{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:36:54Z","timestamp":1776785814795,"version":"3.51.2"},"reference-count":15,"publisher":"American Mathematical Society (AMS)","issue":"255","license":[{"start":{"date-parts":[[2007,5,1]],"date-time":"2007-05-01T00:00:00Z","timestamp":1177977600000},"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":"<p>\n                    The group generated by the Weierstrass points of a smooth curve in its Jacobian is an intrinsic invariant of the curve. We determine this group for all smooth quartics with eight hyperflexes or more. Since Weierstrass points are closely related to moduli spaces of curves, as an application, we get bounds on both the rank and the torsion part of this group for a generic quartic having a fixed number of hyperflexes in the moduli space\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper M 3\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">M<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {M}_{3}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of curves of genus 3.\n                  <\/p>","DOI":"10.1090\/s0025-5718-06-01853-9","type":"journal-article","created":{"date-parts":[[2006,5,24]],"date-time":"2006-05-24T14:43:01Z","timestamp":1148481781000},"page":"1561-1583","source":"Crossref","is-referenced-by-count":3,"title":["The group of Weierstrass points of a plane quartic with at least eight hyperflexes"],"prefix":"10.1090","volume":"75","author":[{"given":"Martine","family":"Girard","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,5,1]]},"reference":[{"key":"1","series-title":"Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-5323-3","volume-title":"Geometry of algebraic curves. Vol. I","volume":"267","author":"Arbarello, E.","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/0387909974"},{"key":"2","unstructured":"W. Bosma and J. Cannon, editors. Handbook of Magma Functions, Sydney, 2002. \\url{http:\/\/magma.maths.usyd.edu.au\/}."},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1006\/jnth.2001.2732","article-title":"G\u00e9om\u00e9trie du groupe des points de Weierstrass d\u2019une quartique lisse","volume":"94","author":"Girard, Martine","year":"2002","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"4","key":"4","doi-asserted-by":"publisher","first-page":"305","DOI":"10.4064\/aa105-4-1","article-title":"Groupe des points de Weierstrass sur une famille de quartiques lisses","volume":"105","author":"Girard, Martine","year":"2002","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"5","unstructured":"M. Girard. Code for computing heights of elliptic curves on number fields, \\url{http:\/\/www.institut.math.jussieu.fr\/ girard\/magma\/}."},{"key":"6","unstructured":"M. Girard. Group of Weierstrass points of a plane quartic with eight hyperflexes or more. Technical report of the Mathematical Institute, Leiden University, June 2002. \\url{http:\/\/www.math.leidenuniv.nl\/reports\/2002-14.shtml}."},{"issue":"3","key":"7","doi-asserted-by":"publisher","first-page":"667","DOI":"10.1090\/S0002-9939-01-06193-7","article-title":"Group generated by the Weierstrass points of a plane quartic","volume":"130","author":"Girard, Martine","year":"2002","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"key":"8","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1210-2","volume-title":"Diophantine geometry","volume":"201","author":"Hindry, Marc","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/0387989757"},{"issue":"166","key":"9","doi-asserted-by":"publisher","first-page":"ix+137","DOI":"10.1090\/memo\/0166","article-title":"Sur les sections analytiques de la courbe universelle de Teichm\u00fcller","volume":"4","author":"Hubbard, John Hamal","year":"1976","journal-title":"Mem. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-9266","issn-type":"print"},{"issue":"3","key":"10","doi-asserted-by":"publisher","first-page":"241","DOI":"10.4064\/aa-74-3-241-257","article-title":"Arithmetic and geometry of the curve \ud835\udc66\u00b3+1=\ud835\udc65\u2074","volume":"74","author":"Klassen, Matthew J.","year":"1996","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"393","DOI":"10.1007\/BF02559578","article-title":"Weierstrass points and gap sequences for families of curves","volume":"32","author":"Laksov, Dan","year":"1994","journal-title":"Ark. Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/0004-2080","issn-type":"print"},{"issue":"4","key":"12","doi-asserted-by":"publisher","first-page":"971","DOI":"10.2307\/2161162","article-title":"On the Jacobian of the Klein curve","volume":"122","author":"Prapavessi, Despina T.","year":"1994","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"issue":"2","key":"13","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1007\/BF01390104","article-title":"Points at infinity on the Fermat curves","volume":"39","author":"Rohrlich, David E.","year":"1977","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"key":"14","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1515\/crll.1983.342.197","article-title":"Heights and the specialization map for families of abelian varieties","volume":"342","author":"Silverman, Joseph H.","year":"1983","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"key":"15","volume-title":"Weierstrass points of weight two on curves of genus three","author":"Vermeulen, Alexius Maria","year":"1983"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01853-9\/S0025-5718-06-01853-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01853-9\/S0025-5718-06-01853-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:37:09Z","timestamp":1776782229000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01853-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,5,1]]},"references-count":15,"journal-issue":{"issue":"255","published-print":{"date-parts":[[2006,7]]}},"alternative-id":["S0025-5718-06-01853-9"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-06-01853-9","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":[[2006,5,1]]}}}