{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T03:13:10Z","timestamp":1783653190809,"version":"3.55.0"},"reference-count":21,"publisher":"American Mathematical Society (AMS)","issue":"356","license":[{"start":{"date-parts":[[2026,6,27]],"date-time":"2026-06-27T00:00:00Z","timestamp":1782518400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Let\n \n \n \n \n \n \n K<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:mrow>\n \\mathbb {{{K}}}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a field of degree\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and discriminant with absolute value\n \n \n \n \n \u0394\n \n <\/mml:mi>\n \\Delta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Under the assumption of the validity of the Generalized Riemann Hypothesis, we provide a new algorithm to compute a set of generators of the class group of\n \n \n \n \n \n \n K<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:mrow>\n \\mathbb {{{K}}}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and prove that the norm of the ideals in that set is\n \n \n \n \n \n \u2264\n \n <\/mml:mo>\n (<\/mml:mo>\n 4<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n (<\/mml:mo>\n 2<\/mml:mn>\n n<\/mml:mi>\n )<\/mml:mo>\n )<\/mml:mo>\n \n log<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n \u2061\n \n <\/mml:mo>\n \n \u0394\n \n <\/mml:mi>\n <\/mml:mrow>\n \\leq (4-1\/(2n))\\log ^2\\Delta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , except for a finite number of fields of degree\n \n \n \n \n n<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n n\\leq 4<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . For those fields, the conclusion holds with the slightly larger limit\n \n \n \n \n (<\/mml:mo>\n 4<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n (<\/mml:mo>\n 2<\/mml:mn>\n n<\/mml:mi>\n )<\/mml:mo>\n +<\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n (<\/mml:mo>\n 2<\/mml:mn>\n \n n<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n )<\/mml:mo>\n \n log<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n \u2061\n \n <\/mml:mo>\n \n \u0394\n \n <\/mml:mi>\n <\/mml:mrow>\n (4-1\/(2n)+1\/(2n^2))\\log ^2\\Delta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . When the cardinality of\n \n \n \n \n \n C<\/mml:mi>\n <\/mml:mrow>\n \n \n \u2113\n \n <\/mml:mi>\n <\/mml:mrow>\n \\mathcal {C}\\!\\ell<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is odd the bounds improve to\n \n \n \n \n (<\/mml:mo>\n 4<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n 2<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n (<\/mml:mo>\n 3<\/mml:mn>\n n<\/mml:mi>\n )<\/mml:mo>\n )<\/mml:mo>\n \n log<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n \u2061\n \n <\/mml:mo>\n \n \u0394\n \n <\/mml:mi>\n <\/mml:mrow>\n (4-2\/(3n))\\log ^2\\Delta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , again with finitely many exceptions in degree\n \n \n \n \n n<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n n\\leq 4<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and to\n \n \n \n \n (<\/mml:mo>\n 4<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n 2<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n (<\/mml:mo>\n 3<\/mml:mn>\n n<\/mml:mi>\n )<\/mml:mo>\n +<\/mml:mo>\n 3<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n (<\/mml:mo>\n 8<\/mml:mn>\n \n n<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n )<\/mml:mo>\n \n log<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n \u2061\n \n <\/mml:mo>\n \n \u0394\n \n <\/mml:mi>\n <\/mml:mrow>\n (4-2\/(3n)+3\/(8n^2))\\log ^2\\Delta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n without exceptions.\n <\/p>","DOI":"10.1090\/mcom\/4114","type":"journal-article","created":{"date-parts":[[2025,6,27]],"date-time":"2025-06-27T10:53:32Z","timestamp":1751021612000},"page":"3145-3163","source":"Crossref","is-referenced-by-count":1,"title":["Breaking the 4 barrier for the bound of a generating set of the class group"],"prefix":"10.1090","volume":"94","author":[{"given":"Lo\u00efc","family":"Greni\u00e9","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Giuseppe","family":"Molteni","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2025,6,27]]},"reference":[{"issue":"191","key":"1","doi-asserted-by":"publisher","first-page":"355","DOI":"10.2307\/2008811","article-title":"Explicit bounds for primality testing and related problems","volume":"55","author":"Bach, Eric","year":"1990","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"262","key":"2","doi-asserted-by":"publisher","first-page":"1185","DOI":"10.1090\/S0025-5718-07-02003-0","article-title":"Small generators of the ideal class group","volume":"77","author":"Belabas, Karim","year":"2008","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"5","key":"3","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1007\/BF01236037","article-title":"Class number parity and unit signature","volume":"59","author":"Berger, Ruth I.","year":"1992","journal-title":"Arch. Math. (Basel)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-889X","issn-type":"print"},{"key":"4","series-title":"Series