{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:43:34Z","timestamp":1776789814906,"version":"3.51.2"},"reference-count":41,"publisher":"American Mathematical Society (AMS)","issue":"266","license":[{"start":{"date-parts":[[2009,9,3]],"date-time":"2009-09-03T00:00:00Z","timestamp":1251936000000},"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 We present a method to solve in an efficient way the problem of constructing the curves given by Torelli\u2019s theorem in dimension\n \n \n \n 3<\/mml:mn>\n 3<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n over the complex numbers: For an absolutely simple principally polarized abelian threefold\n \n \n \n A<\/mml:mi>\n A<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n over\n \n \n \n \n C<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {C}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n given by its period matrix\n \n \n \n \n \n \u03a9\n \n <\/mml:mi>\n ,<\/mml:mo>\n <\/mml:mrow>\n \\Omega ,<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n compute a model of the curve of genus three (unique up to isomorphism) whose Jacobian, equipped with its canonical polarization, is isomorphic to\n \n \n \n A<\/mml:mi>\n A<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n as a principally polarized abelian variety. We use this method to describe the non-hyperelliptic modular Jacobians of dimension 3. We investigate all the non-hyperelliptic new modular Jacobians\n \n \n \n \n \n Jac<\/mml:mtext>\n <\/mml:mrow>\n (<\/mml:mo>\n \n C<\/mml:mi>\n f<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n \\textrm {Jac}(C_f)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of dimension\n \n \n \n 3<\/mml:mn>\n 3<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n which are isomorphic to\n \n \n \n \n A<\/mml:mi>\n f<\/mml:mi>\n <\/mml:msub>\n A_f<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n f<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n S<\/mml:mi>\n 2<\/mml:mn>\n \n new<\/mml:mtext>\n <\/mml:mrow>\n <\/mml:msubsup>\n (<\/mml:mo>\n \n X<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n (<\/mml:mo>\n N<\/mml:mi>\n )<\/mml:mo>\n )<\/mml:mo>\n ,<\/mml:mo>\n <\/mml:mrow>\n f\\in S_2^\\textrm {new}(X_0 (N)),<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n \n \n \n \n N<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n 4000.<\/mml:mn>\n <\/mml:mrow>\n N\\leq 4000.<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n <\/p>","DOI":"10.1090\/s0025-5718-08-02174-1","type":"journal-article","created":{"date-parts":[[2009,12,1]],"date-time":"2009-12-01T13:09:23Z","timestamp":1259672963000},"page":"1173-1191","source":"Crossref","is-referenced-by-count":3,"title":["Non-hyperelliptic modular Jacobians of dimension 3"],"prefix":"10.1090","volume":"78","author":[{"given":"Roger","family":"Oyono","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2008,9,3]]},"reference":[{"key":"1","unstructured":"R. Avanzi, N. Th\u00e9riault, and Z. Wang, Rethinking low genus hyperelliptic Jacobian arithmetic over binary fields: Interplay of field arithmetic and explicit formulae, preprint, 2006."},{"issue":"6","key":"2","doi-asserted-by":"crossref","first-page":"1325","DOI":"10.1353\/ajm.2005.0037","article-title":"Finiteness results for modular curves of genus at least 2","volume":"127","author":"Baker, Matthew H.","year":"2005","journal-title":"Amer. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9327","issn-type":"print"},{"key":"3","isbn-type":"print","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1007\/978-3-540-24847-7_6","article-title":"Implementing the arithmetic of \ud835\udc36_{3,4} curves","author":"Basiri, Abdolali","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/3540221565"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1090\/S1056-3911-02-00307-7","article-title":"Recovering plane curves from their bitangents","volume":"12","author":"Caporaso, Lucia","year":"2003","journal-title":"J. Algebraic Geom.","ISSN":"https:\/\/id.crossref.org\/issn\/1056-3911","issn-type":"print"},{"key":"5","unstructured":"C. Diem and E. Thom\u00e9, Index calculus in class groups of non-hyperelliptic curves of genus 3, 2006, preprint."