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.
Categories: math.AG math.NT