On theta series and the splitting of \(S_ 2({\Gamma}_ 0(q))\) (Q2367464)

This is an interesting paper which considers the dimension of spaces of modular forms which are the span of the theta series in a fixed column of the Brandt matrix series associated to a maximal order in the \((q,\infty)\)-rational quaternion algebra vis-à-vis Hecke's conjecture which asserts the linear independence of such a collection of theta series. For a prime number \(q\), the dimension of the space \(M_ 2(q)\) of modular forms of weight 2 and trivial character for \(\Gamma_ 0(q)\) is \(H\), the class number of maximal orders in the \((q,\infty)\)-rational quaternion algebra. To a maximal order \({\mathcal O}\) with left \({\mathcal O}\)-ideal class representatives \(\ell_ 1,\ell_ 2,\dots, \ell_ H\), one associates theta series \(\theta_{ij}\), \(1\leq i,j\leq H\), where \(\theta_{ij}\) is the theta series attached to the quadratic (norm) form of the quaternion algebra relative to the lattice \(\ell_ j^{-1} \ell_ i\); more precisely, the \(n\)th Fourier coefficient of \(\theta_{ij}\) is the \(ij\)- entry of the Brandt matrix \(B(n)\) associated to \({\mathcal O}\). In 1935 Hecke conjectured that the \(\mathbb{C}\)-linear span of the theta series in any column, \((\theta_{1j}, \theta_{2j},\dots, \theta_{Hj})^ t\), of the matrix \((\theta_{ij})\) was equal to \(M_ 2(q)\). While Eichler showed that \(M_ 2(q)\) is equal to the \(\mathbb{C}\)-linear span of all the \(\theta_{ij}\), Hecke's conjecture is false except for a finite number of small primes [see \textit{A. Pizer}, Pac. J. Math. 79, 541-548 (1978; Zbl 0387.10015)]. Further conjectures suggested that the dimension of the span of a given column of theta series was related either to the class number or the type number of the maximal order. In this paper, the author shows that Hecke's conjecture is false even in its weakest sense. Furthermore, the author compiles data from the explicit computation of the theta series which span \(M_ 2(q)\) for all primes \(q<1000\), and makes a number of interesting observations about these collections of theta series. A splitting of the space of cusp forms is given in which the summands yield new examples: for instance, among the data is an example of two inequivalent rational quadratic forms of rank 4 which are in the same spinor genus and which yield the same theta series.