Frobenius fields for Drinfeld modules of rank 2

Compositio Mathematica

Compositio Mathematica / Volume 144 / Issue 04 / July 2008, pp 827-848
Copyright © Foundation Compositio Mathematica 2008
DOI: http://dx.doi.org/10.1112/S0010437X07003387 (About DOI), Published online: 17 July 2008

Table of Contents - 2008 - Volume 144, Issue 04

Research Article

Frobenius fields for Drinfeld modules of rank 2

Article author query
cojocaru ac [Google Scholar]
david c [Google Scholar]

Alina Carmen Cojocaru^a1^a2 and Chantal David^a3

^a1Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 322 SEO, 851 S. Morgan Street, Chicago, IL 60607, USA (email: cojocaru@math.uic.edu)

^a2Institute of the Romanian Academy, Calea Grivitei 21, 010702, Bucharest, Romania

^a3Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve West, Montréal, Québec, H3G 1M8, Canada (email: cdavid@mathstat.concordia.ca)

Abstract

Let xs03D5 be a Drinfeld module of rank 2 over the field of rational functions $F=\mathbb {F}_q(T)$ , with $\mathrm {End}_{\bar {F}}(\phi ) = \mathbb {F}_q[T]$ . Let K be a fixed imaginary quadratic field over F and d a positive integer. For each prime $\mathfrak {p}$ of good reduction for xs03D5 , let $\pi _{\mathfrak {p}}(\phi )$ be a root of the characteristic polynomial of the Frobenius endomorphism of xs03D5 over the finite field $\mathbb {F}_q[T] / \mathfrak {p}$ . Let Π(K;d) be the number of primes $\mathfrak {p}$ of degree d such that the field extension $F(\pi _{\mathfrak {p}}(\phi ))$ is the fixed imaginary quadratic field K. We present upper bounds for Π(K;d) obtained by two different approaches, inspired by similar ones for elliptic curves. The first approach, inspired by the work of Serre, is to consider the image of Frobenius in a mixed Galois representation associated to K and to the Drinfeld module xs03D5 . The second approach, inspired by the work of Cojocaru, Fouvry and Murty, is based on an application of the square sieve. The bounds obtained with the first method are better, but depend on the fixed quadratic imaginary field K. In our application of the second approach, we improve the results of Cojocaru, Murty and Fouvry by considering projective Galois representations.