Hodge-Tate structures and modular forms
type: Item
instance of: scholarly article
title: Hodge-Tate structures and modular forms
zbMATH Open document ID: 0646.14026
DOI: 10.1007/BF01458064
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164285
review text: 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.
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 4053794
zbMATH Keywords: p-divisible group
zbMATH Keywords: Hodge-Tate structure of weights
zbMATH Keywords: p-adic Galois representation
zbMATH Keywords: p-adic analogue of the Eichler-Shimura isomorphisms
zbMATH Keywords: principal modular curves
reviewed by: Sheldon Kamienny
MaRDI profile type: Publication
cites work: \(p\)-adic étale cohomology
cites work: Les Schémas de Modules de Courbes Elliptiques
cites work: Formes différentielles et modules de Tate des variétés abéliennes sur les corps locaux. (Différential forms and Tate modules of abelian varieties over local fields)
cites work: Universal extensions and one dimensional crystalline cohomology
cites work: Q5539560
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description: scientific article; zbMATH DE number 4053794