8.83 (1982)

In the notation of 8.82, for $r(\pi) > 0$, one can define via Hecke algebras a formal Dirichlet series with Euler multiplication (see G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, 1971). Does there exist an algebraic Hasse–Weil variety whose $\zeta$-function is this Dirichlet series? There are several conjectures.