Semisimple Torsion Fields of Modular Jacobians at Powers of 11 and 13

For \(\ell\in\{11,13\}\) and every \(m\geq1\), we identify the semisimple \(\ell\)-torsion field of the Jacobian of the connected principal modular curve \(X(\ell^m)\) over \(K_m=\mathbb Q(\zeta_{\ell^m})\). It is the compositum of \(K_m\) with the field cut out by the residual Ramanujan Delta representation, and its Galois group over \(K_m\) is \(\operatorname{SL}_2(\mathbb F_\ell)\). The full torsion field has an \(\ell\)-group kernel over this quotient; the prime-to-\(\ell\) part of its degree is exactly 120 or 168, respectively.

The result is an explicit structural corollary of level-one modularity, the modular decomposition of Jacobians, and Hoshi–Iijima's non-pro-\(\ell\) theorem. We give a cyclotomic-field moduli construction of the covering needed in the argument and reproducible exact certificates for the two residual Delta images. We do not determine the full torsion field or claim to resolve the unrestricted AIM splitting-field question. This AI-assisted note is self-audited and unrefereed; absolute priority is not asserted.