Coset Diagram for the Action of Picard Group on Q(i,\surd3)

Details Coset Diagram for the Action of Picard Group on Q(i,\surd3) Coset Diagram for the Action of Picard Group on Q(i,\surd3)

2011-11-16

arxiv.org/abs/1111.3778v2

Mathematics

Group Theory

Scientific paper

The Picard group {\Gamma} is PSL(2,Z[i]). We have defined coset diagram for the Picard group. It has been observed that some elements of Q(i,/surd3) of the form ((a+b/surd3)/c) and their conjugates ((a-b/surd3)/c) over \mathbb{Q}(i) have different signs in the coset diagram for the action of {\Gamma} on the biquadratic field Q(i,/surd3), these are called ambiguous numbers. We have noticed that ambiguous numbers in the coset diagram for the action of {\Gamma} on \mathbb{Q}(i,/surd3) form a unique pattern. It has been shown that there are finite number of ambiguous numbers in an orbit {\Gamma}{\alpha}, where {\alpha} is ambiguous, and they form a closed path and it is the only closed path in the orbit {\Gamma}{\alpha}. We have devised a procedure to obtain ambiguous numbers of the form ((a+k/surd3)/c), where k is a positive integer.

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