On the convergence of certain indefinite theta series (Q6619739)

For a lattice \(L\) with an indefinite integral quadratic form \(Q\), in general, the theta series associated with \((L,Q)\) does not convergent absolutely. Let \(V=L\otimes\mathbb R\). In the case that the quadratic space \((V,Q)\) is of signature \((n,1)\) or \((n,2)\), an indefinite theta series is defined such that it is termwise absolutely convergent. In this article, for \((V,Q)\) of signature \((n,2)\), the author defines a theta series based on the work of Alexandrov, Banerjee, Manshot and Pioline and proves their conjecture on the convergence of the theta series using more elementary methods than those of Funke and Kulda. Let \(\mathcal{C}=\{C_1,\dots,C_N\}\) be the set of non-zero vectors of \(V\) satisfying the conditions: (1)~\(Q(C_j)<0\),~(2)~\(4Q(C_j)Q(C_{j+1})-(C_j,C_{j+1})^2>0\)~,(3)~\(2Q(C_j)(C_{j-1},C_{j+1})-(C_j,C_{j-1})(C_j,C_{j+1})<0\), where \((-,-)\) is the bilinear form associated with \(Q\).\N\NPut \(w(x,\mathcal{C})=\sum_{j=1}^N\mathrm{sgn}((x,C_j))\mathrm{sgn}((x,C_{j+1}))\) for \(x\in V\), where \(C_{N+1}=C_1\) and \(\Phi(x,\mathcal{C})=w(x,\mathcal{C})-w(v,\mathcal{C})\) with a negative vector \(v\). Let \(\mu\) be an element of the dual of \(L\). The author defines the theta series \(\theta_\mu(\tau)=\sum_{x\in \mu+L}\Phi(x,\mathcal{C})q^{Q(x)}~(q=e^{2\pi i\tau})\) and shows that \(\theta_\mu(\tau)\) is termwise absolutely convergent. Further if \(\mathcal{C}\) satisfies only conditions (1),(2) and \(\theta_\mu(\tau)\) is termwise absolutely convergent, then \(\mathcal{C}\) satisfies the condition (3). The essential of the proof is to show that \(w(x,\mathcal{C})\) is locally constant on the set \(\{x\in V\mid (x,C_j)\ne 0 \text{ for any } j\}\) and is constant on the set of negative vectors. This implies that for \(x\in \mu+L\), if \(Q(x)\le 0\), then \(\Phi(x,\mathcal{C})=0\), otherwise \(Q(x)\) is bounded below by some positive definite quadratic form. Assuming only conditions (1) and (2), he obtains similar results for the modified theta series defined correspondingly, as in [\textit{J. Funke} and \textit{S. Kudla}, Pure Appl. Math. Q. 19, No. 1, 191--231 (2023; Zbl 1519.11024)].