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32 changes: 28 additions & 4 deletions tutorial/chap13.html
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Original file line number Diff line number Diff line change
Expand Up @@ -630,9 +630,9 @@ <h4>13.10 <span class="Heading">Bianchi groups</span></h4>

<p>The <em>Bianchi groups</em> are the groups <span class="SimpleMath">G=PSL_2(cal O_-d)</span> where <span class="SimpleMath">d</span> is a square free positive integer and <span class="SimpleMath">cal O_-d</span> is the ring of integers of the imaginary quadratic field <span class="SimpleMath">Q(sqrt-d)</span>. More explicitly,</p>

<p class="pcenter">{\cal O}_{-d} = \mathbb Z\left[\sqrt{-d}\right]~~~~~~~~ {\rm if~} d \equiv 1 {\rm ~mod~} 4\, ,</p>
<p class="pcenter">{\cal O}_{-d} = \mathbb Z\left[\sqrt{-d}\right]~~~~~~~~ {\rm if~} d \equiv 1,2 {\rm ~mod~} 4\, ,</p>

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

<p>These groups act on upper-half space <span class="SimpleMath">frak h^3</span> 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 <span class="SimpleMath">2</span>-dimensional cellular subspace <span class="SimpleMath">cal T⊂ frak h^3</span> where <span class="SimpleMath">cal T</span> is a contractible CW-complex on which <span class="SimpleMath">G</span> acts cellularly. It should be mentioned that the fundamental domain and the contractible <span class="SimpleMath">2</span>-complex <span class="SimpleMath">cal T</span> are not uniquely determined by <span class="SimpleMath">G</span>. Various algorithms exist for computing <span class="SimpleMath">cal T</span> 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 <span class="SimpleMath">d</span> 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 <span class="SimpleMath">d</span> are stored in HAP. The following commands combine data from Schoennenbeck's algorithm with free resolutions for cell stabiliers to compute</p>

Expand Down Expand Up @@ -1034,23 +1034,47 @@ <h4>13.14 <span class="Heading">First homology</span></h4>
[ 2, 2, 4, 5, 7, 16, 29, 43, 157, 179, 1877, 7741, 22037, 292306033,
4078793513671 ]

<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">II:=QuadraticIdeal(OQ,47+61*Sqrt(-1));</span>
ideal of norm 5930 in O(GaussianRationals)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">GG:=HAP_CongruenceSubgroupGamma0(II);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">AbelianInvariants(GG);</span>
[ 0, 0, 0, 0, 0, 0, 0, 0, 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, 4, 4, 4, 4, 4, 4, 4, 4,
4, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 11, 16, 16, 16, 16, 17, 17, 17, 32, 37,
61, 61, 64, 64, 128, 263, 263, 263, 263, 512, 1024, 5099, 5099, 72043,
72043 ]

<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">III:=QuadraticIdeal(OQ,49+69*Sqrt(-1));</span>
ideal of norm 7162 in O(GaussianRationals)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">GGG:=HAP_CongruenceSubgroupGamma0(III);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">AbelianInvariants(GGG);</span>
[ 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 8, 8, 8, 8, 13, 13, 25, 59,
59, 179, 283, 283, 379, 857, 967, 967, 3769, 13537, 25601, 222659,
8180323, 8180323, 11450932001, 11450932001 ]

</pre></div>

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

<p class="pcenter"> \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</p>

<p>as the norm of the prime ideal <span class="SimpleMath">I</span> tends to <span class="SimpleMath">∞</span>. The following approximates <span class="SimpleMath">λ/18π = 0.0161957</span> and <span class="SimpleMath">fraclog |Γ_0(I)_tors^ab|{ Norm(I) = 0.00913432</span> for the above example.</p>
<p>as the norm of the prime ideal <span class="SimpleMath">I</span> tends to <span class="SimpleMath">∞</span>. The following approximates <span class="SimpleMath">λ/18π = 0.0161957</span> and <span class="SimpleMath">fraclog |Γ_0(I)_tors^ab|{ Norm(I) = 0.00913432</span> and <span class="SimpleMath">fraclog |Γ_0(II)_tors^ab|{ Norm(II) = 0.0136594</span> and <span class="SimpleMath">fraclog |Γ_0(III)_tors^ab|{ Norm(III) = 0.0120078</span> for the above example.</p>


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

<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">1.0*Log(Product(AbelianInvariants(F)),10)/Norm(I);</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">1.0*Log(Product(AbelianInvariants(G)),10)/Norm(I);</span>
0.00913432

<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">1.0*Log(Product(Filtered(AbelianInvariants(GG),i-&gt;not i=0)),10)/Norm(II);</span>
0.0136594

<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">1.0*Log(Product(Filtered(AbelianInvariants(GGG),i-&gt;not i=0)),10)/Norm(III);</span>
0.0120078

