The Brumer-Stark Conjecture
This chapter is devoted to the explanation and, in special cases, the proof of a conjecture which generalizes the famous theorem of Stickelberger about the structure of the class group of cyclotomic number fields. This important conjecture, due to A. Brumer and H. Stark, is unresolved in the number field case. The analogous conjecture in function fields is now a theorem due to the efforts of J. Tate and P. Deligne. A short time after Deligne completed Tate’s work on this result, D. Hayes found a proof along completely different lines. We will give a proof for the cyclotomic function fields introduced in Chapter 12. We will do so by using a method of B. Gross which combines the approaches of Tate and Hayes as they apply in this relatively simple special case. The use of 1-motives, which is essential in Deligne’s work, will not be needed here.
KeywordsPrime Ideal Galois Group Function Field Prime Decomposition Effective Divisor
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