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Linear independence result for \(p\)-adic \(L\)-values. (English) Zbl 1464.11071
Let \(\mathbb C_p\) be the completion of a fix algebraic closure of the field of \(p\)-adic numbers \(\mathbb Q_P\). Let \(\overline{\mathbb Q}_p\) be the algebraic closure of \(\mathbb Q\) in \(\mathbb C_p\). Let \(\mathbb K\subseteq\overline{\mathbb Q}_p\) be a number field. For a Dirichlet character \(\chi\), let us write \(\mathbb Q(\chi)\subseteq\overline{\mathbb Q}\) for the smallest subfield of \(\overline{\mathbb Q}\) containing all values \(\chi\). Then the author proves that for every \(\varepsilon >0\) and a sufficiently large positive integer \(s\) we have for a \(p\)-adic \(L\)-function \[\dim_{\mathbb K}(\mathbb K+\sum_{i=2}^s L_p(i,\chi \omega^{1-i})\mathbb K)\geq \frac{(1-\varepsilon)\log s}{2[\mathbb K:\mathbb Q](1+\log 2)},\] where \(\omega\) is the Teichmüller character. The proof makes use of special integrals.
MSC:
11J72 Irrationality; linear independence over a field
11F85 \(p\)-adic theory, local fields
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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