<div class="example"><pre>
gap> Q:=QuadraticNumberField(-5);;
gap> OQ:=RingOfIntegers(Q);;
gap> I:=QuadraticIdeal(OQ,7);;
gap> IsPrime(I);
false

gap> Factors(Discriminant(OQ));
[ -2, 2, 5 ]

</pre></div>

For d < 0 the rings cal O_d=cal O( Q(sqrtd)) are unique factorization domains for precisely

 d = -1, -2, -3, -7, -11, -19, -43, -67, -163.

This result was conjectured by Gauss, and essentially proved by Kurt Heegner, and then later proved by Harold Stark.

The following commands construct the classic example of a prime ideal I that is not principal. They then illustrate reduction modulo I.

<div class="example"><pre>
gap> Q:=QuadraticNumberField(-5);;
gap> OQ:=RingOfIntegers(Q);;
gap> I:=QuadraticIdeal(OQ,[2,1+Sqrt(-5)]);
ideal of norm 2 in O(Q(Sqrt(-5)))

gap> 6 mod I;
0

</pre></div>

<a id="X7D1F72287F14C5E1" name="X7D1F72287F14C5E1"></a>

<h4>13.14 Congruence subgroups for ideals</h4>

Given a ring of integers cal O and ideal I ◃ cal O there is a canonical homomorphism π_I: SL_2(cal O) → SL_2(cal O/I). A subgroup Γ ≤ SL_2(cal O) is said to be a congruence subgroup if it contains ker π_I. Thus congruence subgroups are of finite index. Generalizing the definition in <a href="chap13.html#X79A1974B7B4987DE">13.1</a> above, we define the principal congruence subgroup Γ_1(I)=ker π_I, and the congruence subgroup Γ_0(I) consisting of preimages of the upper triangular matrices in SL_2(cal O/I).

The following commands construct Γ=Γ_0(I) for the ideal I◃ cal O Q(sqrt-5) generated by 12 and 36sqrt-5. The group Γ has index 385 in SL_2(cal O Q(sqrt-5)). The final command displays a tree in a Cayley graph for SL_2(cal O Q(sqrt-5)) whose nodes represent a transversal for Γ.

<div class="example"><pre>
gap> Q:=QuadraticNumberField(-5);;
gap> OQ:=RingOfIntegers(Q);;
gap> I:=QuadraticIdeal(OQ,[36*Sqrt(-5), 12]);;
gap> G:=HAP_CongruenceSubgroupGamma0(I);
CongruenceSubgroupGamma0(ideal of norm 144 in O(Q(Sqrt(-5))))

gap> IndexInSL2O(G);
385

gap> HAP_SL2TreeDisplay(G);

</pre></div>

<img src="images/treesqrt5.gif" align="center" width="350" alt="Information for the cubic tree"/>

The next commands first construct the congruence subgroup Γ_0(I) of index 144 in SL_2(cal O Q(sqrt-2)) for the ideal I in cal O Q(sqrt-2) generated by 4+5sqrt-2. The commands then compute

H_1(\Gamma_0(I),\mathbb Z) = \mathbb Z_3 \oplus \mathbb Z_6 \oplus \mathbb Z_{30} \oplus \mathbb Z^8\, ,

H_2(\Gamma_0(I), \mathbb Z) = (\mathbb Z_2)^9 \oplus \mathbb Z^7\, ,

H_3(\Gamma_0(I), \mathbb Z) = (\mathbb Z_2)^9 \, .

<div class="example"><pre>
gap> Q:=QuadraticNumberField(-2);;
gap> OQ:=RingOfIntegers(Q);;
gap> I:=QuadraticIdeal(OQ,4+5*Sqrt(-2));;
gap> G:=HAP_CongruenceSubgroupGamma0(I);
CongruenceSubgroupGamma0(ideal of norm 66 in O(Q(Sqrt(-2))))

gap> IndexInSL2O(G);
144

gap> R:=ResolutionSL2QuadraticIntegers(-2,4,true);;
gap> S:=ResolutionFiniteSubgroup(R,G);;

gap> Homology(TensorWithIntegers(S),1);
[ 3, 6, 30, 0, 0, 0, 0, 0, 0, 0, 0 ]
gap> Homology(TensorWithIntegers(S),2);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0 ]
gap> Homology(TensorWithIntegers(S),3);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2 ]

</pre></div>

<a id="X85E912617AFE03F4" name="X85E912617AFE03F4"></a>

<h4>13.15 First homology</h4>

The isomorphism H_1(G, Z) ≅ G_ab allows for the computation of first integral homology using computational methods for finitely presented groups. Such methods underly the following computation of

H_1( \Gamma_0(I),\mathbb Z) \cong \mathbb Z_2 \oplus \cdots \oplus
\mathbb Z_{4078793513671}

where I is the prime ideal in the Gaussian integers generated by 41+56sqrt-1.

