Crystalline cohomology and \(GL(2,\mathbb{Q})\)
type: Item
instance of: scholarly article
title: Crystalline cohomology and \(GL(2,\mathbb{Q})\)
zbMATH Open document ID: 0854.14010
DOI: 10.1007/BF02783205
author: Bruce W. Jordan
author: Gerd Faltings
published in: Israel Journal of Mathematics
publication date: 1996-12-19T00:00:00Z
review text: The aim of the paper is to study modular Galois representations attached to automorphic forms \(\text{mod } p\) of weight \(k\geq 2\). For a fixed level \(N\geq 3\) let \(\Gamma (N) \subset \text{SL} (2, \mathbb{Z})\) be the subgroup of matrices which are congruent to the identity modulo \(N\). With the usual notation one has the open modular curve \(Y(N)\) classifying elliptic curves with full level \(N\) structure, and its compactification (i.e. adding the cusps) \(X(N)\). There is a universal elliptic curve \(\pi: E\to Y= Y(N)\) extending to a semi-abelian variety (the semi-stable compactification) \(\overline {\pi}\to \overline {E}\to X= X(N)\). This is all defined over \(\mathbb{Q}_p (e^{2\pi i/N})\). For a prime \(p\), \((p,N)=1\), set \(\mathbb{V}= R^1 \pi_{*,\text{ét}} (\mathbb{Z}_p)\). Similarly, for a congruence subgroup \(\Gamma \supseteq \Gamma (N)\) one has the moduli curves \(Y(\Gamma)= Y(N)/ (\Gamma/ \Gamma (N))\) and its completion \(X(\Gamma)\) defined over some number field contained in \(\mathbb{Q} (e^{2\pi i/N})\). Write \(K\) for the completion of this number field at a prime lying over \(p \nmid N\) and write \(H^1_{\text{par}}= \text{Image} (H^1_!\to H^1)\), where \(H^1_!\) denotes cohomology with compact support. As a first result one has the following theorem \((*)\): With notations as above, let \(k> p\). Then the \(V= H^1_{\text{par}} (Y (\Gamma )_{\overline {K}}, \text{Symm}^{k- 2} (\mathbb{V}))\), \(\text{Gal} (\overline {K}/k)\)-representation, is crystalline, and the \(F\)-crystal corresponding to the dual of \(V\) as a canonical Frobenius filtration \(M= F^0 \supset F^{k-1} \supset 0\), with \(F^{k-1} \cong S_k (\Gamma)\) (cusp forms of weight \(k\) for \(\Gamma\)) and \(F^0/ F^{k-1} \cong S_k (\Gamma)^*\). In particular, for \(\Gamma= \Gamma_1 (N)\) the corresponding moduli curves \(Y= Y(\Gamma)= Y_1 (N)\) and \(X= X(\Gamma)= X_1 (N)\) are defined over \(\mathbb{Q}\). \(Y\) admits correspondences \(\langle d\rangle\), \(d\in (\mathbb…
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 784910
zbMATH Keywords: modular Galois representations
zbMATH Keywords: automorphic forms
zbMATH Keywords: modular curve
zbMATH Keywords: Hecke operators
zbMATH Keywords: Frobenius
zbMATH Keywords: geometric Frobenius
zbMATH Keywords: arithmetic Frobenius
zbMATH Keywords: Eisenstein series
zbMATH Keywords: Eisenstein ideals
zbMATH Keywords: Hecke algebra
zbMATH Keywords: crystalline extensions
zbMATH Keywords: multiplicity one theorem
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 784910