A conjectural product formula for Brumer-Stark units over real quadratic fields
Brumer-Stark units are conjectural elements in global fields whose role is to give algebraic meaning to special values of partial \(\zeta\)-functions. More precisely, the Brumer-Stark conjecture asserts that when \(H/F\) is an abelian extension of global fields, there should exist elements of \(H\) whose valuations are connected in a precise way with the associated partial \(\zeta\)-functions evaluated at \(0\). Those conjectural elements are the Brumer-Stark units. A version of the conjecture given in terms of special values of \(L\)-functions is often treated instead. In the function field case, the conjecture is a theorem thanks to work of Tate and Deligne. Significant progress has been made in the number field case: Greither and Popescu have proven it away from the \(2\)-primary component assuming the vanishing of certain classical Iwasawa \(\mu\)-invariants. The article under review presents two conjectural formulas involving the Brumer-Stark units in the case where \(F\) is a real quadratic field and \(H\) is a narrow ray class field of \(F\). The first formula, the ``product formula'' of the title, appears to be new and expresses a \textit{ratio} of Brumer-Stark units in terms of an infinite product. The product is taken over certain elements of \(F\) that map into a particular Shintani domain. The author points out that when \(F\) has narrow class number \(1\), one of the Brumer-Stark units appearing in the ratio is \(1\) and so the product formula is a formula for the remaining Brumer-Stark unit alone. The second conjectural formula, which the author presented for arbitrary totally real fields \(F\) in [Duke Math. J. 143, No. 2, 225--279 (2008; Zbl 1235.11102)], expresses a Brumer-Stark unit (not a ratio) in terms of a \(p\)-adic integral over the unit group of the completion of \(F\) at a prime \(p\) that is inert in \(F/\mathbb{Q}\). (We suppress, for brevity, the relationship of \(p\) to the extension \(H/F\) and to other arithmetic data.) It is worth noting that the \(p\)-adic integral involves values of partial \(\zeta\)-functions. This is interesting in light of the presence of such integrals -- again involving partial \(\zeta\)-functions -- in the work of P. Cassou-Noguès and Deligne-Ribet. The main theorem of the article establishes that the integral formula implies the product formula. The author remarks that certain parts of the proof are in the spirit of work of Hayes, who considered a similar formula for function fields. The reader wishing to gain useful background may wish to consult [\textit{D. R. Hayes}, Invent. Math. 94, No. 3, 505--527 (1988; Zbl 0666.12009)]. Another important aspect of the proof is the use of formulas of Shintani on the values at \(0\) of functions arising via analytic continuation from certain infinite series. For example, a role is played by the series \(\sum_{m,n=0}^\infty \mathrm{N}(\gamma + m\rho + n\tau)^{-s}\) where \(\gamma,\rho,\tau \in (\mathbb{R}^{>0})^2\) and \(\mathrm{N} : (\mathbb{R}^{>0})^2 \rightarrow \mathbb{R}^{>0}\) is the function that takes the product of the two entries. The value at \(0\) of the analytic continuation of this series involves Bernoulli polynomials, whose distribution relations feature in the proof of the theorem. Along with the articles mentioned above, the article under review will be of interest to anyone wishing to gain a more analytic perspective on Brumer-Stark units. It is largely self-contained in that the results it uses are stated precisely. It would be interesting to investigate how the product formula might generalize when the base field is an arbitrary totally real field.
scientific article; zbMATH DE number 6126841