On the Brumer-Stark conjecture. (English) Zbl 1525.11128

Dasgupta, Samit; Kakde, Mahesh

Ann. Math. (2) 197, No. 1, 289-388 (2023).

L. Stickelberger [Math. Ann. 37, 321–367 (1890; JFM 22.0100.01)] computed the ideal factorization of Gauss sums in cyclotomic fields. When \(K\) is an abelian extension of \(\mathbb Q\), Stickelberger’s result yields an element of the group ring of \(\mathrm{Gal}(K/\mathbb Q)\) that annihilates the ideal class group of \(K\). A.Brumer gave an interpretation in terms of special values of zeta functions and conjectured a similar result for abelian extensions of totally real number fields. J.Tate combined this conjecture with conjectures of H. Stark on existence of units related to values of \(L\)-series, yielding what is known as the Brumer-Stark conjecture. This conjecture, up to the 2-part, is proved in the present paper.
Here is a more precise description of the result. The well-written introduction to the paper gives much more detail and motivation and is highly recommended. Let \(F\) be a totally real number field, let \(H\) be a finite abelian extension of \(F\) that is also a CM field, and let \(G=\mathrm{Gal}(H/K)\). Let \(S, T\) be disjoint finite sets of places of \(F\) with \(S\) containing the set \(S_{\infty}\) of infinite places of \(F\). For a character \(\chi\) of \(G\), the \(S\)-depleted, \(T\)-smoothed \(L\)-function of \(\chi\) is \[ L_{S,T}(\chi, s)= L(s, \chi)\prod_{\mathfrak p \in S\setminus S_{\infty}} (1-\chi(\mathfrak{p})N\mathfrak{p}^{-s}) \prod_{\mathfrak p \in T} (1-\chi(\mathfrak{p})N\mathfrak{p}^{1-s}). \] The Stickelberger element \(\Theta_{S,T}\in \mathbb Q[G]\) satisfies \[ \chi(\Theta_{S,T})=L_{S,T}(\chi^{-1}, 0) \] for all \(\chi\). Now assume \(S\) also contains the set of primes of \(F\) that ramify in \(H/F\), and assume that \(T\) satisfies the following condition: if \(\zeta\in H\) is a root of unity with \(\zeta\equiv 1\pmod {\mathfrak p}\) for all primes \(\mathfrak p\) of \(H\) above primes in \(T\), then \(\zeta=1\). Then \(\Theta_{S,T}\in \mathbb Z[G]\). Let \(Cl^T(H)\) be the ray class group of \(H\) with conductor equal to the product of the primes of \(H\) above those in \(T\). The main theorem of the paper is the following: \[ \Theta_{S,T}\in \mathrm{Ann}_{\mathbb Z[G]}(Cl^T(H))\otimes \mathbb Z[\frac12]. \] In fact, the authors prove a stronger result, conjectured by M.Kurihara, giving a twist of \(\Theta_{S,T}\) as an element of the Fitting ideal of the minus part of the dual of \(Cl^T(H)\), and they obtain an exact formula for this Fitting ideal in terms of Stickelberger elements. From this, they deduce the prime-to-2 part of K. Rubin’s higher rank generalization [Ann. Inst. Fourier 46, No. 1, 33–62 (1996; Zbl 0834.11044)] of the Brumer-Stark conjecture.
The starting point for the proofs is the method used by K.Ribet to prove the converse of Herbrand’s theorem and extended to group ring valued Hilbert modular forms by A.Wiles in his work on the Iwasawa Main Conjecture for totally real fields. However, a key point of the present paper is that the congruences constructed between cusp forms and Eisenstein series are stronger than might be expected and result from trivial zeros of \(p\)-adic \(L\)-functions.

Reviewer: Lawrence C. Washington (College Park)

MSC:

11R23	Iwasawa theory
11R29	Class numbers, class groups, discriminants
11R42	Zeta functions and \(L\)-functions of number fields

Keywords:

class groups; Fitting ideals; Stark conjectures; Hilbert modular forms

Citations:

Zbl 0834.11044; JFM 22.0100.01

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