On missing trace values for the eta multipliers
Let \(\eta\) (z) be the Dedekind eta-function, and for \(\sigma \in SL_ 2({\mathbb{Z}})\) let \(\epsilon\) (\(\sigma)\) be the 24th root of unity such that \[ \eta (\sigma z)=\epsilon (\sigma)\quad (cz+d)^{1/2} \eta (z),\quad \sigma =\left( \begin{matrix} a\\ c\end{matrix} \begin{matrix} b\\ d\end{matrix} \right)\quad. \] This paper treats the question for which pairs of integers s and roots of unity \(\zeta =e^{\pi ik/12}\) there is a \(\sigma \in SL_ 2({\mathbb{Z}})\) with \(trace(\sigma)=s\) and \(\epsilon (\sigma)=\zeta\). Given k, the (''missing trace values'') \(s>2\) are determined which satisfy a certain set of necessary congruences desribed by \textit{T. Asai} [Number theory and combinatorics, Proc. Conf., Tokyo/Jap., Okayama/Jap. and Kyoto/Jap. 1984, 43-51 (1985)], but do not occur as a trace. The result is not easy to state, and the proof requires much perseverance.
scientific article; zbMATH DE number 3987364