{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T09:13:11Z","timestamp":1775466791061,"version":"3.50.1"},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":7862,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1992,9]]},"abstract":"<jats:p>Cet article fait suite \u00e0 [D] dont il utilise les r\u00e9sultats.<\/jats:p><jats:p>Soit if \u2112 langage constitu\u00e9 de deux constantes 0 et 1, et de deux fonctions + et \u00b7; si <jats:italic>A<\/jats:italic> est un sous-ensemble d'une structure de \u2112, nous noterons if \u2112 (<jats:italic>A<\/jats:italic>) le langage obtenu en ajoutant \u00e0 \u2112 if les \u00e9l\u00e9ments de <jats:italic>A<\/jats:italic> comme constantes.<\/jats:p><jats:p>Tous les corps consid\u00e9r\u00e9s seront commutatifs. Nous appellerons <jats:italic>corps de courbe<\/jats:italic> sur un corps <jats:italic>k<\/jats:italic> une extension <jats:italic>K<\/jats:italic> de <jats:italic>k<\/jats:italic> finiment engendr\u00e9e de degr\u00e9 de transcendance 1 sur <jats:italic>k<\/jats:italic> telle que <jats:italic>k<\/jats:italic> soit relativement alg\u00e9briquement clos dans <jats:italic>K<\/jats:italic>.<\/jats:p><jats:p>Nous appellerons <jats:italic>ensemble de coefficients<\/jats:italic> d'une courbe sur un corps <jats:italic>k<\/jats:italic> un sous-ensemble <jats:italic>A<\/jats:italic> de <jats:italic>K<\/jats:italic> tel qu'il existe un syst\u00e8me de g\u00e9n\u00e9rateurs de l'id\u00e9al de cette courbe dont les coefficient sont des \u00e9l\u00e9ments de <jats:italic>A<\/jats:italic>.<\/jats:p><jats:p>Nous nous proposons d'\u00e9tudier les conjectures suivantes:<\/jats:p><jats:p>1. Conjecture. Si <jats:italic>K<\/jats:italic> est un corps de courbe sur un corps alg\u00e9briquement clos <jats:italic>k<\/jats:italic>, il existe un sous-ensemble fini <jats:italic>A<\/jats:italic> de <jats:italic>k<\/jats:italic> tel que tout corps de courbe sur <jats:italic>k<\/jats:italic>\u00e9l\u00e9mentairement \u00e9quivalent \u00e0 <jats:italic>K<\/jats:italic> dans le langage \u2112 (<jats:italic>A<\/jats:italic>), lui est <jats:italic>k<\/jats:italic>-isomorphe.<\/jats:p><jats:p>2. Conjecture. Deux corps de courbe sur un corps alg\u00e9briquement clos <jats:italic>K<\/jats:italic> \u00e9l\u00e9mentairement \u00e9quivalents dans le langage \u2112 sont isomorphes.<\/jats:p><jats:p>Nous d\u00e9montrerons ces conjectures lorsque le genre est diff\u00e9rent de 1, et si le genre est 1, lorsque la caract\u00e9ristique est nulle et le corps de courbe sans multiplication complexe.<\/jats:p><jats:p>Une version plus simple de cet article est parue dans [D\u2032].<\/jats:p>","DOI":"10.2307\/2275432","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:45:35Z","timestamp":1146941135000},"page":"808-823","source":"Crossref","is-referenced-by-count":4,"title":["\u00c9quivalence \u00e9l\u00e9mentaire et isomorphisme des corps de courbe sur un corps alg\u00e9briquement clos"],"prefix":"10.1017","volume":"57","author":[{"given":"Jean-Louis","family":"Duret","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200022362_ref002","first-page":"1","volume-title":"S\u00e9minaire de structures alg\u00e9briques ordonn\u00e9es 1989-1990","author":"Duret","year":"1990"},{"key":"S0022481200022362_ref008","volume-title":"A course of modern analysis","author":"Whittaker","year":"1927"},{"key":"S0022481200022362_ref005","doi-asserted-by":"publisher","DOI":"10.1515\/crll.1873.76.21"},{"key":"S0022481200022362_ref006","volume-title":"Introduction to algebraic geometry","author":"Lang","year":"1972"},{"key":"S0022481200022362_ref003","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107359901"},{"key":"S0022481200022362_ref001","first-page":"948","volume":"51","author":"Duret","year":"1986","journal-title":"Sur la th\u00e9orie \u00e9l\u00e9mentaire des corps de fonctions"},{"key":"S0022481200022362_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3849-0"},{"key":"S0022481200022362_ref007","volume-title":"Elliptic curves","volume":"326","author":"Robert","year":"1973"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200022362","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,16]],"date-time":"2019-05-16T17:01:31Z","timestamp":1558026091000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200022362\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1992,9]]},"references-count":8,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1992,9]]}},"alternative-id":["S0022481200022362"],"URL":"https:\/\/doi.org\/10.2307\/2275432","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1992,9]]}}}