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Logarithmic classes of number fields. (Classes logarithmiques des corps de nombres.) (French) Zbl 0827.11064
Fix a prime $$\ell$$. Let $$K$$ be a number field with associated local field $$K_{\mathfrak p}$$ at the prime $${\mathfrak p}$$. Motivated by an explicit formula for certain Hilbert symbols one is led to consider a $$\mathbb{Z}_\ell$$-valued valuation map $$\widetilde {v}_{\mathfrak p}$$ on $$K_{\mathfrak p}$$ defined via the $$\ell$$-adic (Iwasawa) logarithm $$\text{Log}= \text{Log}_{Iw}$$. This logarithmic valuation $$\widetilde {v}_{\mathfrak p}$$ is actually proportional to the traditional valuation $$v_{\mathfrak p}$$ for $${\mathfrak p}$$ not dividing $$\ell$$. It is a $$\mathbb{Z}_\ell$$-epimorphism of the $$\ell$$-adic compactification $${\mathcal K}^\times_{\mathfrak p}:= \varprojlim K^\times_{\mathfrak p}/ K_{\mathfrak p}^{\times \ell^n}$$ of $$K^\times_{\mathfrak p}$$, with kernel given by the submodule $${\mathcal K}^*_{\mathfrak p}$$ of the cyclotomic norms in $${\mathcal K}^\times_{\mathfrak p}$$. One also defines a logarithmic (absolute) ramification index $$\widetilde {e}_{\mathfrak p}$$, the logarithmetic inertial degree $$\widetilde {f}_{\mathfrak p}$$ of $$K_{\mathfrak p}$$ and the $$\ell$$-adic degree $$\deg {\mathfrak p}$$ of the ideal $${\mathfrak p}$$. Then, by definition, $$\widetilde {v}_{\mathfrak p}=- \text{Log}|\;|_p/ \deg {\mathfrak p}$$, where $$|\;|_{\mathfrak p}$$ is the usual absolute value on $$K^\times_{\mathfrak p}$$. One defines the $$\ell$$-group of logarithmic divisors $${\mathcal D} \ell_K$$ of $$K$$ as the free $$\mathbb{Z}_\ell$$-module on the finite places $${\mathfrak p}$$ of $$K$$: $${\mathcal D}\ell_K:= \bigoplus_{{\mathfrak p}\text{ finite}} \mathbb{Z}_\ell {\mathfrak p}$$. By linear extension of the $$\ell$$-adic degree map one gets a degree map on $${\mathcal D} \ell_K$$. One writes $$\widetilde {{\mathcal D} \ell}_K$$ for the degree zero submodule of $${\mathcal D} \ell_K$$. Write $${\mathcal R}_K= \mathbb{Z}_\ell \otimes_\mathbb{Z} K^\times$$; then the $${\mathcal D} \ell_K$$-valued map $$\widetilde{\text{div}}_K: x\mapsto \sum_{{\mathfrak p}\text{ finite}} \widetilde {v}_{akp} (x){\mathfrak p}$$ maps $${\mathcal R}_K$$ onto a submodule $$\widetilde {{\mathcal P} \ell}_K$$ of $$\widetilde {{\mathcal D} \ell}_K$$. The $$\ell$$-group of logarithmic classes of $$K$$ is defined as the quotient $$\widetilde {{\mathcal C} \ell}_K:= \widetilde {{\mathcal D} \ell}_K/ \widetilde {{\mathcal P} \ell}_K$$. By class field theory it may be identified with the Galois group $$\text{Gal} (K^{lc}/ K^c)$$, where $$K^c$$ is the cyclotomic $$\mathbb{Z}_\ell$$-extension of $$K$$, and $$K^{lc}$$ is the maximal abelian locally cyclotomic pro-$$\ell$$- extension of $$K$$. $$\widetilde {{\mathcal C} \ell}_K$$ is conjectured to be finite (generalized Gross conjecture). Let $$\delta_K$$ be the dimension of the free $$\mathbb{Z}_\ell$$-quotient $$\widetilde {{\mathcal C} \ell}_K/ \widetilde {{\mathcal C} \ell}_K^{\text{tor}}$$. For an extension $$L/K$$ of number fields one has several compatibilities for the $${\mathcal R}$$’s and $${\mathcal D} \ell$$’s, and for a Galois extension $$L/K$$ one can describe the action of $$\text{Gal} (L/K)$$ on $${\mathcal D} \ell_L$$, etc.
