We propose a conjectural explicit isogeny from the Jacobians of hyperelliptic Drinfeld modular curves to the Jacobians of hyperelliptic modular curves of D-elliptic sheaves. The kernel of the isogeny is a subgroup of the cuspidal divisor group constructed by examining the canonical maps from the cuspidal divisor group into the component groups.

We study the existence of rational points on modular curves of D-elliptic sheaves over local fields and the structure of special fibres of these curves. We discuss some applications which include finding presentations for arithmetic groups arising from quaternion algebras, finding the equations of modular curves of D-elliptic sheaves, and constructing curves violating the Hasse principle.

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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.

Let A = F q [T ] be the polynomial ring over F q , and F be the field of fractions of A. Let φ be a Drinfeld A-module of rank r ≥ 2 over F . For all but finitely many primes p ✁ A, one can reduce φ modulo p to obtain a Drinfeld A-module φ ⊗ F p of rank r over F p = A/p. The endomorphism ring E p = End Fp (φ ⊗ F p ) is an order in an imaginary field extension K of F of degree r. Let O p be the integral closure of A in K, and let π p ∈ E p be the Frobenius endomorphism of φ ⊗ F p . Then we have the inclusion of orders A[π p ] ⊂ E p ⊂ O p in K. We prove that if End F alg (φ) = A, then for arbitrary non-zero ideals n, m of A there are infinitely many p such that n divides the index χ(E p /A[π p ]) and m divides the index χ(O p /E p ). We show that the index χ(E p /A[π p ]) is related to a reciprocity law for the extensions of F arising from the division points of φ. In the rank r = 2 case we describe an algorithm for computing the orders A[π p ] ⊂ E p ⊂ O p , and give some computational data.2010 Mathematics Subject Classification. 11G09, 11R58.