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On a conjecture of Brumer. (English) Zbl 0719.11082
In a series of papers, Iwasawa proposed and studied analogues for algebraic number fields $$F$$ of constructions due to Weil for curves over finite fields. In particular he conjectured an analogue of Weil’s theorem relating the zeta function of a nonsingular curve over a finite field with the characteristic polynomial of its Frobenius automorphism.
Iwasawa’s conjecture was proved for $$F=$$ the rationals by B. Mazur and the author in 1984 [Invent. Math. 76, 179–330 (1984; Zbl 0545.12005)]. In 1990 [ibid. 131, No. 3, 493–540 (1990; Zbl 0719.11071)] the author proved Iwasawa’s conjecture for an arbitrary totally real field and in this paper he uses this to recover a version of Brumer’s conjecture connected with a classical theorem of Stickelberger. The key part of this paper is the development of a technique introduced in the author’s 1990 paper of adjoining an auxiliary variable to the $$p$$-adic $$L$$-functions.

MSC:
 11S40 Zeta functions and $$L$$-functions 11G20 Curves over finite and local fields 11R23 Iwasawa theory
Citations:
Zbl 0545.12005; Zbl 0719.11071
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