{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:26:11Z","timestamp":1776831971066,"version":"3.51.2"},"reference-count":34,"publisher":"American Mathematical Society (AMS)","issue":"238","license":[{"start":{"date-parts":[[2002,10,4]],"date-time":"2002-10-04T00:00:00Z","timestamp":1033689600000},"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                    In this paper, we provide tight estimates for the divisor class number of hyperelliptic function fields. We extend the existing methods to any hyperelliptic function field and improve the previous bounds by a factor proportional to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"g\">\n                        <mml:semantics>\n                          <mml:mi>g<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">g<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with the help of new results. We thus obtain a faster method of computing regulators and class numbers. Furthermore, we provide experimental data and heuristics on the distribution of the class number within the bounds on the class number. These heuristics are based on recent results by Katz and Sarnak. Our numerical results and the heuristics imply that our approximation is in general far better than the bounds suggest.\n                  <\/p>","DOI":"10.1090\/s0025-5718-01-01385-0","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"837-861","source":"Crossref","is-referenced-by-count":9,"title":["Explicit bounds and heuristics on class numbers in hyperelliptic function fields"],"prefix":"10.1090","volume":"71","author":[{"given":"Andreas","family":"Stein","sequence":"first","affiliation":[]},{"given":"Edlyn","family":"Teske","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,10,4]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1007\/3-540-58691-1_39","article-title":"A subexponential algorithm for discrete logarithms over the rational subgroup of the Jacobians of large genus hyperelliptic curves over finite fields","author":"Adleman, Leonard M.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/3540586911"},{"key":"2","doi-asserted-by":"crossref","unstructured":"[Art24] E. Artin. Quadratische K\u00f6rper im Gebiete der h\u00f6heren Kongruenzen I, II. Math. Zeitschr., 19:153\u2013206, 1924.","DOI":"10.1007\/BF01181074"},{"key":"3","isbn-type":"print","first-page":"13","article-title":"Improved approximations for Euler products","author":"Bach, Eric","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0821803123"},{"key":"4","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1112\/jlms\/s1-43.1.57","article-title":"How the number of points of an elliptic curve over a fixed prime field varies","volume":"43","author":"Birch, B. J.","year":"1968","journal-title":"J. London Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6107","issn-type":"print"},{"key":"5","doi-asserted-by":"crossref","unstructured":"[BT00] S. R. Blackburn and E. Teske. Baby-step giant-step algorithms for non-uniform distributions. In Algorithmic Number Theory Seminar ANTS-IV, volume 1838 of Lecture Notes in Computer Science, pages 153\u2013168. Springer-Verlag, 2000.","DOI":"10.1007\/10722028_8"},{"key":"6","series-title":"London Mathematical Society Lecture Note Series","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511526084","volume-title":"Prolegomena to a middlebrow arithmetic of curves of genus $2$","volume":"230","author":"Cassels, J. W. S.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0521483700"},{"key":"7","series-title":"Lecture Notes in Mathematics, Vol. 314","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0060944","volume-title":"Lectures on the theory of algebraic functions of one variable","author":"Deuring, Max","year":"1973"},{"issue":"171","key":"8","doi-asserted-by":"publisher","first-page":"223","DOI":"10.2307\/2008060","article-title":"Computation of the class number and class group of a complex cubic field","volume":"45","author":"Dueck, G.","year":"1985","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"9","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1080\/10586458.1995.10504322","article-title":"An investigation of bounds for the regulator of quadratic fields","volume":"4","author":"Jacobson, Michael J., Jr.","year":"1995","journal-title":"Experiment. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"issue":"3","key":"10","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1007\/BF02252872","article-title":"Hyperelliptic cryptosystems","volume":"1","author":"Koblitz, Neal","year":"1989","journal-title":"J. Cryptology","ISSN":"https:\/\/id.crossref.org\/issn\/0933-2790","issn-type":"print"},{"key":"11","series-title":"American Mathematical Society Colloquium Publications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/coll\/045","volume-title":"Random matrices, Frobenius eigenvalues, and monodromy","volume":"45","author":"Katz, Nicholas M.","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/0821810170"},{"issue":"1","key":"12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1090\/S0273-0979-99-00766-1","article-title":"Zeroes of zeta functions and symmetry","volume":"36","author":"Katz, Nicholas M.","year":"1999","journal-title":"Bull. Amer. Math. Soc. (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0273-0979","issn-type":"print"},{"key":"13","isbn-type":"print","first-page":"123","article-title":"On the calculation of regulators and class numbers of quadratic fields","author":"Lenstra, H. W., Jr.","year":"1982","ISBN":"https:\/\/id.crossref.org\/isbn\/0521285135"},{"key":"14","series-title":"Encyclopedia of Mathematics and its Applications","isbn-type":"print","volume-title":"Finite fields","volume":"20","author":"Lidl, Rudolf","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0201135191"},{"key":"15","series-title":"Graduate Studies in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/009","volume-title":"An invitation to arithmetic geometry","volume":"9","author":"Lorenzini, Dino","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0821802674"},{"issue":"17","key":"16","doi-asserted-by":"publisher","first-page":"1687","DOI":"10.1080\/00927878008822538","article-title":"On the theory of congruence function fields","volume":"8","author":"Madan, Manohar L.","year":"1980","journal-title":"Comm. