On a conjecture of Brumer. (English) Zbl 0719.11082

Wiles, A.

Ann. Math. (2) 131, No. 3, 555-565 (1990).

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.

Reviewer: A. R. Rajwade (Chandigarh)

MSC:

11S40	Zeta functions and \(L\)-functions
11G20	Curves over finite and local fields
11R23	Iwasawa theory

Keywords:

Stickelberger theorem; Iwasawa’s conjecture; Brumer’s conjecture; p-adic L-functions

Citations:

Zbl 0545.12005; Zbl 0719.11071