Half-Integer Weight Cusp Forms: Infinite Non-Zero Fourier Coefficients at Fundamental Discriminants — E8 Intelligence Research

FINDING: Recent work on half-integer weight cusp forms (Gun; arXiv:2004.14450) proves infinitely many non-zero Fourier coefficients at fundamental discriminants, refining bounds beyond square-index trivialities. | MATH: For \(f \in S_{k+1/2}^{+}(\Gamma_0(4N))\), coefficients \(a_f(|D|)\) for fundamental discriminants \(D\); key bound: \(a_f(|D|) \ll |D|^{k/2 - 1/4 + \epsilon}\) (Waldspurger-type), with non-vanishing for infinitely many \(D\). The "soft proof" uses the Shimura lift \(\mathcal{S}_t(f) \to S_{2k}(\Gamma_0(2N))\) and the Petersson norm pairing, avoiding deep subconvexity. | CONNECTION: The half-integer weight space decomposes into Kohnen's plus space — a lattice structure isomorphic to a root system of type \(D_n\) in the Hecke eigenbasis. The fundamental discriminants \(D\) correspond to quadratic forms, linking to Gauss composition (base-60 sexagesimal residue patterns in class numbers). The ratio \(1/4\) in the exponent echoes the critical line \(\Re(s)=1/2\) and the go Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com