Local p-Isogeny Kernel Criterion

Translates the abstract Jordan–Hölder criterion from document 35 into a concrete, directly computable isogeny kernel condition. Assuming $E[p]|_{G_{\mathbf Q_p}}$ is reducible, choose a stable cyclic subgroup $C\simeq\mathbf Z/p$ to obtain a local isogeny $\phi:E\to E'$, and let $\sigma(P)=\lambda(\sigma)P$. The document proves a precise chain of equivalences: since $x(P)=x(-P)$ and $p$ is odd, $x(P)\in\mathbf Q_p$ if and only if $\sigma(P)=\pm P$ holds for all $\sigma$, if and only if $\lambda(G_{\mathbf Q_p})\subset\{\pm1\}$, if and only if $\lambda^2=1$—therefore, $\lambda^2=1$ if and only if the kernel polynomial of $\ker\phi$ has a linear factor in $\mathbf Q_p$, which is a concrete condition that can be directly verified using Tate's algorithm or local field computations. The other Jordan–Hölder character $\mu=\omega\lambda^{-1}$ corresponds to the kernel character of the dual isogeny $\widehat\phi:E'\to E$. Thus, $\mathrm{FW17\text{-}H2\ FAIL}$ if and only if the kernel polynomial of $\phi$ or its dual has a linear factor in $\mathbf Q_p$. Finally, the document explains why it is sufficient to check just one isogeny plus its dual, without needing to enumerate all local $p$-isogenies: for a nonsplit reducible extension, the original $E[p]$ has only one stable line, and the quotient constituent will appear in the dual kernel; for a split representation, the two constituents are directly captured by $\phi$ and $\widehat\phi$ respectively.