Very simple 2-adic representations and hyperelliptic jacobians

Abstract

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.

LaTeX 2e, 27 pages Theorem 2.6 (stated in the abstract) was extended to the case of an arbitrary finitely generated field $K$ of characteristic zero. Some references were added, some typos corrected

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