An average Chebotarev density theorem for generic rank 2 Drinfeld modules with complex multiplication
type: Item
instance of: scholarly article
title: An average Chebotarev density theorem for generic rank 2 Drinfeld modules with complex multiplication
zbMATH Open document ID: 1308.11061
author: Alina Carmen Cojocaru
author: Andrew Michael Shulman
published in: Journal of Number Theory
publication date: 2013-01-18T00:00:00Z
review text: In this paper under review, the authors establish an average Chebotarev density theorem in a family of Galois extensions which are division fields of rank 2 Drinfeld modules with complex multiplication over the rational function field over a finite field. The main results of this paper can be viewed as in the context of generalizations of the classical Bombieri-Vinogradov theorem which gives an average of the error terms in the prime number theorem for arithmetic progressions. To state the main result, we let \(q\) be an odd prime power and let \({\mathbb F}_q\) be the finite field of \(q\) elements. Let \(A = {\mathbb F}_q[T]\) and \( k = {\mathbb F}_q(T)\) be the polynomial ring and rational function field over \({\mathbb F}_q\) respectively. Let \(\psi : A \to k\{\tau\}\) be a rank \(r\geq 1\) Drinfeld \(A\)-module of generic characteristic over \(k.\) For a non-zero \(d\in A\), we denote by \(k(\psi[d])\) the \(d\)-division field obtained by adjoining \(k\) the submodule of \(d\)-division points \(\psi[d]\) of \(\psi.\) The paper concerns with the set of primes in \(A\) that split completely in \(k(\psi[d])\) for some monic \(d \in A\). Let \(c_d\) be the degree of constant subfield in \(k(\psi[d])\) over \({\mathbb F}_q.\) Let \(x\) be a positive real number, we set \(c_d(x) = c_d \) if \(c_d \mid x\) and \(c_d(x) = 0\) otherwise. For a monic polynomial \(m\in A\), we denote by \(\ell_m(x)\) the number of primes \(p\in A\) such that \(\deg p = x\) and the ideal \(pA\) splits completely in \(k(\psi[m])\) over \(k\). By the Chebotarev density theorem, for large \(x\), \(\ell_m(x)\) is asymptotically equal to \(\left(c_m(x)/[k(\psi[m]) : k]\right) \pi_A(x)\) where as usual \(\pi_A(x)\) denotes the number of primes of \(A\) whose degrees equal \(x.\) In the case of rank 2 Drinfeld modules with complex multiplication, the authors obtain an average of the errors between \(\ell_m(x)\) and \(\left(c_m(x)/[k(\psi[m]) : k]\right) \pi_A(x)\). Let now \(\psi\) be a Drinfel…
reviewed by: Liang-Chung Hsia
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11R45
zbMATH DE Number: 6126840
zbMATH Keywords: Drinfeld module
zbMATH Keywords: Weil number
zbMATH Keywords: Chebotarev density theorem, prime distribution
MaRDI profile type: Publication
full work available at URL: j.jnt.2012.07.001
OpenAlex ID: W1993472949
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DOI: 10.1016/J.JNT.2012.07.001
description: scientific article; zbMATH DE number 6126840