### Abstract

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_{Falg }(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.

Original language | English (US) |
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Pages (from-to) | 18-39 |

Number of pages | 22 |

Journal | Journal of Number Theory |

Volume | 212 |

DOIs | |

State | Published - Jul 2020 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

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## Cite this

*Journal of Number Theory*,

*212*, 18-39. https://doi.org/10.1016/j.jnt.2019.02.008