<div class="example"><pre>
gap> Q:=QuadraticNumberField(-1);;
gap> OQ:=RingOfIntegers(Q);;
gap> I:=QuadraticIdeal(OQ,41+56*Sqrt(-1));
ideal of norm 4817 in O(GaussianRationals)
gap> G:=HAP_CongruenceSubgroupGamma0(I);;
gap> AbelianInvariants(G);
[ 2, 2, 4, 5, 7, 16, 29, 43, 157, 179, 1877, 7741, 22037, 292306033,
 4078793513671 ]

</pre></div>

We write G^ab_tors to denote the maximal finite summand of the first homology group of G and refer to this as the torsion subgroup. Nicholas Bergeron and Akshay Venkatesh <a href="chapBib.html#biBbergeron">[Ber16]</a> have conjectured relationships between the torsion in congruence subgroups Γ and the volume of their quotient manifold frak h^3/Γ. For instance, for the Gaussian integers they conjecture

 \frac{\log |\Gamma_0(I)_{tors}^{ab}|}{{\rm Norm}(I)} \rightarrow \frac{\lambda}{18\pi},\ \lambda =L(2,\chi_{\mathbb Q(\sqrt{-1})}) = 1 -\frac{1}{9} + \frac{1}{25} - \frac{1}{49} + \cdots

as the norm of the prime ideal I tends to ∞. The following approximates λ/18π = 0.0161957 and fraclog |Γ_0(I)_tors^ab|{ Norm(I) = 0.0210325 for the above example.

<div class="example"><pre>
gap> Q:=QuadraticNumberField(-1);;
gap> Lfunction(Q,2)/(18*3.142);
0.0161957

Loge10:=0.434294481903;; #Log_10(e)
gap> 1.0*Log(Product(AbelianInvariants(G)),10)/(Loge*Norm(I));
0.0210325

</pre></div>

The link with volume is given by the Humbert volume formula


{\rm Vol} ( {\frak h}^3 / PSL_2( {\cal O}_{d} ) )
= \frac{|D|^{3/2}}{24} \zeta_{ \mathbb Q( \sqrt{d} ) }(2)/\zeta_{\mathbb Q}(2) 

valid for square-free d<0, where D is the discriminant of Q(sqrtd). The volume of a finite index subgroup Γ is obtained by multiplying the right-hand side by the index |PSL_2(cal O_d): Γ|.

The following commands produce a graph of fraclog |Γ_0(I)_tors^ab|{ Norm(I) against Norm(I) for prime ideals I of norm 49 ≤ Norm(I) ≤ 4357 (where one ideal for each norm is taken).

<div class="example"><pre>
gap> Q:=QuadraticNumberField(-1);;
gap> OQ:=RingOfIntegers(Q);;
gap> N:=QuadraticIntegersByNorm(OQ,20000);;

gap> #########################################
gap> fn:=function(x);
gap> if IsRat(x) then return x; fi;
gap> return x!.rational+x!.irrational*Sqrt(-1);
gap> end;
gap> #########################################

gap> NN:=List(N,fn);
gap> P:=Filtered(NN,x->IsPrime(QuadraticIdeal(OQ,x)));
gap> PP:=Classify(P,x->Norm(Q,x));
gap> PP:=List(PP,x->x[1]);;
gap> PP:=Filtered(PP,x->not x=0);

gap> Loge:=0.434294481903;; ###Log_10(e)
gap> #########################################
gap> ffn:=function(x)
gap> local I, G, A, S, F;
gap> I:=QuadraticIdeal(OQ,x);
gap> G:=HAP_CongruenceSubgroupGamma0(I);;
gap> A:=AbelianInvariants(G);
gap> A:=Filtered(A,x->not x=0);
gap> return [Norm(Q,x),1.0*Log(Product(Filtered(AbelianInvariants(G),i->not i=0)),10)/(Loge*Norm(I))];
gap> end;
gap> #########################################

gap> S:=List(PP{[9..267]},ffn);
gap> ScatterPlot(S);

</pre></div>

<img src="images/primesTorsion.png" align="center" width="1200" alt="graph of torsion versus norm"/>

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