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Based on comparison theorems for Hecke algebras and universal deformation rings with strong restrictions at the critical prime l , as provided by the results of Wiles, Taylor, Diamond, et al., we prove under rather general conditions that the corresponding universal deformation spaces with no restrictions at l can be identified with certain Hecke algebras of l -adic modular forms as conjectured by Gouvêa, thus generalizing previous work of Gouvêa and Mazur. Along the way, we show that the universal deformation spaces we consider are complete intersections, flat over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] l of relative dimension three, in which the modular points form a Zariski dense subset. Furthermore the fibers above [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] l of these spaces are generically smooth.

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On a locally Noetherian scheme X over a field of positive characteristic p we study the category of coherent O X -modules M equipped with a p −e -linear map, i.e. an additive mapThe notion of nilpotence, meaning that some power of the map C is zero, is used to rigidify this category. The resulting quotient category, called Cartier crystals, satisfies some strong finiteness conditions. The main reasult in this paper states that, if the Frobenius morphism on X is a finite map, i.e. if X is F -finite, then all Cartier crystals have finite length. We further show how this and related results can be used to recover and generalize other finiteness results of Hartshorne-Speiser [HS77], Lyubeznik [Lyu97], Sharp [Sha07], Enescu-Hochster [EH08], and Hochster [Hoc07] about the structure of modules with a left action of the Frobenius. For example, we show that over any regular F -finite scheme X Lyubeznkik's F -finite modules [Lyu97] have finite length.

Let X be a smooth, projective, geometrically connected curve over a finite field Fq, and let G be a split semisimple algebraic group over Fq. Its dual group G is a split reductive group over Z. Conjecturally, any l-adic G-local system on X (equivalently, any conjugacy class of continuous homomorphisms π1(X) → G(Q l )) should be associated to an everywhere unramified automorphic representation of the group G.We show that for any homomorphism π1(X) → G(Q l ) of Zariski dense image, there exists a finite Galois cover Y → X over which the associated local system becomes automorphic.