## 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)].

### MSC:

 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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