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arXiv · 1111.3778

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

Qaiser Mushtaq·Saima Anis
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Abstract

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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Qaiser Mushtaq, Saima Anis. 2011-11-16. Coset Diagram for the Action of Picard Group on Q(i,\surd3). https://arxiv.org/abs/1111.3778

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