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Modular forms and \(p\)-adic Hodge theory. (English) Zbl 0877.11034
For an elliptic modular form, P. Deligne constructs an associated \(\ell\)-adic representation of the absolute Galois group \(\text{Gal} (\overline\mathbb{Q}/ \mathbb{Q})\) [Sémin. Bourbaki 1968/69, No. 355, 139-172 (1971; Zbl 0206.49901)]. In the paper, it is shown that the construction is compatible with the local Langlands correspondence at \(p= \ell\) in the sense of Fontaine using a result of Carayol for \(p\neq\ell\).
More precisely, the restriction to the local Galois group \(\text{Gal} (\overline \mathbb{Q}_p/ \mathbb{Q}_p)\) of the \(p\)-adic representation \(V_p(f)\) associated to a modular form \(f\) is potentially semi-stable as follows from a theorem of T. Tsuji [Invent. Math. 137, No. 2, 233–411 (1999; Zbl 0945.14008)]. J.-M. Fontaine has associated a representation of the Weil-Deligne group to a potentially semi-stable representation by using the ring \(B_{st}\) [Représentations \(\ell\)-adiques potentiellement semi-stables. Périodes \(p\)-adiques, Astérisque 223, 321-347 (1994)]. On the other hand, an automorphic representation \(\pi_f\) of the adele group \(GL_2 (\mathbb{A})\) is associated to a modular form \(f\). By factoring the tensor product \(\pi_f= \bigotimes \pi_{p,f}\), we obtain an irreducible admissible representation \(\pi_{p,f}\) of \(GL_2 (\mathbb{Q}_p)\). The main result asserts that the representation of the Weil-Deligne group associated to \(V_p(f)\) corresponds to the irreducible admissible representation \(\pi_{p,f}\) of \(GL_2 (\mathbb{Q}_p)\) in the sense of the local Langlands correspondence.
The proof goes as follows. Since the correspondence is established for \(\ell \neq p\) by H. Carayol [Ann. Sci. Éc. Norm. Supér., IV. Sér. 19, 409-468 (1986; Zbl 0616.10025)], it suffices to compare \(p\) and \(\ell\). The comparison is made by reducing to the Lefschetz trace formula, which is the same for \(p\) and \(\ell\), by using the weight spectral sequences of Steenbrink, M. Rapoport, and T. Zink for \(\ell\) [Invent. Math. 68, 21-101 (1982; Zbl 0498.14010)] and of A. Mokrane for \(p\) [Duke Math. J. 72, 301-377 (1993; Zbl 0834.14010)].

11F85 \(p\)-adic theory, local fields
11S37 Langlands-Weil conjectures, nonabelian class field theory
14F30 \(p\)-adic cohomology, crystalline cohomology
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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