Moduli problem and points on some twisted Shimura varieties of PEL type

In this article we describe the moduli problem of a “twist” of some simple Shimura varieties of PEL type that appear in Kottwitz's papers [R. Kottwitz, Shimura varieties and λ-adic representations, in: Automorphic Forms, Shimura Varieties and L-Functions, part 1, in: Perspect. Math., vol. 10, Academic Press, San Diego, CA, 1990, pp. 161–209; R. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (2) (1992) 373–444] and [R. Kottwitz, On the λ-adic representations associated to some simple Shimura varieties, Invent. Math. 108 (1992) 653–665] 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.