[Abélien est égal à A-fini pour les A-modules d’Anderson]
Anderson introduced $t$-modules as higher dimensional analogs of Drinfeld modules. Attached to such a $t$-module, there are its $t$-motive and its dual t-motive. The $t$-module gets the attribute “abelian” when the $t$-motive is a finitely generated module, and the attribute “$t$-finite” when the dual $t$-motive is a finitely generated module. The main theorem of this article is the affirmative answer to the long standing question whether these two attributes are equivalent. The proof relies on an invariant of the $t$-module and a condition for that invariant which is necessary and sufficient for both being abelian and being $t$-finite. We further show that this invariant also provides the information whether the t-module is pure or not. Moreover, we conclude that also over general coefficient rings A, i.e. for Anderson A-modules, the attributes of being abelian and being A-finite are equivalent.
Anderson a introduit les $t$-modules en tant qu’analogues de dimension supérieure des modules de Drinfeld. Attachés à un tel $t$-module, il y a son $t$-motif et son t-motif dual. Le $t$-module obtient l’attribut « abélien » lorsque le $t$-motif est un module de génération finie, et l’attribut « $t$-fini » lorsque le $t$-motif dual est un module de génération finie. Le théorème principal de cet article est la réponse affirmative à la question de longue date de savoir si ces deux attributs sont équivalents. La preuve repose sur un invariant du $t$-module et une condition pour cet invariant qui est nécessaire et suffisante pour être à la fois abélien et $t$-fini. Nous montrons en outre que cet invariant fournit également l’information si le $t$-module est pur ou non. De plus, nous concluons que également sur les anneaux de coefficients généraux A, c’est-à-dire pour les A-modules d’Anderson, les attributs d’être abélien et d’être A-fini sont équivalents.
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Keywords: abelian, $t$-module, $t$-motive, skew field, Newton polygon
Mots-clés : abélien, $t$-module, $t$-motif, corps gauches, polygone de Newton
Maurischat, Andreas 1
@unpublished{AIF_0__0_0_A42_0,
author = {Maurischat, Andreas},
title = {Abelian equals {A-finite} for {Anderson} {A-modules}},
journal = {Annales de l'Institut Fourier},
year = {2026},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
doi = {10.5802/aif.3754},
language = {en},
note = {Online first},
}
Maurischat, Andreas. Abelian equals A-finite for Anderson A-modules. Annales de l'Institut Fourier, Online first, 46 p.
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