ON SOME $\ell$-ADIC REPRESENTATIONS OF $\mathop{\rm Gal}(\overline{/mathbb{Q}})$ ATTACHED TO NONCONGRUENCE SUBGROUPS

Abstract

The $\ell$-adic parabolic cohomology groups attached to noncongruence subgroups of $\mathop{\rm SL_2(\mathbb{Z})$ are finite-dimensional $\ell$-adic representations of $\mathop{\rm Gal}(\overline{/mathbb{K}})$ for some number field $K$. We exhibit examples (with $K=\overline{\mathbb{Q}}$) for which the primitive parts give Galois representations whose images are open subgroups of the full group of symplectic similitudes (of arbitrary dimension). The determination of the image of the Galois group relies on Katz's classification theorem for semisimple subalgebras of $\mathfrak{sl}_n$ containing a principal nilpotent element, for which we give a short conceptual proof, suggested by I. Grojnowski.