</pre></div>

<p>The link with volume is given by the Humbert volume formula</p>
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37 changes: 31 additions & 6 deletions tutorial/chap13.txt
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Original file line number Diff line number Diff line change
Expand Up @@ -821,14 +821,14 @@
quadratic field Q(sqrt-d). More explicitly,


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



Expand Down Expand Up @@ -1275,6 +1275,24 @@
[ 2, 2, 4, 5, 7, 16, 29, 43, 157, 179, 1877, 7741, 22037, 292306033, 
 4078793513671 ]

gap> II:=QuadraticIdeal(OQ,47+61*Sqrt(-1));
ideal of norm 5930 in O(GaussianRationals)
gap> GG:=HAP_CongruenceSubgroupGamma0(II);;
gap> AbelianInvariants(GG);
[ 0, 0, 0, 0, 0, 0, 0, 0, 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, 4, 4, 4, 4, 4, 4, 4, 4, 
 4, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 11, 16, 16, 16, 16, 17, 17, 17, 32, 37, 
 61, 61, 64, 64, 128, 263, 263, 263, 263, 512, 1024, 5099, 5099, 72043, 
 72043 ]

gap> III:=QuadraticIdeal(OQ,49+69*Sqrt(-1));
ideal of norm 7162 in O(GaussianRationals)
gap> GGG:=HAP_CongruenceSubgroupGamma0(III);;
gap> AbelianInvariants(GGG);
[ 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 8, 8, 8, 8, 13, 13, 25, 59, 
 59, 179, 283, 283, 379, 857, 967, 967, 3769, 13537, 25601, 222659, 
 8180323, 8180323, 11450932001, 11450932001 ]



We write G^ab_tors to denote the maximal finite summand of the first
Expand All @@ -1291,17 +1309,24 @@


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) = 0.00913432 for the
above example.
λ/18π = 0.0161957 and fraclog |Γ_0(I)_tors^ab|{ Norm(I) = 0.00913432 and
fraclog |Γ_0(II)_tors^ab|{ Norm(II) = 0.0136594 and fraclog
|Γ_0(III)_tors^ab|{ Norm(III) = 0.0120078 for the above example.

 Example 
gap> Q:=QuadraticNumberField(-1);;
gap> Lfunction(Q,2)/(18*3.142);
0.0161957

gap> 1.0*Log(Product(AbelianInvariants(F)),10)/Norm(I);
gap> 1.0*Log(Product(AbelianInvariants(G)),10)/Norm(I);
0.00913432

gap> 1.0*Log(Product(Filtered(AbelianInvariants(GG),i->not i=0)),10)/Norm(II);
0.0136594

gap> 1.0*Log(Product(Filtered(AbelianInvariants(GGG),i->not i=0)),10)/Norm(III);
0.0120078



The link with volume is given by the Humbert volume formula
Expand Down
32 changes: 28 additions & 4 deletions tutorial/chap13_mj.html
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Original file line number Diff line number Diff line change
Expand Up @@ -633,9 +633,9 @@ <h4>13.10 <span class="Heading">Bianchi groups</span></h4>

<p>The <em>Bianchi groups</em> are the groups <span class="SimpleMath">\(G=PSL_2({\cal O}_{-d})\)</span> where <span class="SimpleMath">\(d\)</span> is a square free positive integer and <span class="SimpleMath">\({\cal O}_{-d}\)</span> is the ring of integers of the imaginary quadratic field <span class="SimpleMath">\(\mathbb Q(\sqrt{-d})\)</span>. More explicitly,</p>

<p class="center">\[{\cal O}_{-d} = \mathbb Z\left[\sqrt{-d}\right]~~~~~~~~ {\rm if~} d \equiv 1 {\rm ~mod~} 4\, ,\]</p>
<p class="center">\[{\cal O}_{-d} = \mathbb Z\left[\sqrt{-d}\right]~~~~~~~~ {\rm if~} d \equiv 1,2 {\rm ~mod~} 4\, ,\]</p>

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

<p>These groups act on upper-half space <span class="SimpleMath">\({\frak h}^3\)</span> 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 <span class="SimpleMath">\(2\)</span>-dimensional cellular subspace <span class="SimpleMath">\({\cal T}\subset {\frak h}^3\)</span> where <span class="SimpleMath">\({\cal T}\)</span> is a contractible CW-complex on which <span class="SimpleMath">\(G\)</span> acts cellularly. It should be mentioned that the fundamental domain and the contractible <span class="SimpleMath">\(2\)</span>-complex <span class="SimpleMath">\({\cal T}\)</span> are not uniquely determined by <span class="SimpleMath">\(G\)</span>. Various algorithms exist for computing <span class="SimpleMath">\({\cal T}\)</span> and its cell stabilizers. One algorithm due to Swan <a href="chapBib_mj.html#biBswan">[Swa71a]</a> has been implemented by Alexander Rahm <a href="chapBib_mj.html#biBrahmthesis">[Rah10]</a> and the output for various values of <span class="SimpleMath">\(d\)</span> 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_mj.html#biBschoennenbeck">[BCNS15]</a> and, again, its output for various values of <span class="SimpleMath">\(d\)</span> are stored in HAP. The following commands combine data from Schoennenbeck's algorithm with free resolutions for cell stabiliers to compute</p>

