Transcendental values of class group $L$-functions, II

Abstract:

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.