### 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
_{
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.

Original language | English (US) |
---|---|

Journal | Journal of Number Theory |

DOIs | |

State | Published - Jan 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

}

**Endomorphism rings of reductions of Drinfeld modules.** / Garai, Sumita; Papikian, Mihran.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Endomorphism rings of reductions of Drinfeld modules

AU - Garai, Sumita

AU - Papikian, Mihran

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85063754752&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85063754752&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2019.02.008

DO - 10.1016/j.jnt.2019.02.008

M3 - Article

AN - SCOPUS:85063754752

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -