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The various faces of a pairing on p-units
The talk will begin with a review of some central objects in algebraic number theory, such as class groups, cyclotomic fields, and irregular primes. We will then discuss a certain pairing on p-units of the cyclotomic field K of pth roots of unity, for an irregular prime p. This pairing takes values in essentially the class group of K modulo p. McCallum and I have conjectured that it is surjective, which I have now proven for p < 1000. We will discuss a variety of applications of this pairing: to the structure of class groups of Kummer extensions, to the K-theory of integer rings, to a Lie algebra associated to the geometric fundamental group of the projective line minus three points, and to cuspidal eigenforms congruent to Eisenstein series modulo p.
Friday, January 23, 2004, 245 Altgeld Hall, 4:00 p.m.
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