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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
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References:
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[3] 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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