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Hodge-Tate structures and modular forms. (English) Zbl 0646.14026
The author proves the existence of a Hodge-Tate structure of weights $$(0,k+1)$$ for the p-adic Galois representation associated to a weight $$k+2\geq 2$$ eigenform in $$S_{k+2}(\Gamma_ 0(N),\epsilon)$$. As a consequence he gives a p-adic analogue of the Eichler-Shimura isomorphisms relating spaces of modular forms and cohomology of the corrresponding modular curves.
More precisely, let N be a prime $$\geq 3$$, p another prime, V the $${\mathfrak p}$$-adic completion of the integer ring of a number field containing $${\mathbb{Q}}(\zeta_ N)$$ at a prime $${\mathfrak p}$$ dividing p, and K its field of fractions. Let Y and X be the semistable regular models over V of the principal modular curves Y(N) and X(N), and let $$\phi: E\to X$$ be the universal semi-abelian variety. Then $${\mathcal V}_ k=S^ k(R^ 1\phi_*{\mathbb{Z}}_ p)(k+1)$$ (the Tate twist of the k-th symmetric power of $$R^ 1\phi_*{\mathbb{Z}}_ p)$$ is a locally constant étale sheaf on $$Y\otimes_ V\bar K$$. The author shows that the étale cohomology group $$F_ 2=H^ 1(Y\otimes_ V\bar K,{\mathcal V}_ k)\otimes_{{\mathbb{Z}}_ p}\bar K\hat {\;}$$ has a filtration $$0=F_ 0\subset F_ 1\subset F_ 2$$, where the successive quotients are isomorphic to certain étale cohomology groups of X that are the p-adic versions of the groups occurring in the classical Eichler-Shimura theory. He also proves that $$F_ 1$$ is contained in parabolic cohomology, and that the quotient is the p-adic version of the space $$S_{k+2}(\Gamma (N))$$. From this he concludes the desired analogues of the Eichler-Shimura isomorphisms.
Reviewer: S.Kamienny

##### MSC:
 14H25 Arithmetic ground fields for curves 14L05 Formal groups, $$p$$-divisible groups 11F11 Holomorphic modular forms of integral weight
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##### References:
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