Abstract
We compute the \(\mathbb{Z}/\ell\) and \(\mathbb{Z}_{\ell}\) monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the \(\mathbb{Z}/\ell\) monodromy of the moduli space of hyperelliptic curves of genus g is the symplectic group \({\rm S}_{P2g}(\mathbb{Z}/\ell)\). We prove that the \(\mathbb{Z}/\ell\) monodromy of the moduli space of trielliptic curves with signature (r,s) is the special unitary group \({\rm SU}_{(r,s)}(\mathbb{Z}/\ell\otimes\mathbb{Z}[\zeta_{3}])\).
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Rachel Pries was partially supported by NSF grant DMS-04-00461.
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Achter, J.D., Pries, R. The integral monodromy of hyperelliptic and trielliptic curves. Math. Ann. 338, 187–206 (2007). https://doi.org/10.1007/s00208-006-0072-0
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DOI: https://doi.org/10.1007/s00208-006-0072-0