Brumer-Stark units and explicit class field theory (Q6566410)

Let \(F\) be a totally real field of degree \(n\) over \(\mathbf Q\), let \(H\) be a finite abelian extension of \(F\) that is a CM field, and let \(G=\text{ Gal}(H/F)\). Let \(S\) and \(T\) be finite nonempty disjoint sets of places of \(F\) such that \(S\) contains the real places of \(F\) and the finite primes of \(F\) that ramify in \(H\). For any character \(\chi :G\rightarrow {\mathbf C^*}\) we have the Artin \(L\)-function\N\[\NL_S(\chi,s)=\prod_{\mathfrak p \notin S}\frac{1}{1-\chi(\mathfrak p)\text {N}\mathfrak p^{-s}},\hspace{.25 in} \text{Re}(s)>1,\N\]\Nand the function \N\[\NL_{S,T}(\chi,s)=L_S(\chi,s)\prod_{\mathfrak p \in T}(1-\chi(\mathfrak p)\text {N}\mathfrak p^{1-s}),\N\]\Nthe latter of which can be analytically continued to a holomorphic function on the complex plane. Define the Stickelberger elements\N\[\N\Theta^{H/F}_S(s), \hspace{.25 in} \Theta^{H/F}_{S,T}(s) \in \mathbf{C}[G]\N\]\Nby (dropping the superscript \(H/F\)) \N\[\N\chi(\Theta_S(s))=L_S(\chi^{-1},s), \hspace{.25 in} \chi( \Theta_{S,T}(s))=L_{S,T}(\chi^{-1},s)\N\]\Nfor all characters \(\chi\) of \(G\). Then \(\Theta_S:=\Theta_S(0)\in {\mathbf Q}[G]\), and, under a mild condition on \(T\) which the authors assume throughout the present paper, \(\Theta_{S,T}:=\Theta_{S,T}(0)\in {\mathbf Z}[G]\). Now suppose \(\mathfrak p\not\in S\cup T\) is a prime of \(F\) that splits completely in \(H\), and let \(U^-_{\mathfrak p}\subset H^*\) be the group of elements \(u\) such that \(|u|_v=1\) for all places \(v\) of \(H\) not lying above \(\mathfrak p\), including the complex places. Let \(U^-_{{\mathfrak p},T}\subset U^-_{\mathfrak p}\) be the subgroup of elements \(u\) such that \(u\equiv 1\pmod{ {\mathfrak q}\mathcal O_H}\) for all \({\mathfrak q}\in T\). In [Ann. Math. (2) 197, No. 1, 289--388 (2023; Zbl 1525.11128)] the authors of the present paper obtained a proof of the following conjecture away from 2.\N\NConjecture 1.1 (Tate-Brumer-Stark) Fix a prime \(\mathfrak P\) of \(H\) above \(\mathfrak p\). There exists an element \(u_{\mathfrak p}\in U^-_{{\mathfrak p},T}\) such that \N\[\N\text{ord}_G(u_{\mathfrak p}):=\sum_{\sigma \in G}\text{ord}_{\mathfrak P}(\sigma(u_{\mathfrak p}))\sigma^{-1}=\Theta_{S,T}\N\]\Nin \({\mathbf Z}[G]\).\N\NIn the present paper the authors generalize their work on Conjecture 1.1 away from 2 to obtain a proof of their first main result, Theorem 1.4. To state this theorem keep the notation above and let \(S_{\mathfrak p}=S\cup \{\mathfrak p\}\). Let \(L\) be a finite abelian CM extension of \(F\) containing \(H\) that is ramified over \(F\) only at the places in \(S_{\mathfrak p}\). Let \({\mathfrak g}=\text{Gal}(L/F)\) and \(\Gamma =\text{Gal}(L/H)\), so \({\mathfrak g}/\Gamma \cong G\). Let \(I\) denote the kernel of the canonical projection \N\[\N\text{Aug}^{\mathfrak g}_G: {\mathbf Z}[\mathfrak g]\twoheadrightarrow {\mathbf Z}[G].\N\]\NSince \({\mathfrak p}\) splits completely in \(H\) we have \(\Theta^{L/F}_{S_{\mathfrak p},T}\in I\). Now let \N\[\N\text{rec}_{\mathfrak P}: H^*_{\mathfrak P}\rightarrow \Gamma\N\]\Ndenote the composition of the inclusion \(H^*_{\mathfrak P}\hookrightarrow {\mathbf A}^*_H\) with the global Artin reciprocity map \({\mathbf A}^*_H\twoheadrightarrow \Gamma\). We have the following conjecture.