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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.

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The paper https://arxiv.org/abs/1403.7399 shows that the monodromy of the moduli stack of trigonal curves of genus g over $\mathbb C$ is as big as possible (equal to $Sp_{2g}(\mathbb Z)$ in the topological setting, or equal to $Sp_{2g}(\widehat{\mathbb Z})$ in the arithmetic setting, by the comparison theorem). An alternate, (and to my taste, geometrically enlightening,) explanation that the mod-2 monodromy is the full symplectic group is also sketched in Anand Patel's thesis https://www2.bc.edu/anand-p-patel/Research/ThesisTheGeometryofHurwitzSpace.pdf in Proposition 1.11 and the paragraphs following its proof.

To explain this implication in more detail, the main theorems on the first two pages of the above cited paper shows that the lowest Maroni strata, (which is dense in the locus of trigonal curves) has monodromy equal to the full symplectic group. Since the moduli stack of trigonal curves of a given Maroni invariant is smooth, the generic point also has maximal monodromy, equal to the full symplectic group. Technically speaking, the above cited paper only applies when the genus is at least 5, but the same results holds in genus 3 and 4 because trigonal curves are dense in $M_g$ for genus $3$ and $4$, and $M_g$ is smooth with monodromy equal to the full symplectic group by a classical result (see for example 5.12 of Deligne and Mumford's "The irreducibility of the moduli stack of curves of given genus").

Since in your question you ask about $GSp_{2g}(\mathbb F_2)$, you may be asking the question not only over an algebraically closed field, but also over a number field. In the case of a number field $K$ with $K \cap \mathbb Q^{cyc} = \mathbb Q$, we can use the geometric statement above to deduce that the monodromy group of the trigonal locus is all of $GSp_{2g}(\widehat{\mathbb Z})$. To see this, let $\chi:Gal(\overline K/K) \rightarrow \widehat{\mathbb Z}^\times$ denote the cyclotomic character, let $H_C$ denote the monodromy of a family of curves C, and let T denotes the moduli stack of trigonal curves. Then, the monodromy $H_T$ for $T$ over $K$ fits in an exact sequence \begin{align*} 0 \rightarrow (H_T)_{\overline K} \rightarrow H_T \xrightarrow{\operatorname{mult}} \chi(K) \rightarrow 0, \end{align*} Since $(H_T)_{\mathbb C} = Sp_{2g}(\widehat{\mathbb Z}),$ and includes into $(H_T)_{\overline K}$, it follows that $(H_T)_{\overline K} = Sp_{2g}(\widehat{\mathbb Z})$. Therefore, since $\chi(K) = \widehat{\mathbb Z}^\times$ when $K \cap \mathbb Q^{cyc} = \mathbb Q$, we obtain $H_T$ is a subgroup of $GSp_{2g}(\widehat{\mathbb Z})$ which contains the symplectic group and has surjective mult map, so $H_T = GSp_{2g}(\widehat{\mathbb Z}).$ In general, the monodromy over $K$ will be the preimage of $\chi(K)$ under the mult map.

It is also interesting to ask this question not only in the case of the generic trigonal curve (i.e., the generic point of the moduli space), but also in the case of a general trigonal curve over a number field. For simplicitly, let's work over $\mathbb Q$, although analogously to the previous paragraph, we can easily generalize this to any number field. While I do not think the monodromy will be maximal for a general trigonal curve, it does hold for a density-$1$ subset of the $\mathbb Q$ points of the moduli stack, with density ordered by height. That is, a density-$1$ subset of trigonal curves over $\mathbb Q$ have monodromy equal to $GSp_{2g}(\widehat{\mathbb Z})$ (as is shown in the statement and proof of Corollary 1.3 in http://arxiv.org/pdf/1608.05371v1.pdf).

Thanks to David Zureick-Brown and Jordan Ellenberg for help with answering this question.

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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.

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