Genus of abstract modular curves with level-ℓ structures

Abstract

Abstract We prove – in arbitrary characteristic – that the genus of abstract modular curves associated to bounded families of continuous geometrically perfect {\mathbb{F}_{\ell}} -linear representations of étale fundamental groups of curves goes to infinity with {\ell} . This applies to the variation of the Galois image on étale cohomology groups with coefficients in {\mathbb{F}_{\ell}} in 1-dimensional families of smooth proper schemes or, under certain assumptions, to specialization of first Galois cohomology groups.