Two-dimensional families of hyperelliptic Jacobians with big monodromy (Q2790603)

Let \(K\) be a field of characteristic \(\neq 2\), and \(f(x)\), a polynomial of degree \(n\geq 5\) over \(K\) without multiple roots. Let \(C : y^2=f(x)\) be the hyperelliptic curve, and \(J\), its Jacobian variety over \(K\). Let \(\rho\) be the representation \(\mathrm{Gal}(\bar K/K) \to \mathrm{Aut}_{\mathbb{Z}_\ell}(T_\ell(J))\) where \(\ell\) is a prime number \(\neq \mathrm{char}(K)\) and \(T_\ell(J)\) is the Tate module of \(J\). Let \(G\) denote its image.NEWLINENEWLINELet \( e_\ell\) be the Galois-equivariant alternating bilinear Riemann form \(T_\ell(J)\times T_\ell(J)\to\mathbb{Z}_\ell\), obtained via the canonical principal polarization on \(J\), so that \( e(\sigma(x),\sigma(y)) = \chi_\ell(\sigma)\, e(x,y)\) for all \(x,y\in T_\ell(J)\) and \(\sigma\in \mathrm{Gal}(\bar K/K)\) where \(\chi\) is the cyclotomic character \(\mathrm{Gal}(\bar K/K) \to\mathbb{Z}_\ell^*\) deteremined by the Galois action on \(\ell\)-th power roots of unity in \(\bar K\). This implies \(G \subset \mathrm{Gp}(T_\ell(J), e_\ell) \subset \mathrm{Aut}_{\mathbb{Z}_\ell}(T_\ell(J))\) where \(\mathrm{Gp}(T_\ell(J), e_\ell)\) is the subgroup consisting of \( \tau \in \mathrm{Aut}_{\mathbb{Z}_\ell}(T_\ell(J))\) such that \( e_\ell( \tau(x),\tau(y) ) = c_\tau\, e_\ell(x,y),\;\forall x,y\in T_\ell(J)\). Let us denote by \(\mathrm{Sp}(T_\ell(J),e_\ell)\) the subgroup consisting of \(\tau\in\mathrm{Gp}(T_\ell(J), e_\ell)\) such that \(c_\tau=1\).NEWLINENEWLINEThe following result is derived easily from the author's earlier work [Proc. Lond. Math. Soc. (3) 100, No. 1, 24--54 (2010; Zbl 1186.14031)]: Suppose that \(K\) is a field that is finitely generated over \(\mathbb{Q}\) and \(n=2g+2\geq 12\) is even. Assume that \(f(x) = (x-t_1)(x-t_2) u(x)\) with \(t_1,t_2\in K,\;t_1\neq t_2,\;u(x)\in K[x], \mathrm{deg}(u)=n-2\), and the Galois group of the splitting field of \(u(x)\) is either \(S_{2g}\) or \(A_{2g}\). Then, \(G\) is an open subgroup of finite index in \(\mathrm{Gp}(T_\ell(J), e_\ell)\). Inspired by [\textit{C. Hall}, Bull. Lond. Math. Soc. 43, No. 4, 703--711 (2011; Zbl 1225.11083)], the author proves under certain arithmetic conditions that if \(K\) is a field that is finitely generated over its prime subfield, then \(\mathrm{Sp}(T_\ell(J),e_\ell)\subset G\) for all but finitely many primes \(\ell\neq\mathrm{char}(K)\). If, in addition, \(\mathrm{char}(K)=0\), then \(G=\mathrm{Gp}(T_\ell(J))\) for all but finitely many primes \(\ell\). The author also presents similar results for the representation \(\mathrm{Gal}(\bar K/K) \to \mathrm{Aut}_{\mathbb{Z}/m}(J[m])\) for \(m>0\) not divisible by the characteristic of \(K\).