},{"issue":"3","key":"6","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1016\/0001-8708(87)90010-7","article-title":"On the projective invariants of quartic plane curves","volume":"64","author":"Dixmier, J.","year":"1987","journal-title":"Adv. in Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0001-8708","issn-type":"print"},{"key":"7","unstructured":"I. Dolgachev, Topics in classical algebraic geometry, part I, Available on \\url{http:\/\/www.math.lsa.umich.edu\/ idolga\/lecturenotes.html}, 2003."},{"key":"8","isbn-type":"print","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1007\/978-3-540-24632-9_5","article-title":"Fast arithmetic on Jacobians of Picard curves","author":"Flon, St\u00e9phane","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/3540210180"},{"key":"9","unstructured":"S. Flon, R. Oyono, and C. Ritzenthaler, Fast addition on non-hyperelliptic genus 3 curves, Preprint, available on \\url{http:\/\/eprint.iacr.org\/2004}, 2004."},{"key":"10","isbn-type":"print","doi-asserted-by":"publisher","first-page":"346","DOI":"10.1007\/11792086_25","article-title":"Classification of genus 3 curves in special strata of the moduli space","author":"Girard, Martine","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540360759"},{"issue":"3","key":"11","doi-asserted-by":"publisher","first-page":"263","DOI":"10.4064\/aa116-3-3","article-title":"Abelian surfaces of \ud835\udc3a\ud835\udc3f\u2082-type as Jacobians of curves","volume":"116","author":"Gonz\u00e1lez, Josep","year":"2005","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"12","unstructured":"E. Gonz\u00e1lez-Jim\u00e9nez and R. Oyono, Non-hyperelliptic modular curves of genus 3, work in progress, 2007."},{"issue":"241","key":"13","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1090\/S0025-5718-02-01458-8","article-title":"Modular curves of genus 2","volume":"72","author":"Gonz\u00e1lez-Jim\u00e9nez, Enrique","year":"2003","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"14","isbn-type":"print","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1007\/3-540-45455-1_15","article-title":"Computations on modular Jacobian surfaces","author":"Gonz\u00e1lez-Jim\u00e9nez, Enrique","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/3540438637"},{"key":"15","unstructured":"E. Gonz\u00e1lez-Jim\u00e9nez and J. 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Ohno, Invariant subring of ternary quartics I - generators and relations, preprint."},{"key":"27","unstructured":"R. Oyono, Arithmetik nicht-hyperelliptischer Kurven des Geschlechts 3 und ihre Anwendung in der Kryptographie, Ph.D. thesis, Essen, 2006."},{"issue":"3","key":"28","doi-asserted-by":"publisher","first-page":"809","DOI":"10.1215\/S0012-7094-94-07634-5","article-title":"The hyperelliptic locus","volume":"76","author":"Poor, Cris","year":"1994","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"},{"key":"29","isbn-type":"print","first-page":"13","article-title":"Efficient reduction on the Jacobian variety of Picard curves","author":"Reinaldo Barreiro, Ernesto","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/3540662480"},{"key":"30","unstructured":"B. Riemann, Sur la th\u00e9orie des fonctions ab\u00e9liennes, Oeuvres de Riemann, 2nd edition (1898), 487."},{"key":"31","unstructured":"C. Ritzenthaler, Probl\u00e8mes arithm\u00e9tiques relatifs \u00e0 certaines familles de courbes sur les corps finis, Ph.D. thesis, Universit\u00e9 Paris 7, 2003."},{"key":"32","doi-asserted-by":"crossref","unstructured":"T. Shaska and J. L. Thompson, On the generic curve of genus 3, Contemporary. Math. 369 (2005), 233\u2013244.","DOI":"10.1090\/conm\/369\/06814"},{"key":"33","doi-asserted-by":"publisher","first-page":"523","DOI":"10.2969\/jmsj\/02530523","article-title":"On the factors of the jacobian variety of a modular function field","volume":"25","author":"Shimura, Goro","year":"1973","journal-title":"J. Math. Soc. 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Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"key":"40","first-page":"33","article-title":"Zum Beweis des Torellischen Satzes","volume":"1957","author":"Weil, Andr\u00e9","year":"1957","journal-title":"Nachr. Akad. Wiss. G\\\"{o}ttingen. Math.-Phys. Kl. IIa."},{"issue":"3","key":"41","doi-asserted-by":"publisher","first-page":"443","DOI":"10.2307\/2118559","article-title":"Modular elliptic curves and Fermat\u2019s last theorem","volume":"141","author":"Wiles, Andrew","year":"1995","journal-title":"Ann. of Math. 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