in Pure Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1142\/0663","volume-title":"Class number parity","volume":"8","author":"Conner, P. E.","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/9971506696"},{"key":"5","series-title":"Graduate Texts in Mathematics","isbn-type":"print","volume-title":"Multiplicative number theory","volume":"74","author":"Davenport, Harold","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/0387950974","edition":"3"},{"key":"6","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-01512-1","volume-title":"On the class number of Abelian number fields","author":"Hasse, Helmut","year":"2019","ISBN":"https:\/\/id.crossref.org\/isbn\/9783030015107"},{"issue":"240","key":"7","doi-asserted-by":"publisher","first-page":"1745","DOI":"10.1090\/S0025-5718-01-01373-4","article-title":"Bounds for the smallest norm in an ideal class","volume":"71","author":"de la Maza, Ana-Cecilia","year":"2002","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"313","key":"8","doi-asserted-by":"publisher","first-page":"2483","DOI":"10.1090\/mcom\/3281","article-title":"Explicit bounds for generators of the class group","volume":"87","author":"Greni\u00e9, Lo\u00efc","year":"2018","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"9","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1112\/jlms\/s1-35.2.241","article-title":"On the product of \ud835\udc5b complex homogeneous linear forms","volume":"35","author":"Mulholland, H. P.","year":"1960","journal-title":"J. London Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6107","issn-type":"print"},{"key":"10","unstructured":"A. M. Odlyzko, Discriminant bounds, 1976, available at \\url{http:\/\/www.dtc.umn.edu\/ odlyzko\/unpublished\/index.html}."},{"key":"11","unstructured":"A. M. Odlyzko, Tables of zeros of the Riemann zeta function, 1998, available at \\url{https:\/\/www-users.cse.umn.edu\/ odlyzko\/zeta_{t}ables\/index.html}."},{"issue":"3","key":"12","doi-asserted-by":"publisher","first-page":"275","DOI":"10.4064\/aa-29-3-275-297","article-title":"Lower bounds for discriminants of number fields","volume":"29","author":"Odlyzko, A. M.","year":"1976","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"2","key":"13","doi-asserted-by":"publisher","first-page":"209","DOI":"10.2748\/tmj\/1178240652","article-title":"Lower bounds for discriminants of number fields. II","volume":"29","author":"Odlyzko, A. M.","year":"1977","journal-title":"T\\^{o}hoku Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0040-8735","issn-type":"print"},{"key":"14","unstructured":"PARI Group, Univ. Bordeaux. package nftables.tgz, 2008, available at \\url{http:\/\/pari.math.u-bordeaux.fr\/packages.html}."},{"key":"15","unstructured":"PARI Group, Bordeaux, PARI\/GP, version 2.15.5, 1985\u20132024, Available at \\url{http:\/\/pari.math.u-bordeaux.fr\/}."},{"key":"16","unstructured":"PlaFRIM, Bordeaux, Experimental testbed, 2022, \\url{https:\/\/www.plafrim.fr}."},{"key":"17","isbn-type":"print","first-page":"Exp. No. 6, 18","article-title":"Sur les petits discriminants","author":"Poitou, Georges","year":"1977","ISBN":"https:\/\/id.crossref.org\/isbn\/2859262563"},{"key":"18","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1007\/BF02398277","article-title":"The product of \ud835\udc5b real homogeneous linear forms","volume":"82","author":"Rogers, C. A.","year":"1950","journal-title":"Acta Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0001-5962","issn-type":"print"},{"issue":"134","key":"19","doi-asserted-by":"publisher","first-page":"337","DOI":"10.2307\/2005976","article-title":"Sharper bounds for the Chebyshev functions \ud835\udf03(\ud835\udc65) and \ud835\udf13(\ud835\udc65). II","volume":"30","author":"Schoenfeld, Lowell","year":"1976","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"16","key":"20","doi-asserted-by":"publisher","first-page":"4991","DOI":"10.1093\/imrn\/rnx031","article-title":"A Chebotarev variant of the Brun-Titchmarsh theorem and bounds for the Lang-Trotter conjectures","author":"Thorner, Jesse","year":"2018","journal-title":"Int. Math. Res. Not. IMRN","ISSN":"https:\/\/id.crossref.org\/issn\/1073-7928","issn-type":"print"},{"issue":"3","key":"21","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1007\/BF01394249","article-title":"Ideale kleiner Norm in Idealklassen und eine Regulatorabsch\u00e4tzung","volume":"62","author":"Zimmert, Rainer","year":"1981","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2025-94-356\/S0025-5718-2025-04114-0\/S0025-5718-2025-04114-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:52:22Z","timestamp":1776837142000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2025-94-356\/S0025-5718-2025-04114-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,27]]},"references-count":21,"journal-issue":{"issue":"356","published-print":{"date-parts":[[2025,11]]}},"alternative-id":["S0025-5718-2025-04114-0"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/4114","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2025,6,27]]}}}