Frobenius distributions of Drinfeld modules of any rank
type: Item
instance of: scholarly article
title: Frobenius distributions of Drinfeld modules of any rank
zbMATH Open document ID: 1012.11048
author: Chantal David
published in: Journal of Number Theory
publication date: 2002-01-02T00:00:00Z
reviewed by: Ernst-Ulrich Gekeler
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11F80
zbMATH DE Number: 1684890
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: Lang-Trotter conjecture
MaRDI profile type: Publication
full work available at URL: jnth.2000.2664
OpenAlex ID: W2046292647
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DOI: 10.1006/JNTH.2000.2664
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review text: Let \(A =\mathbb{F}_q[T]\) be the polynomial ring in an indeterminate \(T\) over the finite field \(\mathbb{F}_q\), with quotient field \(F =\mathbb{F}_q(T)\). Drinfeld \(A\)-modules, and in particular Drinfeld modules of rank two, offer striking similarities with elliptic curves over the rationals. For example, if \(E/\mathbb{Q}\) is an elliptic curve without complex multiplication, \textit{S. Lang} and \textit{H. Trotter} [Lect. Notes Math. 504, Springer, Berlin (1976; Zbl 0329.12015)] formulated a precise conjecture about the growth of the function \(x\longmapsto \#\{p \leq x \mid \text{tr}(E,p) = t\}\), where \(t \in \mathbb{Z}\) is fixed and \(\text{tr}(E,p)\) is the Frobenius trace of the reduction of \(E\) at the prime \(p \in {\mathbb{N}},\) neglecting the finite number of \(p\) with bad reduction of \(E\). The Lang-Trotter conjecture, suitably adapted, may be translated to Drinfeld modules. The present paper is dedicated to a proof of a result toward the ``Lang-Trotter'' conjecture on Drinfeld modules. It is as follows. Let \(\Phi\) be a Drinfeld \(A\)-module of rank \(r \in {\mathbb{N}}\) over \(F\), \(t \in A\), and put NEWLINE\[NEWLINE\pi_t(k) := \# \{\text{primes }\mathfrak p \subset A~|\text{ deg } \mathfrak p = k\text{ and tr}(\Phi,\mathfrak p) = t\}.NEWLINE\]NEWLINE Then the author's result Theorem 1.1 states that NEWLINE\[NEWLINE\pi_t(k) \ll \frac{r}{k}q^{k\theta(r)},NEWLINE\]NEWLINE where \(\theta(r) = 1-\frac{1}{2(r^2+2r)}\) depends only on \(\Phi\), which is supposed to possess no ``complex multiplications'' other than the operators in \(A\). The result, which is the first in this direction, is much weaker than the actual conjecture, both since \(k\cdot \theta(r)\) is larger than the expected correct exponent \(k(1-\frac{1}{r})\) and since it doesn't provide lower estimates of \(\pi_t(k)\). The proof is modelled after \textit{J.-P. Serre}'s proof of a corresponding result for elliptic curves [see Publ. Math., Inst. Hautes Étud. Sci. 54, 123-202 (…
description: scientific article; zbMATH DE number 1684890