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Very simple 2-adic representations and hyperelliptic jacobians

Let $K$ be a number field, $n>4$ an integer, $f(x)$ an irreducible polynomial over $K$ of degree $n$, whose Galois group is either the full symmetric group $S_n$ or the alternating group $A_n$. Suppose $C:y^2=f(x)$ is the corresponding hyperelliptic curve and $J$ its jacobian defined over $K$. For each prime $\ell$ we write $V_{\ell}(J)$ for the $Q_{\ell}$-Tate module of $J$ and $e_{\ell}$ for the Riemann form on $V_{\ell}(J)$ attached to the theta divisor. (Here $Q_{\ell}$ is the field of $\ell$-adic numbers.) We write $sp(V_{\ell}(J))$ for the $Q_{\ell}$-Lie algebra of the symplectic group of $e_{\ell}$. We write $g_{\ell}$ for the Lie algebra of the image of the Galois group $Gal(K)$ of $K$ in $Aut(V_{\ell}(J))$. We prove that $g_{\ell}$ coincides with the direct sum $Q_{\ell}I \oplus sp(V_{\ell}(J))$ where $I$ is the identity operator.