Next, define, for a set $$S$$ of non-archimedean primes of $$K$$ the $$\ell$$- group of logarithmic $$S$$-units $$\widetilde {\mathcal E}_K^S$$ by $$\widetilde {\mathcal E}^S_K:= \{\varepsilon\in {\mathcal R}_K\mid \widetilde {v}_{\mathfrak p} (\varepsilon) =0$$, $$\forall {\mathfrak p}\not\in S\}$$. For $$S= \emptyset$$ one writes $$\widetilde {\mathcal E}_K$$ for $$\widetilde {\mathcal E}_K^\emptyset$$, the $$\ell$$-group of logarithmic units of $$K$$. These coincide with the cyclotomic norms. Write $$r_K$$ (resp. $$c_K$$) for the number of real (resp. complex) places of $$K$$. Then $$\widetilde {\mathcal E}_K \simeq \mu_K \times \mathbb{Z}_\ell^{r_K+ c_K+ \delta K}$$, and assuming the validity of the generalized Gross conjecture, for $$s_K= \# (S)< \infty$$, one has $$\widetilde {\mathcal E}_K^S \simeq {\mathcal E}_K^S \simeq \mu_K \times \mathbb{Z}_\ell^{r_K+ c_K+ s_K -1}$$, where $${\mathcal E}^S_K$$ is the $$\ell$$-group of ordinary $$S$$-units of $$K$$. Without difficulty one obtains a logarithmic analogue of Herbrand’s representation theorem on characters.
One can also develop a logarithmic analogue of Chevalley’s formula for his ‘classes ambiges’. It gives an exact sequence for $$\widetilde {{\mathcal C} \ell}^G_L$$, where $$G= \text{Gal} (L/K)$$ for the Galois extension $$L/K$$, in particular, for $$L/K$$ a primitively ramified cyclic $$\ell$$-extension and assuming the truth of the Gross conjecture, one gets an explicit formula for $$\#(\widetilde {{\mathcal C} \ell}^G_L)$$ as the product of $$\#(\widetilde {{\mathcal C} \ell}_K)$$, the local degrees of $$L/K$$ at infinity and the logarithmic ramification indices at the finite places of $$L/K$$, divided by the product of $$[L^c: K^c]$$ and $$(\widetilde {\mathcal E}_K: \widetilde {\mathcal E}_K \cap {\mathcal N}_{L/K})$$. A similar result can be derived for the number of $$G$$- invariant logarithmic $$S$$-classes.
One defines the $$\ell$$-group of logarithmic genera of the finite extension $$L/K$$ as the Galois group $$\widetilde {{\mathcal G} \ell}_{L/K}= \text{Gal} (L^{lc}\cap LK^{ab}/ L^c)$$. One gets the remarkable result for $$\# (\widetilde {{\mathcal G} \ell}_{L/K})$$: a formula analogous to the one for $$\#(\widetilde {{\mathcal C}\ell}^G_L)$$ mentioned above, but holding for any finite extension $$L/K$$ of number fields and without assuming the Gross conjecture. For an abelian $$\ell$$-extension $$L/K$$ the formula for $$\#(\widetilde {{\mathcal G} \ell}_{L/K})$$ is the same as the one for $$\#(\widetilde {{\mathcal C} \ell}^G_L)$$ with $${\mathcal N}_{L/K}$$ replaced by $${\mathcal N}_{L/K}^{\text{loc}}$$. Finally, one may define the $$\ell$$-group of central logarithmic classes of the (Galois) $$\ell$$-extension $$L/K$$ of number fields as the biggest quotient $${}^G\widetilde {{\mathcal C} \ell}_L= \widetilde {{\mathcal C} \ell}_L/ \widetilde {{\mathcal C} \ell}_L^{I_G}$$ of $$\widetilde {{\mathcal C} \ell}_L$$ with trivial $$G$$-action. Again, one may derive a formula for $$\#({}^G \widetilde {{\mathcal C} \ell}_L)$$.

##### MSC:
 11R23 Iwasawa theory 12J20 General valuation theory for fields
##### Keywords:
logarithmic classes; logarithmic genera
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##### References:
  Federer, L.J. & Gross, B.H. (with an appendix by Sinnot, W.)” Regulators and Iwasawa modules, Invent. Math.62 (1981), 443-457. · Zbl 0468.12005  Jaulent, J.-F., L’arithmétique des l-extensions, (Thèse d’Etat) Pub. Math. Fac. Sci. Besançon Théor. Nombres 1984-85 & 1985-86, fasc. 1 (1986), 1-349. · Zbl 0601.12002  Tate, J., Les conjectures de Stark sur les fonctions L d’Artin en s = 0, Prog. in Math.47, (1984), Birkhäuser. · Zbl 0545.12009
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