zbMATH — the first resource for mathematics

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

14H25 Arithmetic ground fields for curves
14L05 Formal groups, \(p\)-divisible groups
11F11 Holomorphic modular forms of integral weight
Full Text: DOI EuDML
[1] Bloch, S., Kato, K.: P-adic etale cohomology. Preprint 1984 · Zbl 0613.14017
[2] Deligne, P.: Formes modulaires et représentations l-adiques. Seminaire Bourbaki355 (1969)
[3] Deligne, P., Rapoport, M.: Les schémas de modules de courbes elliptiques. Lect. Notes Math. 349, 143-316. Berlin, Heidelberg, New York, Springer 1973 · Zbl 0281.14010
[4] Fontaine, J.-M.: Formes différentielles et modules de Tate des variétés abeliennes sur les corps locaux. Invent. Math.65, 379-409 (1982) · Zbl 0502.14015
[5] Mazur, B., Messing, W.: Universal extensions and one dimensional cohomology. Lect. Notes Math. 370, Berlin, Heidelberg, New York: Springer 1974 · Zbl 0301.14016
[6] Ribet, A.: On l-adic representations attached to modular forms. II. Preprint 1984
[7] Serre, J-P.: Abelian L-adic representations and elliptic curves. New York: Benjamin 1968
[8] Tate, J.T.: P-Divisible groups. Conference on Local Fields, Driebergen, Berlin, Heidelberg, New York: Springer 1967 · Zbl 0157.27601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.