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Transcendental values of class group \(L\)-functions. II. (English) Zbl 1282.11082
Summary: Let \(K\) be an imaginary quadratic field and \(\mathfrak{f}\) an integral ideal. Denote by \(Cl(\mathfrak{f})\) the ray class group of \(\mathfrak{f}\). For every non-trivial character \(\chi\) of \(Cl(\mathfrak{f})\), we show that \(L(1,\chi )/\pi \) is transcendental. If \(\mathfrak{f}=\overline{\mathfrak{f}}\), then complex conjugation acts on the character group of \(Cl(\mathfrak{f})\). Denoting by \(\widehat {Cl(f)}^+\) the orbits of the group of characters, we show that the values \(L(1,\chi )\) as \(\chi\) ranges over elements of \(\widehat {Cl(\mathfrak{f})}^+\) are linearly independent over \(\overline{\mathbb{Q}}\). We give applications of this result to the study of transcendental values of Petersson inner products and certain special values of Artin \(L\)-series attached to dihedral extensions.
Part I, cf. Math. Ann. 351, No. 4, 835–855 (2011; Zbl 1281.11071).

MSC:
11J81 Transcendence (general theory)
11M32 Multiple Dirichlet series and zeta functions and multizeta values
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