Hodge-Tate structures and modular forms. (English) Zbl 0646.14026

Faltings, Gerd

Math. Ann. 278, 133-149 (1987).

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

Keywords:

p-divisible group; Hodge-Tate structure of weights; p-adic Galois representation; p-adic analogue of the Eichler-Shimura isomorphisms; principal modular curves

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