Treffer: On Gauss sum characters of finite groups and generalized Bernoulli numbers

Let \(f: X\to Y\) be a Galois covering of compact Riemann surfaces with \(G= \text{Gal}(X/Y)\) (finite) and \(V= H^0(X, \Omega^1_X)\) denote the space of holomorphic differentials on \(X\). The problem addressed in this paper is the decomposition of \(V\) into irreducible characters of \(G\). Previously, results were obtained by several mathematicians (e.g., Hecke, Shih, Weintrab, the author and others) for some specific groups: \(G= \text{PSL}_2(\mathbb{F}_p)\), or when \(G\) has a pair of characters whose values generate an imaginary quadratic field. In this paper, the problem is solved in most generality, that is, for a group \(G\) without any assumptions. The result is formulated in terms of an explicit description of certain inner products associated to Gauss sum characters and the specific character, \(\mu\). Let \(G\) be a finite group. For a character \(\chi\) of \(G\), let \(\mathbb{Q}(\chi)= \mathbb{Q}(\chi(g)\mid g\in G)\subset \mathbb{C}\) be the value field of \(\chi\), let \(\Gamma_\chi= \text{Gal}(\mathbb{Q}(\chi)/ \mathbb{Q})\), and \((\Gamma_\chi)^\wedge= \text{Him}(\Gamma_\chi, \mathbb{C}^\times)\). For a character \(\chi\) of \(G\) and \(\lambda\in (\Gamma_\chi)^\wedge\), define a Gauss sum character \(\alpha(\chi, \lambda)\) of \(G\) by \(\alpha(\chi, \lambda)= \sum_{\gamma\in \Gamma_\chi} \lambda(\gamma)\chi^\gamma\). Now let \(X\), \(Y\) be connected compact Riemann surfaces of genus \(g_X\) and \(g_Y\), respectively. Denote by \(\mathcal F\) a locally free \({\mathcal O}_Y\)-module of finite rank. If \(f: X\to Y\) is a finite Galois cover with Galois group \(G\), let \(f^* {\mathcal F}\) denote the pull-back of \(\mathcal F\) to \(X\). Define the character \(\mu= \text{ch}(H^0(X, f^* {\mathcal F}))- \text{ch}(H^1(X, f^* {\mathcal F}))\), where \(\text{ch}(V)\) stands for the character of \(G\) determined by \(V\). Further, define the inner product \[ m(\chi, \lambda)= \langle \alpha(\chi, \lambda), \mu\rangle_G= \sum_{\gamma\in \Gamma_\chi} \lambda(\gamma) \langle \chi^\gamma, \mu\rangle_G. \] The main result of the paper is formulated as follows. Theorem. (1) If \(\lambda\) is the trivial character, then \[ \begin{aligned} {1\over |\Gamma_\chi|} m(\chi, \lambda)=&(\deg({\mathcal F})- \text{rank}({\mathcal F}) (g_Y- 1)) \deg(\chi)-\\ -&{1\over 2} \text{ rank} ({\mathcal F}) \sum_{P\in X} {|G_P|\over |G|} (\deg (\chi)- \langle \chi|_{G_P}, \text{\textbf{1}}_{G_P}\rangle_{G_P}).\end{aligned} \] Here \(G_P= \{g\in G\mid g\cdot P= P\}\) is the stabilizer of \(P\) and \(\text{\textbf{1}}_{G_P}\) is the trivial character of \(G_P\). (2) If \(\lambda\) is even and non-trivial, then \(m(\chi, \lambda)= 0\). (3) If \(\lambda\) is odd, then \(m(\chi, \lambda)\) is a multiple of the generalized Bernoulli number \(B_{1, \lambda}\). In particular, if \(\lambda\) is the Dirichlet character corresponding to the extension \(\mathbb{Q}(\sqrt{- p})/ \mathbb{Q}\) with \(p\equiv 3\pmod 4\), then \(B_{1, \lambda}= h(\mathbb{Q} (\sqrt{- p}))\) (the class number of \(\mathbb{Q}(\sqrt{- p})\)), and the theorem is just Hecke's classical result \(m(\chi)- m(\overline \chi)= h(\mathbb{Q}(\sqrt{- p}))\), with \(X= X(p)\) and \(G= \text{PSL}_2(\mathbb{F}_p)\).