Let \(C\) be a smooth connected projective curve of genus \(g \geq 1\) over an algebraically closed field \(k\) of characteristic \(p > 0\). As it is well known, the Jacobian \(\text{Pic}^0(C)\) is a principally polarized abelian variety of dimension \(g\) and the number of \(p\)-torsion points of \(\text{Pic}^0(C)\) is equal to \(p^f\) for some integer \(f\), called \(p\)-rank of \(C\), with \(0 \leq f \leq g\). The \(p\)-rank induces a stratification of the moduli space \({\mathcal M}_g\) of smooth connected projective genus \(g\) curves \({\mathcal M}_g = \bigcup {\mathcal M}^f_g\) by locally closed reduced substacks \({\mathcal M}^f_g\), whose geometric points correspond to curves of genus \(g\) and \(p\)-rank \(f\).
The main result of the paper is the calculation of the \({\mathbb Z}/{\ell}\)-monodromy and \({\mathbb Z}_{\ell}\)-monodromy of every irreducible component of the stratum \({\mathcal M}^f_g\). Namely, let \(S\) be a connected stack over \(k\), and let \(s\) be a geometric point of \(S\). Let \(C \to S\) be a relative smooth proper curve of genus \(g\) over \(S\). Then \(\text{Pic}^0(C)[l]\) is an étale cover of S with geometric fiber isomorphic to \(({\mathbb Z}/{\ell})^{2g}\). The fundamental group \(\pi_1 (S, s)\) acts linearly on \(\text{Pic}^0(C)[{\ell}]_s\), and the monodromy group \(\text{M}_{\ell} := \text{M}_{\ell} (C \to S, s)\) is the image of \(\pi_1 (S, s)\) in \(\text{Aut}(\text{Pic}^0 (C)[{\ell}]_s)\).
To determine \(\text{M}_{\ell}(S)\), for an irreducible component \(S\) of \({\mathcal M}^f_g\) the authors use induction and a recent result of C.-L. Chai [Pure Appl. Math. Q. 1, No. 2, 291–303 (2005; Zbl 1140.11031)] about the integral monodromy of the \(p\)-rank strata \({\mathcal A}^f_g\) of principally polarized abelian varieties of dimension \(g\). The result of Chai covers the cases \(g \leq 3\), while for \(g \geq 4\) the authors show that the closure of every component \(S\) of \({\mathcal M}^f_g\) in \({\bar {\mathcal M}}_g\) contains moduli points of chains of curves of specific genera and \(p\)-rank, which on its turn implies that the monodromy group \(\text{M}_{\ell}(S)\) contains two non-identical copies of \(\text{Sp}_{2g-2}({\mathbb Z}/{\ell})\), and is thus isomorphic to \(\text{Sp}_{2g}({\mathbb Z}/{\ell})\). By considering the powers of \(\ell\) and taking the inverse limit, the \(\ell\)-adic monodromy turns out to be \(\text{Sp}_{2g}({\mathbb Z}_\ell)\).
In the last part of the paper the authors apply the results about the monodromy of components of \({\mathcal M}^f_g\) to show existence of curves with trivial automorphism group, absolutely simple Jacobian, as well as to estimate the proportion of such curves with a rational point of order \(\ell\) on the Jacobian or for which the numerator of the zeta function has large splitting field.