Fouquet–Wan Supersingular Source Audit

The actual audit results from Referee C, re-verifying FW-H1, FW-H2, and FW-H3 for any odd good supersingular prime $p$ of $E_q$. The approach this time is to directly cross-reference the normalization in the original Fouquet–Wan paper, rather than trusting the previously self-derived version. H1: FW requires the global residual representation to be absolutely irreducible; the mod-$p$ image of the base 696.e1 is maximal, and the quadratic twist preserves absolute irreducibility, PASS. H2: FW Theorem 1.7 excludes the form $\bar\rho|_{G_{\mathbf Q_p}}^{ss}=\chi\oplus\chi_{\rm cyc}\chi$; the good supersingular local residual representation is an irreducible niveau-2 type and cannot be a character direct sum, PASS. The audit of H3 is the most detailed and the focal point of the entire document: instead of guessing the representation normalization, it directly locates the explicit definition of Assumption 3 near FW Theorem 1.1 in the original text—the local automorphic representation is special Steinberg, the twist by an unramified character sends $\ell$ to $(-1)\ell^{k/2-1}$, and the residual representation is ramified. Substituting weight $k=2$: $(-1)\ell^0=-1$, which for an elliptic curve newform means $a_\ell=-1$, i.e., nonsplit multiplicative. Taking $\ell=29$, 696.e1 is exactly nonsplit multiplicative at $29$, and an admissible $q$ makes $29$ split in $\mathbf Q(\sqrt q)$, rendering the local quadratic twist trivial; furthermore, $v_{29}(\Delta)=1$, so for any odd prime $p\ne29$ the residual remains ramified, PASS. Finally, the document strings together the BSD conclusion: Theorem 2.14 first gives $L(E_q,1)\ne0$, and Fouquet–Wan Corollary 1.10 thus provides the corresponding $p$-part BSD, with the period issue explicitly left for the next audit to handle.