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The Langlands-Kottwitz approach for the modular curve. (English) Zbl 1309.14019
The object of this paper is to determine Hasse-Weil zeta functions of modular curves. The local factors at the places of good reduction were previously determined by R. E. Kottwitz [Math. Ann. 269, 287–300 (1984; Zbl 0533.14009)] using the trace formula for the Frobenius automorphism. Their methods are extended in this paper to the places of bad reduction. The methods of this paper have been further extended by the author to obtain the zeta functions of some other Shimura varieties [P. Scholze, Invent. Math. 192, No. 3, 627–661 (2013; Zbl 1309.14020)].

##### MSC:
 14G35 Modular and Shimura varieties 11G18 Arithmetic aspects of modular and Shimura varieties 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 22E50 Representations of Lie and linear algebraic groups over local fields
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