Difference between revisions of "ApCoCoA-1:Picard group"
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StrohmeierB (talk | contribs) (New page: === <div id="Picard_group">Picard group</div> === ==== Description ==== ==== Reference ==== Discontinuous Groups and Riemann Surfaces: Proceedings,...) |
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| − | === <div id="Picard_group">[[:ApCoCoA:Symbolic data#Picard_group|Picard | + | === <div id="Picard_group">[[:ApCoCoA:Symbolic data#Picard_group|Picard Group]]</div> === |
==== Description ==== | ==== Description ==== | ||
| − | + | The Picard Group is PSL_{2}(O_{1}) where O_{1} is the ring of integers in the | |
| + | quadratic imaginary number field Q sqrt(-1). | ||
==== Reference ==== | ==== Reference ==== | ||
| − | Discontinuous Groups and Riemann Surfaces: Proceedings, | + | L. Greenberg - "Discontinuous Groups and Riemann Surfaces: Proceedings" - Princeton University Press (1974) 151; B. Fine - "THE EUCLIDEAN BIANCHI GROUPS" - COMMUNICATIONS IN ALGEBRA, 18(8), 2461-2484 (1990) 2461 |
==== Computation ==== | ==== Computation ==== | ||
| − | /*Use the ApCoCoA package ncpoly.*/ | + | /*Use the ApCoCoA package ncpoly.*/ |
| − | + | Use ZZ/(2)[a,l,t,u]; | |
| − | + | NC.SetOrdering("LLEX"); | |
| − | + | Define CreateRelationsPicard() | |
Relations:=[]; | Relations:=[]; | ||
| Line 45: | Line 46: | ||
Gb:=NC.GB(Relations,31,1,100,1000); | Gb:=NC.GB(Relations,31,1,100,1000); | ||
Gb; | Gb; | ||
| + | |||
==== Example in Symbolic Data Format ==== | ==== Example in Symbolic Data Format ==== | ||
<FREEALGEBRA createdAt="2014-03-26" createdBy="strohmeier"> | <FREEALGEBRA createdAt="2014-03-26" createdBy="strohmeier"> | ||
Latest revision as of 21:06, 22 April 2014
Description
The Picard Group is PSL_{2}(O_{1}) where O_{1} is the ring of integers in the quadratic imaginary number field Q sqrt(-1).
Reference
L. Greenberg - "Discontinuous Groups and Riemann Surfaces: Proceedings" - Princeton University Press (1974) 151; B. Fine - "THE EUCLIDEAN BIANCHI GROUPS" - COMMUNICATIONS IN ALGEBRA, 18(8), 2461-2484 (1990) 2461
Computation
/*Use the ApCoCoA package ncpoly.*/
Use ZZ/(2)[a,l,t,u];
NC.SetOrdering("LLEX");
Define CreateRelationsPicard()
Relations:=[];
// add the relation a^2 = 1
Append(Relations,[[a,a],[1]]);
// add the relation l^2 = 1
Append(Relations,[[l,l],[1]]);
// add the relation (al)^2 = 1
Append(Relations,[[a,l,a,l],[1]]);
// add the relation (tl)^2 = 1
Append(Relations,[[t,l,t,l],[1]]);
// add the relation (ul)^2 = 1
Append(Relations,[[u,l,u,l],[1]]);
// add the relation (at)^3 = 1
Append(Relations,[[a,t,a,t,a,t],[1]]);
// add the relation (ual)^3 = 1
Append(Relations,[[u,a,l,u,a,l,u,a,l],[1]]);
//add the relation tu = ut
Append(Relations,[[t,u],[u,t]]);
Return Relations;
EndDefine;
Relations:=CreateRelationsPicard();
Relations;
Gb:=NC.GB(Relations,31,1,100,1000);
Gb;
Example in Symbolic Data Format
<FREEALGEBRA createdAt="2014-03-26" createdBy="strohmeier"> <vars>a,l,t,u</vars> <uptoDeg>9</uptoDeg> <basis> <ncpoly>a*a-1</ncpoly> <ncpoly>l*l-1</ncpoly> <ncpoly>a*l*a*l-1</ncpoly> <ncpoly>t*l*t*l-1</ncpoly> <ncpoly>u*l*u*l-1</ncpoly> <ncpoly>a*t*a*t*a*t-1</ncpoly> <ncpoly>u*a*l*u*a*l*u*a*l-1</ncpoly> <ncpoly>t*u-u*t</ncpoly> </basis> <Comment>The partial LLex Gb has 104 elements</Comment> <Comment>Picard_group</Comment> </FREEALGEBRA>
