Drinfeld–Stuhler modules

Abstract

We study D-elliptic sheaves in terms of their associated modules, which we call Drinfeld-Stuhler modules. First, we prove some basic results about Drinfeld-Stuhler modules and give explicit examples. Then we examine the existence and properties of Drinfeld-Stuhler modules with large endomorphism rings, which are analogous to CM and supersingular Drinfeld modules. Finally, we examine the fields of moduli of Drinfeld-Stuhler modules.

Drinfeld-Stuhler modules and the Hasse principle

“…It can be shown that the kernel of any nonzero morphism u is a finite group scheme over K (i.e., a nonzero morphism is an isogeny), and Hom K (φ, ψ) is a free Amodule of rank ≤ d 2 ; cf. [25]. We denote End K (φ) = Hom K (φ, φ) and Aut K (φ) = End K (φ) × .…”

“…Therefore, the Tate module T p (ψ) contains a nonzero unramified submodule. Let M(ψ) be the O D -motive associated to ψ; see Section 3 of [25]. The F -vector space M(ψ) ⊗ A F is a D opp -vector space of dimension 1, where D opp denotes the opposite algebra of D. The fact that T p (ψ) contains a nonzero unramified submodule, implies that M(ψ) ⊗ F contains a nonzero unramified vector subspace W .…”

“…Over the years, the third author of this paper have studied the arithmetic properties of D-elliptic sheaves and their modular varieties, trying to extend to this setting the rich theory of abelian surfaces with quaternionic multiplication and Shimura curves; see, for example, [19], [20], [21], [22], [23], [24], [25]. The current paper is a natural continuation of [25].…”

“…These conditions are listed in Theorems 6.6, 6.10, 6.13, which are the main results of this paper. A technical issue arises here from the fact that X D is only a coarse moduli scheme, so the K-rational points on X D may not be represented by Drinfeld-Stuhler O D -modules defined over K. Fortunately, a result from [25] partly resolves this: If X D (K) = ∅, then there is a Drinfeld-Stuhler O D -module defined over K if and only if D ⊗ F K = M d (K).…”

“…Let H K be the maximal unramified abelian extension of K in which ∞ splits completely (i.e., H K is the Hilbert class field of K). The Galois group Gal(H K /K) is isomorphic to the class group of the integral closure B of A in K. By developing a theory of "complex multiplication" for Drinfeld-Stuhler modules, the third author has shown in [25,Thm. 4.10] that X D (H K ) = ∅.…”