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

Abstract

The l-adic parabolic cohomology groups attached to noncongruence subgroups of SL2(Z) are finite-dimensional l-adic representations of Gal ( Q ¯ / K ) for some number field K. We exhibit examples (with K = 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 sln containing a principal nilpotent element, for which we give a short conceptual proof, suggested by I. Grojnowski. 2000 Mathematics Subject Classification 11F80 (primary), 11G18 (secondary).