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0092-7872","issn-type":"print"},{"issue":"226","key":"17","doi-asserted-by":"publisher","first-page":"807","DOI":"10.1090\/S0025-5718-99-01040-6","article-title":"Computing discrete logarithms in real quadratic congruence function fields of large genus","volume":"68","author":"M\u00fcller, Volker","year":"1999","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"18","isbn-type":"print","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1007\/3-540-61581-4_63","article-title":"Computational aspects of curves of genus at least 2","author":"Poonen, Bjorn","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/3540615814"},{"issue":"227","key":"19","doi-asserted-by":"publisher","first-page":"1233","DOI":"10.1090\/S0025-5718-99-01066-2","article-title":"Real and imaginary quadratic representations of hyperelliptic function fields","volume":"68","author":"Paulus, Sachar","year":"1999","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"20","doi-asserted-by":"crossref","unstructured":"[Sch31] F. K. Schmidt. Analytische Zahlentheorie in K\u00f6rpern der Charakteristik \ud835\udc5d. Mathematische Zeitschrift, 33:1\u201332, 1931.","DOI":"10.1007\/BF01174341"},{"issue":"9","key":"21","first-page":"397","article-title":"Sur le nombre des points rationnels d\u2019une courbe alg\u00e9brique sur un corps fini","volume":"296","author":"Serre, Jean-Pierre","year":"1983","journal-title":"C. R. Acad. Sci. Paris S\\'{e}r. I Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0249-6291","issn-type":"print"},{"key":"22","unstructured":"[Ser99] J. P. Serre, 1999. Personal communications, Aug. 27, Aug. 28, Sept. 7, Sept. 11."},{"issue":"1-2","key":"23","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1007\/BF00125081","article-title":"Key-exchange in real quadratic congruence function fields","volume":"7","author":"Scheidler, R.","year":"1996","journal-title":"Des. Codes Cryptogr.","ISSN":"https:\/\/id.crossref.org\/issn\/0925-1022","issn-type":"print"},{"key":"24","unstructured":"[STa] A. Stein and E. Teske. Optimized baby step\u2013giant step methods and applications to hyperelliptic function fields. Unpublished manuscript."},{"key":"25","unstructured":"[STb] A. Stein and E. Teske. The parallelized Pollard kangaroo method in real quadratic function fields. Math. Comp., posted on October 4, 2001, PII 50025-5718(01)01343-6 (to appear in print)."},{"key":"26","series-title":"Universitext","isbn-type":"print","volume-title":"Algebraic function fields and codes","author":"Stichtenoth, Henning","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/3540564896"},{"key":"27","isbn-type":"print","doi-asserted-by":"publisher","first-page":"607","DOI":"10.1007\/BFb0054896","article-title":"An improved method of computing the regulator of a real quadratic function field","author":"Stein, Andreas","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/3540646574"},{"issue":"2","key":"28","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1080\/10586458.1999.10504394","article-title":"Some methods for evaluating the regulator of a real quadratic function field","volume":"8","author":"Stein, Andreas","year":"1999","journal-title":"Experiment. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"key":"29","first-page":"93","article-title":"Algebraic cycles and poles of zeta functions","author":"Tate, John T.","year":"1965"},{"key":"30","volume-title":"Gesammelte Abhandlungen. B\\\"{a}nde I, II, III, IV","author":"Weyl, Hermann","year":"1968"},{"issue":"2","key":"31","first-page":"261","article-title":"Artins Theorie der quadratischen Kongruenzfunktionenk\u00f6rper und ihre Anwendung auf die Berechnung der Einheiten- und Klassengruppen","volume":"12","author":"Weis, Bosco","year":"1991","journal-title":"Mitt. Math. Ges. Hamburg","ISSN":"https:\/\/id.crossref.org\/issn\/0340-4358","issn-type":"print"},{"key":"32","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1007\/BF01425416","article-title":"On an analogue of the Sato conjecture","volume":"19","author":"Yoshida, Hiroyuki","year":"1973","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"issue":"4","key":"33","first-page":"425","article-title":"Ambiguous classes and 2-rank of class group of quadratic function field","volume":"17","author":"Zhang, Xian Ke","year":"1987","journal-title":"J. China Univ. Sci. Tech.","ISSN":"https:\/\/id.crossref.org\/issn\/0253-2778","issn-type":"print"},{"issue":"2","key":"34","doi-asserted-by":"publisher","first-page":"563","DOI":"10.1006\/jabr.1996.6985","article-title":"The continued fraction algorithm and regulator for quadratic function fields of characteristic 2","volume":"190","author":"Zuccherato, Robert J.","year":"1997","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2002-71-238\/S0025-5718-01-01385-0\/S0025-5718-01-01385-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-238\/S0025-5718-01-01385-0\/S0025-5718-01-01385-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:56:57Z","timestamp":1776725817000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-238\/S0025-5718-01-01385-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,10,4]]},"references-count":34,"journal-issue":{"issue":"238","published-print":{"date-parts":[[2002,4]]}},"alternative-id":["S0025-5718-01-01385-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-01-01385-0","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":[[2001,10,4]]}}}