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Mathematics Colloquium


Date:  Tuesday, April 17, 2012

Title:  Small generators for S-arithmetic groups

Abstract:  Pell's equation for a square-free integer d > 0 is x^2 - d y^2 = 1. One expects (though it has not been proved) that there is no polynomial in d which bounds the sizes of the minimal positive solutions x and y to this equation. This illustrates the fact that in general, one expects any generating set for the unit group of a number field to include a unit of large height relative to the discriminant of the field. A surprising discovery of H. W. Lenstra, Jr., was that one can find generators of small height for groups of S-units. These groups lie between the units and the full multiplicative group of the field. I will discuss joint work with Matt Stover on generalizing Lenstra's results from S-units to the S-integral points of linear algebraic groups. This has applications to finding presentations for such groups and to the congruence subgroup problem.

Speaker:  Ted Chinburg
Institution:  Univ of Penn


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