On the Brumer--Stark conjecture

Abstract

In the setting of Brumer-Stark, ρ 1 is the trivial character and ρ 2 is a totally odd character χ. When p is odd, χ(c) = −1 ≡ 1 (mod p), where c denotes complex conjugation. Ribet's method therefore carries through. However when p = 2, we may have χ ≡ 1 (mod 2), and a new construction is necessary. This issue was handled in our paper [12], whose main theorem is a general version of Ribet's Lemma that applies in the residually indistinguishable case.The next issue at p = 2 concerns the Ritter-Weiss module ∇ Σ ′ Σ (H) appearing in our previous work. It turns out that this module is not what appears in the Galois representations that we study locally at 2. We must alter the module ∇ Σ ′ Σ (H) by modifying the local conditions at the archimedean places. On the analytic side this modification is reflected in the power of 2 in the denominator of the constant terms of Eisenstein series for F .These modifications have an important effect on our constructions with group-ring valued Hilbert modular forms. In the case that the extension H/F is not ramified at any primes above 2, the set Σ that one would naturally like to take is empty; however it is an important assumption in our study of Ritter-Weiss modules that Σ is non-empty. One is therefore led to introduce an auxiliary prime l that does not divide the conductor of the extension H/F into the set Σ. The inclusion of this prime leads to the presence of oldforms in our constructions with Hilbert modular forms, which in turn yields a Hecke algebra T that is not reduced. Working with a non-reduced Hecke algebra involves new ideas to show that the Galois representations we construct have the desired properties. Therefore, the main technical contribution of this paper is the introduction of our new generalized Ritter-Weiss module ∇ Σ ′ Σ (H) in which the depletion set Σ need not contain the archimedean primes, as well as a delicate handling of the constructions with modular forms and Galois representations that ensue; the connection between our Ritter-Weiss modules and Galois cohomology is handled in a more conceptual way than in [11], and requires new results on class formations.We conclude the introduction by stating precisely our main arithmetic results. First we recall the statement of the Brumer-Stark conjecture, and next we describe our stronger results involving Ritter-Weiss modules.

Fitting ideals and various notions of equivalence for modules

“…Let us stress however that the results in [10] still concern the dualized module Cl T,∨,− K and assume ETNC. Finally, Dasgupta and Kakde [4] proved the same formula on the Fitting ideal of Cl T,∨,− K without assuming ETNC. In a very recent paper [1], Atsuta and the second author determine the Fitting ideal of non-dualized T -smoothed class group Cl T,− K , still assuming the relevant instance of ETNC to hold.…”

“…Before explaining our new notion of equivalence more closely, we quickly review the shape of known formulas for the Fitting ideals of arithmetic modules. In the previous work [4,6,10], etc., it is observed that the Fitting ideals of class groups can be described as a product θ J , where J is the so-called algebraic factor and θ is the analytic factor. Roughly speaking, this implies that the computation of the Fitting ideals of class groups splits up into determining the analytic factor θ , and the algebraic factor J , as two separate tasks.…”

Modular algorithms for Gross-Stark units and Stark-Heegner points

“…Note that these are all G-conjugate: σ(u A ) = u AAσ . The Brumer-Stark conjecture, proven up to powers of 2 in [DK23], implies that u e A , where e = #µ(H), gives an element of O H [1/p] × . More precisely, there exists an element ∈ O H [1/p] × such that ⊗ 1 = e • u and such H( e √ )/F is an abelian extension.…”

“…His conjecture related special values of derivatives of p-adic L-functions to local norms of Gross-Stark units, and was proved in [DDP11]. Recent work of Dasgupta and Kakde [DK23] on the Brumer-Stark conjecture refines this by removing the norm. The computation of Gross-Stark units over quadratic fields has been studied in [TY13] when p splits in F , and [Das07], [DK21], and [FL22] for p inert.…”

Comparing two formulas for the Gross–Stark units

“…This unit has P-order equal to the value of a partial zeta function at 0 for a prime ideal P, of H, lying above p. The unit u is only non-trivial when F is totally real and H is totally complex containing a complex multiplication (CM) subfield, we assume this for the remainder of the paper. The ground-breaking work of Dasgupta-Kakde in [7] has shown that the Brumer-Stark conjecture holds away from 2.…”