On the equivariant Tamagawa number conjecture for Tate motives. II. (English) Zbl 1156.11042
This paper is a continuation of that of D. Burns and C. Greither [Invent. Math. 153, 303–359 (2003; Zbl 1142.11076)], in view of proving the Equivariant Tamagawa Number Conjecture (ETNC) for the pair \((h^0 (\text{Spec}(L)(r)),\;{\mathbb Z}[\text{Gal}(L/K)]),\) where \(r \in {\mathbb Z},\) \(L\) is an abelian number field and \(K\) a subfield of \(L.\) The ETNC was proved in op. cit. for \(r \leq 0\) for \(p \not= 2,\) the case \(p = 2\) being dealt with in [M. Flach, Contemp. Math. 358, 79–125 (2004; Zbl 1070.11025)].
To complete the proof, it is sufficient to show the compatibility between the ETNC and the functional equation of Artin \(L\)-functions. The most succinct formulation of the ETNC for Tate motives asserts the vanishing of a certain element \(T \Omega ({\mathbb Q}(r)_L, {\mathbb Z} [G]) \in K_0 ({\mathbb Z}[G], {\mathbb R}),\) where \(G = \text{Gal}(L/K).\) The functional equation is reflected by an equality \[ T \Omega ({\mathbb Q}(r)_L, {\mathbb Z}[G]) + \psi^\ast (T \Omega {\mathbb Q}(1-r)_L, {\mathbb Z} [G] ^{op})) = T \Omega^{loc} ({\mathbb Q} (r)_L, {\mathbb Z}[G]), \] where \(\psi^\ast\) is a natural isomorphism \[ K_0 ({\mathbb Z} [G]^{op},{\mathbb R}) \simeq K_0 ({\mathbb Z} [G], {\mathbb R}) \] and \(T \Omega^{loc} ({\mathbb Q}(r)_L, {\mathbb Z} [G]) \in K_0 ({\mathbb Z} [G], {\mathbb R})\) is the sum of an “analytic” element constructed from the archimedean Euler factors and epsilon constants associated to both \({\mathbb Q} (r)_L\) and \({\mathbb Q} (1-r)_L,\) of an element reflecting sign differences between regulator maps used in defining \(T \Omega (\cdot),\) and of an “algebraic” element constructed from the various realisations of \({\mathbb Q}(r)_L.\) The proof of the vanishing of \(T \Omega^{loc} ({\mathbb Q}(r)_L, {\mathbb Z}[G])\) for \(r > 0\) combines classical computations of Hasse (concerning Gauss sums) and Leopoldt-Lettl (concerning integer rings in cyclotomic fields) with a systematic use of the Iwasawa theory of complexes in the spirit of Kato-Nekovář, as well as the generalization by Perrin-Riou of the fundamental exact sequence of Coleman theory. Note that much effort is required to deal with the subtleties introduced by the prime 2.
The compatibility between the functional equation and the Tamagawa number conjecture (equivariant or not) can actually be put in the context of Kato’s “local epsilon” conjecture for a \(p\)-adic representation \(V\) of \(G_{{\mathbb Q}_p}\): if \(V\) comes from a Tate motif, Kato’s conjecture is true for a cyclotomic field \({\mathbb Q}_p (\xi_p n)\) [D. Benois and the reviewer, Ann. Sci. Éc. Norm. Supér. (4), 35, No. 5, 641–672 (2002; Zbl 1125.11351)] and more generally for an abelian extension \(L/{\mathbb Q}_p\) (this article) ; if \(V\) is any crystalline representation, it is true for \({\mathbb Q}_p (\xi_p n)\) [D. Benois and L. Berger, Comment. Math. Helv. 83, No. 3, 603–677 (2008; Zbl 1157.11041)]. See also the appendix B of A. Huber and G. Kings, [Duke Math. J., 119, 393–464 (2003; Zbl 1044.11095)] for Dirichlet motives.
To complete the proof, it is sufficient to show the compatibility between the ETNC and the functional equation of Artin \(L\)-functions. The most succinct formulation of the ETNC for Tate motives asserts the vanishing of a certain element \(T \Omega ({\mathbb Q}(r)_L, {\mathbb Z} [G]) \in K_0 ({\mathbb Z}[G], {\mathbb R}),\) where \(G = \text{Gal}(L/K).\) The functional equation is reflected by an equality \[ T \Omega ({\mathbb Q}(r)_L, {\mathbb Z}[G]) + \psi^\ast (T \Omega {\mathbb Q}(1-r)_L, {\mathbb Z} [G] ^{op})) = T \Omega^{loc} ({\mathbb Q} (r)_L, {\mathbb Z}[G]), \] where \(\psi^\ast\) is a natural isomorphism \[ K_0 ({\mathbb Z} [G]^{op},{\mathbb R}) \simeq K_0 ({\mathbb Z} [G], {\mathbb R}) \] and \(T \Omega^{loc} ({\mathbb Q}(r)_L, {\mathbb Z} [G]) \in K_0 ({\mathbb Z} [G], {\mathbb R})\) is the sum of an “analytic” element constructed from the archimedean Euler factors and epsilon constants associated to both \({\mathbb Q} (r)_L\) and \({\mathbb Q} (1-r)_L,\) of an element reflecting sign differences between regulator maps used in defining \(T \Omega (\cdot),\) and of an “algebraic” element constructed from the various realisations of \({\mathbb Q}(r)_L.\) The proof of the vanishing of \(T \Omega^{loc} ({\mathbb Q}(r)_L, {\mathbb Z}[G])\) for \(r > 0\) combines classical computations of Hasse (concerning Gauss sums) and Leopoldt-Lettl (concerning integer rings in cyclotomic fields) with a systematic use of the Iwasawa theory of complexes in the spirit of Kato-Nekovář, as well as the generalization by Perrin-Riou of the fundamental exact sequence of Coleman theory. Note that much effort is required to deal with the subtleties introduced by the prime 2.
The compatibility between the functional equation and the Tamagawa number conjecture (equivariant or not) can actually be put in the context of Kato’s “local epsilon” conjecture for a \(p\)-adic representation \(V\) of \(G_{{\mathbb Q}_p}\): if \(V\) comes from a Tate motif, Kato’s conjecture is true for a cyclotomic field \({\mathbb Q}_p (\xi_p n)\) [D. Benois and the reviewer, Ann. Sci. Éc. Norm. Supér. (4), 35, No. 5, 641–672 (2002; Zbl 1125.11351)] and more generally for an abelian extension \(L/{\mathbb Q}_p\) (this article) ; if \(V\) is any crystalline representation, it is true for \({\mathbb Q}_p (\xi_p n)\) [D. Benois and L. Berger, Comment. Math. Helv. 83, No. 3, 603–677 (2008; Zbl 1157.11041)]. See also the appendix B of A. Huber and G. Kings, [Duke Math. J., 119, 393–464 (2003; Zbl 1044.11095)] for Dirichlet motives.
Reviewer: Thong Nguyen Quang Do (Besançon)
