Étale cohomology, cofinite generation, and \(p\)-adic \(L\)-functions (Q5962644)

Let \(p\) be prime, let \(k\) be a number field, and let \(E\) be a finite extension of \(\mathbb Q_p\) with valuation ring \(\mathcal O_E\). Let \(\eta\) be the character of an Artin representation of \(G_k=\text{Gal}(\overline{k}/k)\) defined over \(E\). There is an \(\mathcal O_E\)-lattice \(M(E,\eta)\) on which \(G_k\) acts such that \(M(E,\eta)\otimes_{\mathcal O_E} E\) gives this representation. Let \(W(E, \eta)=M(E,\eta)\otimes_{\mathbb Z_p}\mathbb Q_p/\mathbb Z_p\). Let \(S\) be a finite set of primes of \(k\) that includes all primes above \(p\) and those at which \(\eta\) is ramified, and let \(\mathcal O_{k, S}\) be the ring of integers of \(k\) with the primes in \(S\) inverted. The authors give a subset of \(E\) containing \(\mathcal O_E\) and define twists \(W(E,\eta)\langle 1-e\rangle\) for \(e\) in this set. They prove that the étale cohomology groups \(H^i(\mathcal O_{k, S}, W(E,\eta)\langle 1-e\rangle)\) are cofinitely generated when \(i\geq 0\), and there is a constant \(D=D(S, \eta, k)\) that is independent of \(e, E\), and the choice of \(M(\eta, E)\) such that each of these cohomology groups is cogenerated by at most \(D\) elements. Let \(r_i\) be the \(\mathcal O_E\)-corank of \(H^i(\mathcal O_{k, S}, W(E,\eta)\langle 1-e\rangle)\). They express \(r_0-r_1+r_2\) in terms of the behavior at the Archimedean primes. Moreover, \(r_0=0\) if \(e\neq 1\), and \(r_0\) is the multiplicity of the trivial character in \(\eta\) when \(e=1\). When \(p\neq 2\), the group \(H^2(\mathcal O_{k, S}, W(E,\eta)\langle 1-e\rangle)\) is \(p\)-divisible. When \(p=2\), the group is \(2\)-divisible plus an elementary \(2\)-group.NEWLINENEWLINENow assume that \(k\) is totally real and that \(\eta\) is even. Let \(L_{p, S}(s, \eta, k)\) be the \(p\)-adic \(L\)-function. For each \(i\geq 0\), \(H^i(\mathcal O_{k, S}, W(E,\eta)\langle 1-e\rangle)\) is finite for all but finitely many \(e\). The authors prove that the following are equivalent: (a) \(L_{p, S}(e, \eta, k)\neq 0\), (b) \(H^1(\mathcal O_{k, S}, W(E,\eta)\langle 1-e\rangle)\) is finite, (c) \(H^2(\mathcal O_{k, S}, W(E,\eta)\langle 1-e\rangle)\) is finite, (d) \(H^2(\mathcal O_{k, S}, W(E,\eta)\langle 1-e\rangle)\) is trivial. If these hold (and, when \(p=2\), if we assume a weak form of the Main Conjecture of Iwasawa theory), then NEWLINE\[NEWLINE | L_{p, S}(e, \eta, k))|_p = \left(\frac{\# H^0(\mathcal O_{k, S}, W(E,\eta)\langle 1-e\rangle)}{\# H^1(\mathcal O_{k, S}, W(E,\eta)\langle 1-e\rangle)} \right)^{1/[E : \mathbb Q_p]}. NEWLINE\]NEWLINE This result is used to give a short proof of the equivariant Tamagawa number conjecture at negative integers for Artin motives of the appropriate parity over a totally real number field with coefficients in a maximal order. This extends work of \textit{A. Huber} and \textit{G. Kings} [Duke Math. J. 119, No. 3, 393--464 (2003; Zbl 1044.11095)] for Dirichlet motives over \(\mathbb Q\) and \(p\) odd. The authors also give a proof of a generalization of a conjecture of \textit{J. Coates} and \textit{S. Lichtenbaum} [Ann. Math. (2) 98, 498--550 (1973; Zbl 0279.12005)].