Generators for groups of units in orders in rational quaternion algebras and a problem in number theory

09/07/2010 - 12:15pm
09/07/2010 - 1:10pm
Mehrdad Shahshahani (Institute for Research in Fundamental Sciences (IPM), Tehran, Iran)

Groups of units in orders in quaternion algebras de finite over Q and split over R are simple examples of arithmetic Fuchsian groups. A natural question is how to explicitly calculate a set of generators for such a group. The problem can viewed as a noncommutative generalization of finding the generator for the solutions of Pell's equation (x^2 - ay^2 = 1) where the number of variables is now four (x_0^2 - ax_1^2 - px_2^2 + apx_3^2 = 1). The solutions to this equation form a noncommutative group commensurable with the group of units in an order in a quaternion algebra. While Schur's theorem provides a bound for the "size" of the generator for Pell's equation, no similar result is known for this case. Obtaining numerical data for these arithmetic Fuchsian groups presents interesting problems. The purpose of this work is to obtain numerical data for the "size" of a set of generators for such groups. Just as in the case of Pell's equations the solutions exhibit strong fluctuations. The theoretical problem is unresolved but may lend itself to investigation using automorphic L-functions. A paper on this work will appear shortly in the International Journal of Number Theory and this presentation is based on the work of my student Majid Jahangiri.

Millikan 208 (Pomona College)
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