On reducible $\mathrm{\ell }$-torsion representations of Picard curves

Let $C\mathrm{/}\mathbb{Q}$ be a Picard curve and let $\mathrm{\ell }\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}3)$ be a prime such that the $\lambda$-torsion representation of the Jacobian of $C$ is reducible for a prime $\lambda$ of $\mathbb{Q}({\zeta }_3)$ above $\mathrm{\ell }$. We describe the constraints imposed by complex conjugation on the $2$-dimensional subquotient $\rho$, in particular, showing that if it is absolutely irreducible, then up to twist it descends to $\mathbb{Q}$. We then give criteria for ruling out the irreducible case, using Dickson’s classification of subgroups of ${\mathrm{GL}}_2({\mathbb{F}}_{\mathrm{\ell }})$, under the natural hypothesis that the characteristic polynomials of all elements in the image of $\rho$ split. We also develop algorithms to compute the set of $1$-dimensional Jordan–Hölder constitutents of the $\mathrm{\ell }$-torsion representation over $\mathbb{Q}({\zeta }_3)$, and Frobenius characteristic polynomials of the $2$-dimensional subquotient $\rho$. We apply these methods to $381$ Picard curves with possibly reducible $7$-torsion representations, and to five corresponding $13$-torsion examples. Under the reducibility and splitting hypotheses, we rule out an irreducible $2$-dimensional subquotient in all of the examples considered, showing that the semisimplification of the $\mathrm{\ell }$-torsion representation is a direct sum of $6$ characters.