\N\NConjecture 1.3 (Gross) Define \N\[\N\text{rec}_G(u_{\mathfrak p})=\sum_{\sigma\in G}(\text{rec}_{\mathfrak P}\sigma(u_{\mathfrak p})-1)\tilde{\sigma}^{-1}\in I/I^2,\N\]\Nwhere \(\tilde{\sigma}\in {\mathfrak g}\) is any lift of \(\sigma \in G\). Then \N\[\N\text{rec}_G(u_{\mathfrak p})\equiv \Theta^{L/F}_{S_{\mathfrak p},T}\N\]\Nin \(I/I^2\).\N\NNow let \(p\) be the rational prime below \({\mathfrak p}\), and assume \(p\not = 2\). The authors prove the following theorem.\N\NTheorem 1.4 Let \(p\) be an odd prime, and suppose that \({\mathfrak p}\) lies above \(p\). Gross's Conjecture 1.3 holds in \((I/I^2)\otimes {\mathbf Z}_p\).\N\NThe second main result of the present paper, Theorem 1.6, provides a proof of the following conjecture stated in [\textit{S. Dasgupta}, Duke Math. J. 143, No. 2, 225--279 (2008; Zbl 1235.11102)]. We continue with the notation established above. Let \(\mathfrak n\) be a nonzero ideal of \({\mathcal O}_F\) and denote by \(F(\mathfrak n)\) the narrow ray class field of \(F\) associated to the conductor \(\mathfrak n\). Let \(H\) be the maximal CM subfield of \(F(\mathfrak n)\) in which \(\mathfrak p\) splits completely. Let \(f\) be the order of \(\mathfrak p\) in the narrow ray class group of conductor \(\mathfrak n\), and let \({\mathfrak p}^f=(\pi)\) for a totally positive element \(\pi \in 1+\mathfrak n\). Assume that \(T\) contains a prime whose norm is a rational prime in \(\mathbf Z\). Now let \({\mathcal O}_{\mathfrak p}\) denote the completion of \({\mathcal O}_F\) at \(\mathfrak p\), let \({\mathbf O}={\mathcal O}_{\mathfrak p}-\pi{\mathcal O}_{\mathfrak p}\), and let \(\mathfrak b\) denote an integral ideal of \({\mathcal O}_F\) that is relatively prime to \(\mathfrak n\). Then for a certain Shintani domain \(\mathcal D\) we have a totally positive unit \(\epsilon({\mathfrak b},{\mathcal D}, \pi)\in {\mathcal O}^*_F\) congruent to 1 modulo \({\mathfrak n}\), and a \({\mathbf Z}\)-valued measure \(\nu({\mathfrak b}, \mathcal D)\) on \({\mathcal O}_{\mathfrak p}\).\N\NConjecture 2.1 Let \(\sigma_{\mathfrak b}\in \text{ Gal}(H/F)\) denote the Frobenius element associated with \(\mathfrak b\). Let \(\mathfrak P\) and \(u_{\mathfrak p}\) be as in Conjecture 1.1, and consider \(H\) as a subfield of \(F_{\mathfrak p}\) via \(H\subset H_{\mathfrak P}\cong F_{\mathfrak p}\). We then have \[\sigma_{\mathfrak b}(u_{\mathfrak p})=\epsilon({\mathfrak b},{\mathcal D}, \pi)\cdot \pi^{\zeta_{S,T}(F(\mathfrak n)/F,{\mathfrak b},0)}\times \int_{\mathbf O}xd\nu({\mathfrak b}, \mathcal D,x)\in F^*_{\mathfrak p}.\hspace{.5 in}(14)\]\N\NTheorem 1.6 Let \(p\) denote the rational prime below \(\mathfrak p\). Suppose that\N\begin{itemize}\N\item[\((\ast)\)] \(p\) is odd and \(H\cap F(\mu_{p^\infty})\subset H^+\), the maximal totally real subfield of \(H\).\N\end{itemize}\NThen equation (7), or more precisely its generalization (14) to the general setting, holds up to multiplication by a root of unity in \(F^*_{\mathfrak p}\).\N\NBy a previous result of the first named author, Theorem 1.4 implies Theorem 1.6. As a consequence of the latter theorem, the authors of the present paper obtain an effective method of generating the maximal abelian extension of any totally real field.