On the coefficients of modular polynomials for Drinfeld modules
type: Item
instance of: scholarly article
title: On the coefficients of modular polynomials for Drinfeld modules
zbMATH Open document ID: 0915.11035
author: Liang-Chung Hsia
published in: Journal of Number Theory
publication date: 1999-07-06T00:00:00Z
review text: Let \(h(\phi)\) be the logarithmic height of the polynomial \(\phi\) (i.e., the maximum of logarithms of absolute values of the coefficients of \(\phi)\). A result of \textit{P. Cohen} [Math. Proc. Camb. Philos. Soc. 95, 389-402 (1984; Zbl 0541.10026)] states that the modular polynomial \(\phi_m\) \((m\in\mathbb{N})\) of elliptic curves satisfies \[ h(\phi_m)= 6\psi(m)(\log m- 2\kappa(m)+ O(1)) \] with well-known arithmetic functions \(\psi(m)\) and \(\kappa(m)\). Following the analogy between elliptic curves and Drinfeld \(A\)-modules of rank two (here \(A\) is the polynomial ring \(\mathbb{F}_q[T]\) over a finite field \(\mathbb{F}_q\)), the author proves a similar statement for the height of the Drinfeld modular polynomial \(\phi_m(x,y)\) (where \(m\in A\) is a non-constant monic polynomial over \(\mathbb{F}_q\)). Main ingredients are Farey sequences for \(A\) (the theory of which is partially developed), estimates on the Drinfeld invariant \(j(t)\) in terms of the imaginary distance \(| z|_i\), and some non-archimedean function theory.
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11T55
Mathematics Subject Classification ID: 11B57
zbMATH DE Number: 1236189
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: logarithmic height
zbMATH Keywords: Drinfeld modular polynomial
zbMATH Keywords: Farey sequences
zbMATH Keywords: Drinfeld invariant
reviewed by: Ernst-Ulrich Gekeler
MaRDI profile type: Publication
OpenAlex ID: W2074603345
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DOI: 10.1006/JNTH.1998.2271
description: scientific article; zbMATH DE number 1236189