Moduli problem and points on some twisted Shimura varieties of PEL type. (English) Zbl 1213.11128
Summary: We describe the moduli problem of a “twist” of some simple Shimura varieties of PEL type that appear in R. Kottwitz’s papers [Automorphic forms, Shimura varieties, and L-functions. Vol. I, Proc. Conf., Ann Arbor/MI (USA) 1988, Perspect. Math. 10, 161–209 (1990; Zbl 0743.14019), J. Am. Math. Soc. 5, No. 2, 373–444 (1992; Zbl 0796.14014) and Invent. Math. 108, No. 3, 653–665 (1992; Zbl 0765.22011)] and then, using the moduli problem, we compute the cardinality of the set of points over finite fields of the twisted Shimura varieties. Using this result, we compute the zeta function of the twisted varieties. The twist of the Shimura varieties is done by a mod \(q\) representation of the absolute Galois group of the reflex field of the Shimura varieties.
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