Question

For a hyperelliptic curve H, the mod 2 monodromy is smaller than $GSp_{2g}(F_2)$ -- since the two torsion of the Jacobian H is generated by differences of Weierstrass point, the monodromy of a generic curve is the symmetric group.

The degree 3 map $C \to \mathbb{P}^1$ associated to a trigonal curve may only have ramification of type (2,1), and thus may not give any "easy to see" 3 torsion points on the Jacobian. So it is reasonable to ask if a generic such curve has maximal mod l monodromy for every l.

Answer 1

Set $\tilde{C}:=\overline{C\times_{\mathbb{P}^1}C\smallsetminus\mathrm{diagonal}}$, then $J\tilde{C}$ is a quotient of $JC\times JC$, and $JC$ maps into $J\tilde{C}$, where the kernel is a subgroup of the $2$-torsion points. Moreover, $J\tilde{C}/\mathbb{P}^1$ is Galois (with Galois group $S_3$). Now let $C'$ be the unique intermediate cover of $\tilde{C}/\mathbb{P}^1$ which is degree $2$ over the $\mathbb{P}^1$, then $\tilde{C}/C'$ is a cyclic degree $3$ cover, so you get some $3$ torsion data.