Deligne's congruence and supersingular reduction of Drinfeld modules
type: Item
instance of: scholarly article
title: Deligne's congruence and supersingular reduction of Drinfeld modules
zbMATH Open document ID: 0931.11019
author: Gunther Cornelissen
published in: Archiv der Mathematik
publication date: 1999-11-30T00:00:00Z
review text: Let \(A\) be the polynomial ring \(\mathbb{F}_q [T]\) over the finite field \(\mathbb{F}_q\). Drinfeld \(A\)-modules \(\varphi\) of rank two over the quotient field \(\mathbb{F}_q(T)\) share many properties with elliptic curves over the field \(\mathbb{Q}\) of rationals. In particular, there is the notion of supersingular reduction of \(\varphi\) at a prime \({\mathfrak p}\) of \(A\). It is known that, for given \(\varphi\), the number of supersingular primes \({\mathfrak p}\) is infinite with density zero, if \(\varphi\) is neither of CM-type nor ``exceptional'' [\textit{M. L. Brown}, Invent. Math. 110, 419-439 (1992; Zbl 0767.11028); \textit{C. David}, Duke Math. J. 78, 399-412 (1995; Zbl 0842.11025); \textit{B. Poonen}, Math. Res. Notices 3, 151-159 (1998; Zbl 0898.11024)]. The author proves estimates (with explicit constants) for the number \(S(k,\varphi)\) of supersingular \({\mathfrak p}\) of degree less or equal to \(k\in \mathbb{N}\). These estimates are rather weak asymptotically, where they yield \(S(k,\varphi)= O(q^{k-2}/k)\), but supply meaningful information if \(k\) is small. The proof relies on the theory of Drinfeld modular forms, notably on ``Deligne's congruence'' between the Hasse invariant of \(\varphi\) reduced at \({\mathfrak p}\) and the Eisenstein series of weight \(q^{\deg {\mathfrak p}}-1\).
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11F52
Mathematics Subject Classification ID: 11F33
zbMATH DE Number: 1309372
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: supersingular reduction
zbMATH Keywords: Drinfeld modular forms
reviewed by: Ernst-Ulrich Gekeler
MaRDI profile type: Publication
full work available at URL: s000130050342
OpenAlex ID: W2001373015
DOI: 10.1007/S000130050342
description: scientific article; zbMATH DE number 1309372