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Generalising Stickelberger: Annihilators (and more) for class groups of number fields

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Stickelberger’s Theorem (from 1890) gives an explicit ideal in the Galois group-ring which annihilates the minus-part of the class group of a cyclotomic field. In the 1980s Tate and Brumer proposed a generalisation (the `Brumer-Stark conjecture’) for any abelian extension of number fields K/k, with K of CM type and k totally real.

Both the theorem and the conjecture leave certain questions unanswered: Is the (generalised) Stickelberger ideal the full annihilator, the Fitting ideal or what? And, at a more basic level, what can we say in the plus part, e.g. for a real abelian field? I shall discuss possible answers, some still conjectural, to pieces of these puzzles, using two new p-adic ideals of the group ring.

This talk is part of the Number Theory Seminar series.

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