The image of the generalized Dedekind sum

Abstract

The newform Dedekind sum $S_{\ddot{I}{‡}_1,\ddot{I}{‡}_2}$ associated to a pair of primitive Dirichlet characters $\ddot{I}{‡}_1$, $\ddot{I}{‡}_2$ of respective conductors $q_1$, $q_2$, is a group homomorphism from $\hat{I}{“}_1(q_1{q}_2)$ into the number field $F_{\ddot{I}{‡}_1,\ddot{I}{‡}_2}$ generated by the values of the characters. It is a basic question to identify the image of this map, which is known to be a lattice $L_{\ddot{I}{‡}_1,\ddot{I}{‡}_2}$ in $F_{\ddot{I}{‡}_1,\ddot{I}{‡}_2}$. It has recently been conjectured that when $\ddot{I}{‡}_1$ and $\ddot{I}{‡}_2$ are quadratic, then $L_{\ddot{I}{‡}_1,\ddot{I}{‡}_2}=2\mathbb{Z}$. In this paper, we make some progress towards this conjecture by exhibiting an explicit lattice in which $L_{\ddot{I}{‡}_1,\ddot{I}{‡}_2}$ is contained; in particular, when the characters are quadratic, the $q_i$ are coprime, odd, and sufficiently large, then $L_{\ddot{I}{‡}_1,\ddot{I}{‡}_2}\subseteq \mathbb{Z}$.

9 pages