| Popis: |
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 F p ( 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. |
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