Expand Down Expand Up @@ -1037,23 +1037,47 @@ <h4>13.14 <span class="Heading">First homology</span></h4>
[ 2, 2, 4, 5, 7, 16, 29, 43, 157, 179, 1877, 7741, 22037, 292306033,
4078793513671 ]

<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">II:=QuadraticIdeal(OQ,47+61*Sqrt(-1));</span>
ideal of norm 5930 in O(GaussianRationals)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">GG:=HAP_CongruenceSubgroupGamma0(II);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">AbelianInvariants(GG);</span>
[ 0, 0, 0, 0, 0, 0, 0, 0, 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, 4, 4, 4, 4, 4, 4, 4, 4,
4, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 11, 16, 16, 16, 16, 17, 17, 17, 32, 37,
61, 61, 64, 64, 128, 263, 263, 263, 263, 512, 1024, 5099, 5099, 72043,
72043 ]

<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">III:=QuadraticIdeal(OQ,49+69*Sqrt(-1));</span>
ideal of norm 7162 in O(GaussianRationals)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">GGG:=HAP_CongruenceSubgroupGamma0(III);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">AbelianInvariants(GGG);</span>
[ 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 8, 8, 8, 8, 13, 13, 25, 59,
59, 179, 283, 283, 379, 857, 967, 967, 3769, 13537, 25601, 222659,
8180323, 8180323, 11450932001, 11450932001 ]

</pre></div>

<p>We write <span class="SimpleMath">\(G^{ab}_{tors}\)</span> to denote the maximal finite summand of the first homology group of <span class="SimpleMath">\(G\)</span> and refer to this as the <em>torsion subgroup</em>. Nicholas Bergeron and Akshay Venkatesh <a href="chapBib_mj.html#biBbergeron">[Ber16]</a> have conjectured relationships between the torsion in congruence subgroups <span class="SimpleMath">\(\Gamma\)</span> and the volume of their quotient manifold <span class="SimpleMath">\({\frak h}^3/\Gamma\)</span>. For instance, for the Gaussian integers they conjecture</p>

<p class="center">\[ \frac{\log |\Gamma_0(I)_{tors}^{ab}|}{{\rm Norm}(I)} \rightarrow \frac{\lambda}{18\pi},\ \lambda =L(2,\chi_{\mathbb Q(\sqrt{-1})}) = 1 -\frac{1}{9} + \frac{1}{25} - \frac{1}{49} + \cdots\]</p>

<p>as the norm of the prime ideal <span class="SimpleMath">\(I\)</span> tends to <span class="SimpleMath">\(\infty\)</span>. The following approximates <span class="SimpleMath">\(\lambda/18\pi = 0.0161957\)</span> and <span class="SimpleMath">\(\frac{\log |\Gamma_0(I)_{tors}^{ab}|}{{\rm Norm}(I)} = 0.00913432\)</span> for the above example.</p>
<p>as the norm of the prime ideal <span class="SimpleMath">\(I\)</span> tends to <span class="SimpleMath">\(\infty\)</span>. The following approximates <span class="SimpleMath">\(\lambda/18\pi = 0.0161957\)</span> and <span class="SimpleMath">\(\frac{\log |\Gamma_0(I)_{tors}^{ab}|}{{\rm Norm}(I)} = 0.00913432\)</span> and <span class="SimpleMath">\(\frac{\log |\Gamma_0(II)_{tors}^{ab}|}{{\rm Norm}(II)} = 0.0136594\)</span> and <span class="SimpleMath">\(\frac{\log |\Gamma_0(III)_{tors}^{ab}|}{{\rm Norm}(III)} = 0.0120078\)</span> for the above example.</p>


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

<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">1.0*Log(Product(AbelianInvariants(F)),10)/Norm(I);</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">1.0*Log(Product(AbelianInvariants(G)),10)/Norm(I);</span>
0.00913432

<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">1.0*Log(Product(Filtered(AbelianInvariants(GG),i-&gt;not i=0)),10)/Norm(II);</span>
0.0136594

<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">1.0*Log(Product(Filtered(AbelianInvariants(GGG),i-&gt;not i=0)),10)/Norm(III);</span>
0.0120078

</pre></div>

<p>The link with volume is given by the Humbert volume formula</p>
Expand Down

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