Zeta-functions of binary Hermitian forms and special values of Eisenstein series
The work under review is a systematic investigation of zeta-functions of binary Hermitian forms over the ring \(\mathfrak o\) of integers of an imaginary quadratic number field \(K={\mathbb{Q}}(\sqrt{D})\) with discriminant \(D<0\). First we develop a theory of representations of numbers by binary Hermitian forms over \(\mathfrak o\) which parallels the classical theory of representations of integers by binary quadratic forms over \(\mathbb{Z}.\) A key result is a bijection described in Theorem 2.3. This bijection means that the sum of the numbers of representations (modulo \(\mathrm{SL}(2,\mathfrak o)\)-units) of an integer \(k\neq 0\) by a set of representatives of the \(\mathrm{SL}(2,\mathfrak o)\)-classes of binary Hermitian forms over \({\mathfrak o}\) with discriminant \(\Delta\) is equal to the number of cosets \(\lambda +k{\mathfrak o}\) with \(\lambda\in {\mathfrak o}\), \(\lambda \bar{\lambda}+\Delta \equiv 0 \bmod k\). The number of these cosets is explicitly computed, and the result is applied to the study of zeta-functions of (definite or indefinite) Hermitian forms. For simplicity we restrict in this review to the case of positive definite forms over \(\mathfrak o\). Let \(f\) be a positive definite binary Hermitian form over \(\mathfrak o\) with \(\mathrm{SL}(2,\mathfrak o)\)-unit group \(\mathcal E_ 1(f)\) and define \[ Z_{\mathfrak o}(f,s):=\frac{1}{| {\mathcal E}_ 1(f)|}\sum_{u,v\in \mathfrak o, <u,v>=\mathfrak o}f(u,v)^{-1-s},\quad Z(\Delta,s):=\sum_{[f]}Z_{\mathfrak o}(f,s), \] where the latter sum extends over a representative system of the \(\mathrm{SL}(2,\mathfrak o)\)-orbits of positive definite binary Hermitian forms over \(\mathfrak o\) with discriminant \(\Delta\). Then the analytic version of the results from above says that \[ Z(\Delta,s)=\theta (\Delta,s) \zeta (s) L(s+1,\chi_ D)^{-1}, \] where \(\theta (\Delta,s)\) is a finite Euler product whose factors are explicitly known. Second we consider the genus \(\mathcal G(f)\) of the positive definite form \(f\) and its associated zeta-function \(\hat Z(\mathcal G(f),s)\). The main result of \textit{H. Braun} [Abh. Math. Semin. Hansische Univ. 14, 61--150 (1941; Zbl 0025.01603)] expresses \(\hat Z(\mathcal G(f),s)\) in terms of a Dirichlet series whose coefficients are determined by certain local densities. We explicitly compute those local densities in all cases. The results are most easily formulated in terms of the Hilbert symbol, but the mere length of the theorem is still considerable (see Theorem 6.1). This enables us to evaluate \(\hat Z(\mathcal G(f),s)\) in terms of sums of certain products \(L(s,\chi)L(s+1,\chi)\) for suitable \(\chi\); the very lengthy results are written out in Theorems 7.4--7.6. Third we reformulate the preceding results in terms of the Eisenstein series for \(\mathrm{PSL}(2,\mathfrak o)\) considered by the authors in [J. Reine Angew. Math. 360, 160--213 (1985; Zbl 0555.10012)]. In particular, we compile tables of the various \(\mathrm{SL}(2,\mathfrak o)\)-classes of positive definite forms over \(\mathfrak o\) for some small discriminants \(\Delta\) and for \(D=-4,-8,-8\). Then we apply our general results to these examples. This yields a wealth of results on special values of Eisenstein series and on explicit representation numbers. In particular, we obtain several explicit representation numbers which were already communicated by Liouville more than 100 years ago, and we obtain some representation numbers which we could not locate in the literature. In some cases the formulae for the representation numbers involve Fourier coefficients of cusp forms of weight 2, and the Dirichlet series associated with these cusp forms are known to be equal to \(L\)-series of certain elliptic curves. This yields some intriguing formulae for the periods of the elliptic curves in question.
zeta-functions of binary Hermitian forms
Fourier coefficients of cusp forms of weight 2
scientific article; zbMATH DE number 4077330
scientific article