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On the special values of Artin \(L\)-functions for dihedral extensions. (English) Zbl 1322.11111

The author studies special values at negative integers of Artin \(L\)-functions attached to dihedral extensions. Let \(p\) and \(l\) be two distinct odd prime numbers. Let \(L^+\) be a totally real dihedral extension of a totally real field \(F^+\), \(\mathrm{Gal}(L^+ / F^+ ) \cong D_{2l}\). For a totally positive algebraic number \(r \in F^+\), let \(L = L^+ (\sqrt{-r})\) and \(F = F^+ (\sqrt{-r})\). Take a character \(\chi\) of \(\mathrm{Gal}(L/F^+)\) and denote by \(\mathcal L(L/F^+ , \chi, s)\) the corresponding Artin \(L\)-function, with \(d_\chi = [\mathbb Z_p [Im(\chi)]:\mathbb Z_p]\). One says that \(\chi\) is even if it is the inflation of a character of \(\mathrm{Gal}(L^+ / F^+ )\), odd if it is the product of an even character with the inflation of the non trivial character of \(\mathrm{Gal}(F/F^+ )\).
The main theorem states that for \(n \geq 2\) and for an irreducible character \(\chi\) having the same parity as \(n\), one has a \(p\)-adic equivalence \[ \mathcal L(L/F^+ , \bar{\chi}, 1 - n )^{\chi (1)d_ \chi}\sim_p\frac{\#K_{2n-2}(O_L)^\chi_{\mathrm{tors}}}{\#K_{2n-1}(O_L)^\chi_{\mathrm{tors}}}.\tag{\(*\)} \] Here the \(\chi\)-part \(M^\chi\) of a module \(M\) is defined as \(M^\chi=e_\chi(M \otimes\mathbb Z_p [Im(\chi)])\), where \(e_\chi=\frac{\chi(1)}{2l} \sum_{g\in D_{2l}} \chi(g^{-1})g\).
The proof deals separately with the irreducible representations of \(\mathrm{Gal}(L^+/F^+)\) (which are known) according to their dimensions. In dimension 2, they identify \(\mathrm{Gal}(L^+/F^+)\) with \(D_{2l} =\langle a, b \rangle\), \(a^l=b^2=1\), and “descend” to the field \(K^+\) fixed by \(a\) in \(L^+\); they apply then the formula giving \(\mathcal L(L/K^+\), \(\varphi^{-1} , 1 - n )^{d \varphi}\) (where \(\varphi\) is a character of \(\mathrm{Gal}(L/K^+)\) having the same parity as \(n\)) similarly to (\(*\)), which was shown by Kolster, Fleckinger and the reviewer [M. Kolster et al., Duke Math. J. 84, No. 3, 679–717 (1996; Zbl 0863.19003)] in their proof of the Lichtenbaum conjecture for abelian number fields. In dimension 1, the authors proceed analogously with \(F/ F^+\) and \(K/ F^+\).

MSC:

11R23 Iwasawa theory
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)

Citations:

Zbl 0863.19003
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References:

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