Endomorphism rings of reductions of Drinfeld modules

Sumita Garai, Mihran Papikian

Research output: Contribution to journalArticle

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 languageEnglish (US)
JournalJournal of Number Theory
DOIs
StatePublished - Jan 1 2019

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Drinfeld Modules
Endomorphism Ring
Divides
Reciprocity Law
Integral Closure
Module
Field extension
Endomorphism
Polynomial ring
Frobenius
Modulo
Division
Inclusion
Computing
Arbitrary

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

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

In: Journal of Number Theory, 01.01.2019.

Research output: Contribution to journalArticle

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