The \(\chi\)-part of the analytic class number formula, for global function fields (Q2880827)

Let \(k\) be a congruence function field with field of constants \({\mathbb F}_q\). Let \(\infty\) be a fixed place of \(k\) and let \(k_{\infty}\) be the completion of \(k\) at \(\infty\). Let \(F\subseteq k_{\infty}\) be a finite abelian extension of \(k\) with Galois group \(G\) of order \(g\). The author considers a group of Stark units \({\mathcal E}_F\) having finite index in \({\mathcal O}_F^{\ast}\) where \({\mathcal O}_F\) is the ring of functions of \(F\) which are regular outside the places of \(F\) sitting above \(\infty\). In [Bull. Lond. Math. Soc. 43, No. 3, 523--535 (2011; Zbl 1248.11091)] \textit{H. Oukhaba} and the author proved, using Euler systems, that \({\mathcal E}_F\) satisfies the Gras conjecture NEWLINE\[NEWLINE \#({\mathbb Z}_p\otimes_{\mathbb Z}({\mathcal O}_F^{\ast}/{\mathcal E}_F))_{\psi} =\#({\mathbb Z}_p\otimes_{\mathbb Z} Cl({\mathcal O}_F))_{\psi} NEWLINE\]NEWLINE for every nontrivial irreducible character \(\psi\) of \(G\) and for all prime numbers \(p\nmid qg\), where the subscript \(\psi\) means the \(\psi\)--part and \(Cl({\mathcal O}_F)\) is the ideal class group of \({\mathcal O}_F\). They were not able to prove the conjecture when \(p| \# Cl({\mathcal O}_k)\), \(\psi\) is a conjugate of the Teichmüller character, \(\mu_p\not\subseteq k\) and \(\mu_p \subseteq F\) where \(\mu_p\) is the group of \(p\)-th roots of unit.NEWLINENEWLINEIn this paper the author proves that for every nontrivial complex character \(\chi\) of \(G\), NEWLINE\[NEWLINE \roman{Fit}_\vartheta(\vartheta\otimes_{\mathbb Z}({\mathcal O}_F^{\ast}/ {\mathcal E}_F))_{\chi} =\roman{Fit}_\vartheta(\vartheta\otimes_{\mathbb Z} Cl({\mathcal O}_F))_{\chi}, NEWLINE\]NEWLINE where \(\vartheta\) is the integral closure of \({\mathbb Z}[g^{-1}]\) in \({\mathbb Q}(\mu_g)\) and \(\roman{Fit}_{\vartheta}(M)\) is the Fitting ideal of \(M\).NEWLINENEWLINEFrom this result it follows that the classical Gras conjecture holds for all prime numbers \(p\nmid g\). Finally, combining this complex Gras conjecture with computations made by the author in [Manuscr. Math. 136, No. 3--4, 445--460 (2011; Zbl 1264.11093)], he derives a ``\(\chi\)--part'' version of the analytic class number formula NEWLINE\[NEWLINE R({\mathcal O}_F)_{\chi} \roman{Fit}_\vartheta((\vartheta\otimes_{\mathbb Z}Cl({\mathcal O}_F))_{\chi}) =\roman{Fit}_\vartheta((\vartheta\otimes_{\mathbb Z} \mu(F))_{\chi})L_{ {\mathfrak f}_{\chi}}(0,\bar{\chi}_{\text{pr}}), NEWLINE\]NEWLINE as \(\vartheta\)--submodules of \({\mathbb Q}(\mu_g)\) where \(\chi\neq 1\) is a complex irreducible character of \(G\), \(R({\mathcal O}_F)_{\chi}\) is the ``\(\chi\)--part'' of the regulator and \(L_{ {\mathfrak f}_{\chi}}(0,\bar{\chi}_{\text{pr}})\) is the value at \(0\) of the \(L\)--function attached to \(\bar{\chi}_{\text{pr}}\).