Madison Experimental Mathematics Lab

Starting from a curve $C\colon y^3 = f_4(x)$ with $f_4$ a degree $4$ polynomial defined over $\mathbb{Q}$, the $\ell$-torsion of the Jacobian $J(C)$ yields a representation $\rho_\ell\colon G_\mathbb{Q} \to \operatorname{GSp}(6,\Z/\ell)$. For primes $\ell \equiv 1\pmod{3}$, the ring $\mathcal{O}_K$ contains prime ideals $\lambda$ and $\bar{\lambda}$ above $\ell$. Restricting $\rho_\ell$ to $G_K$ produces matrix representations consisting of two $3\times3$ blocks. For certain curves the upper $3\times 3$ block $M$ is itself reducible, its semisimplification is the direct sum of a character $\varepsilon_1$ and a $2$-dimensional subrepresentation $\rho_\lambda: G_K \to \operatorname{GL}(2,\mathbb{F}_\ell)$.

The central objects of study are the character $\varepsilon_1$, the determinant character of $\rho_\lambda$, and the traces $\operatorname{tr}(\rho_\lambda(\operatorname{Frob}_\mathfrak{p}))$. We present algorithms to compute these from the Euler factors of $C$. We also establish a conductor bound $N(\rho_\lambda) \leq \frac{\sqrt{N(\rho_\ell)}}{N(\varepsilon_1)}$

Unexpectedly, the representation proves to be completely reducible over $\mathbb{Q}$. Consequently, it admits no 2-dimensional subrepresentations.

Link to the code: https://github.com/xiyaochen2002/Serre-Modularity-Over-IQF

Link to MXM: https://mxm.math.wisc.edu/

Poster: https://drive.google.com/file/d/1Ekz1xZzJCYXtQJ6vxjEZJ-91Is-9TWTA/view?usp=drive_link