Difference between Galois representations in automorphism and outer-automorphism groups of a fundamental group

Abstract:

Let $C$ be a proper smooth geometrically connected curve over a number field of $K$ of genus $g\geq 3$. For a fixed $\ell$, let $\Pi ^{\ell }$ denote the pro-$\ell$ completion of the geometric fundamental group of $C$. For an $L$-rational point $x$ of $C$, we have $\rho _{A,x}:G_L \to \operatorname {Aut} \Pi ^\ell$ associated to the base point $x$, and its quotient by the inner automorphism group $\rho _O:G_L \to \operatorname {Out} \pi ^\ell :=\operatorname {Aut}/\operatorname {Inn}$, which is independent of the choice of $x$. We consider whether the equality $\operatorname {Ker} \rho _{A,x}=\operatorname {Ker} \rho _{O,x}$ holds or not. Deligne and Ihara showed the equality when the curve is the projective line minus three points with a choice of tangential basepoint. The result here is: Fix an $\ell$ dividing $2g-2$. Then there are infinitely many curves of genus $g$ such that for any $L$-rational point $x$ with $[L:K]$ finite and coprime to $\ell$, the index $[\operatorname {Ker} \rho _{O,x}:\operatorname {Ker} \rho _{A,x}]$ is infinite.