Dedekind sums and signatures of intersection forms
Let \(\varphi: \Gamma\to {1\over 3}\mathbb{Z}\) be the Rademacher function on the modular group \(\Gamma= \text{SL}_ 2 \mathbb{Z}\). This function is uniquely characterized by the property \[ \varphi(A_ 1)+ \varphi(A_ 2)+ \varphi(A_ 3)=- \text{sign} (c_ 1 c_ 2 c_ 3) \quad \text{for}\quad A_ i= \left( \begin{smallmatrix} * &*\\ c_ i &* \end{smallmatrix} \right)\in \Gamma \quad \text{with } A_ 1 A_ 2 A_ 3=1. \] For an arbitrary sequence of integers \(b_ 1, b_ 2,\dots, b_ n\), let \[ A= \left( \begin{smallmatrix} 0 &-1\\ 1 &0\end{smallmatrix} \right) \left( \begin{smallmatrix} b_ 1 &-1\\ 1 &0\end{smallmatrix} \right) \dots \left( \begin{smallmatrix} b_ n &-1\\ 1 &0\end{smallmatrix} \right). \] Finally, denote by \(I\) the \(n\times n\) matrix whose \((i,j)\) entry is 1 if \(| i-j| =1\), \(b_ i\) if \(i=j\), and 0 otherwise. This matrix represents the intersection form of certain 4-manifolds related to lens spaces. There is an unexpected relation (first observed by Hirzebruch around 1970) between the signature of these 4-manifolds and Dedekind sums (which appear in an explicit expression for the Rademacher function). Among the most beautiful manifestations of this relation is the identity \[ \varphi(A)= {\textstyle {1\over 3}} \text{trace}(I)- \text{sign}(I). \] In this paper the author gives a purely arithmetic proof of three such identities. The proof is based on the 1-cocycle property of the following modification of the Rademacher \(\varphi\)-function: \[ \psi(A)(x)= \varphi(A)+ \rho(A)(x), \quad \rho \left( \begin{smallmatrix} a &b\\ c &d\end{smallmatrix} \right) (x)= \text{sign} (cx_ 2 (cx_ 1- ax_ 1)), \quad x\in \mathbb{R}^ 2\setminus \{0\}. \] Contrary to the Rademacher function, \(\psi\) satisfies the property \[ \psi(AB) (x)= \psi(A)(x)+ \psi(B) (A^{-1} x) \text{ for all }A,B\in \Gamma. \] In other words, \(\psi\) is a 1-cocycle on \(\Gamma\) with values in the \(\Gamma\)-module of functions on \(\mathbb{R}^ 2\setminus \{0\}\) equipped with the \(\Gamma\) action \((Af)(x)= f(A^{-1} x)\).
scientific article; zbMATH DE number 683143