In the previous part of this paper, we constructed a large family of Hecke algebras on some classical groups G defined over p-adic fields in order to understand their admissible representations. Each Hecke algebra is associated to a pair (JΣ, ρΣ) of an open compact subgroup JΣ and its irreducible representation ρΣ which is constructed from given data Σ = (Γ, P′0, ϱ). Here, Γ is a semisimple element in the Lie algebra of G, P′0 is a parahoric subgroup in the centralizer of Γ in G, and ϱ is a cuspidal representation on the finite reductive quotient of P′0. In this paper, we explicitly describe those Hecke algebras when P′0 is a minimal parahoric subgroup, ϱ is trivial and ρΣ is a character.

We describe birational models and decide the rationality/unirationality of moduli spaces $\cal A$d (and $\cal A$levd) of (1, d)-polarized Abelian surfaces (with canonical level structure, respectively) for small values of d. The projective lines identified in the rational/unirational moduli spaces correspond to pencils of Abelian surfaces traced on nodal threefolds living naturally in the corresponding ambient projective spaces, and whose small resolutions are new Calabi–Yau threefolds with Euler characteristic zero.