{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:53:54Z","timestamp":1776844434231,"version":"3.51.2"},"reference-count":44,"publisher":"American Mathematical Society (AMS)","issue":"249","license":[{"start":{"date-parts":[[2005,5,18]],"date-time":"2005-05-18T00:00:00Z","timestamp":1116374400000},"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 This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank\u00a0\n \n \n \n 0<\/mml:mn>\n 0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n abelian varieties\u00a0\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 that are optimal quotients of\n \n \n \n \n \n J<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n (<\/mml:mo>\n N<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n J_0(N)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n attached to newforms. We prove theorems about the ratio\n \n \n \n \n L<\/mml:mi>\n (<\/mml:mo>\n \n A<\/mml:mi>\n f<\/mml:mi>\n <\/mml:msub>\n ,<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n \n \u03a9\n \n <\/mml:mi>\n \n \n A<\/mml:mi>\n f<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n L(A_f,1)\/\\Omega _{A_f}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , develop tools for computing with\u00a0\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 , and gather data about certain arithmetic invariants of the nearly\n \n \n \n \n 20<\/mml:mn>\n ,<\/mml:mo>\n 000<\/mml:mn>\n <\/mml:mrow>\n 20,000<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n abelian varieties\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 of level\n \n \n \n \n \n \u2264\n \n <\/mml:mo>\n 2333<\/mml:mn>\n <\/mml:mrow>\n \\leq 2333<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Over half of these\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 have analytic rank\u00a0\n \n \n \n 0<\/mml:mn>\n 0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and for these we compute upper and lower bounds on the conjectural order of\u00a0\n \n \n \n \n \n \u0428<\/mml:mo>\n <\/mml:mrow>\n (<\/mml:mo>\n \n A<\/mml:mi>\n f<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n \\Sha (A_f)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We find that there are at least\n \n \n \n 168<\/mml:mn>\n 168<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n such\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 for which the Birch and Swinnerton-Dyer conjecture implies that\n \n \n \n \n \n \u0428<\/mml:mo>\n <\/mml:mrow>\n (<\/mml:mo>\n \n A<\/mml:mi>\n f<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n \\Sha (A_f)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is divisible by an odd prime, and we prove for\n \n \n \n 37<\/mml:mn>\n 37<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of these that the odd part of the conjectural order of\n \n \n \n \n \n \u0428<\/mml:mo>\n <\/mml:mrow>\n (<\/mml:mo>\n \n A<\/mml:mi>\n f<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n \\Sha (A_f)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n really divides\n \n \n \n \n \n #\n \n <\/mml:mi>\n \n \u0428<\/mml:mo>\n <\/mml:mrow>\n (<\/mml:mo>\n \n A<\/mml:mi>\n f<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n \\#\\Sha (A_f)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n by constructing nontrivial elements of\n \n \n \n \n \n \u0428<\/mml:mo>\n <\/mml:mrow>\n (<\/mml:mo>\n \n A<\/mml:mi>\n f<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n \\Sha (A_f)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n using visibility theory. We also give other evidence for the conjecture. The appendix, by Cremona and Mazur, fills in some gaps in the theoretical discussion in their paper on visibility of Shafarevich-Tate groups of elliptic curves.\n <\/p>","DOI":"10.1090\/s0025-5718-04-01644-8","type":"journal-article","created":{"date-parts":[[2004,9,20]],"date-time":"2004-09-20T09:57:53Z","timestamp":1095674273000},"page":"455-484","source":"Crossref","is-referenced-by-count":20,"title":["Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero"],"prefix":"10.1090","volume":"74","author":[{"given":"Amod","family":"Agashe","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"William","family":"Stein","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2004,5,18]]},"reference":[{"issue":"5","key":"1","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1016\/S0764-4442(99)80173-6","article-title":"On invisible elements of the Tate-Shafarevich group","volume":"328","author":"Agash\u00e9, Amod","year":"1999","journal-title":"C. R. Acad. Sci. Paris S\\'{e}r. I Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-4442","issn-type":"print"},{"key":"2","unstructured":"[AS] A. Agashe and W. A. Stein, Appendix to Joan-C. Lario and Ren\u00e9 Schoof: Some computations with Hecke rings and deformation rings, to appear in J. Exp. Math."},{"key":"3","doi-asserted-by":"crossref","unstructured":"[AS02] A. Agashe and W. A. Stein, Visibility of Shafarevich-Tate Groups of Abelian Varieties, to appear in J. of Number Theory (2002).","DOI":"10.1006\/jnth.2002.2810"},{"key":"4","unstructured":"[AS04] A. Agashe and W. A. Stein, The Manin constant, congruence primes, and the modular degree, preprint, (2004). http:\/\/modular.fas.harvard.edu\/papers\/manin-agashe\/"},{"key":"5","doi-asserted-by":"crossref","unstructured":"[BCP97] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235\u2013265, Computational algebra and number theory (London, 1993).","DOI":"10.1006\/jsco.1996.0125"},{"key":"6","first-page":"396","article-title":"Elliptic curves over \ud835\udc44: A progress report","author":"Birch, B. J.","year":"1971"},{"key":"7","series-title":"Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-51438-8","volume-title":"N\\'{e}ron models","volume":"21","author":"Bosch, Siegfried","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/3540505873"},{"key":"8","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1112\/plms\/s3-12.1.259","article-title":"Arithmetic on curves of genus 1. III. The Tate-\u0160afarevi\u010d and Selmer groups","volume":"12","author":"Cassels, J. W. S.","year":"1962","journal-title":"Proc. London Math. Soc. (3)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6115","issn-type":"print"},{"key":"9","isbn-type":"print","volume-title":"Algorithms for modular elliptic curves","author":"Cremona, J. E.","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/0521598206","edition":"2"},{"key":"10","doi-asserted-by":"crossref","unstructured":"[CM] J. E. Cremona and B. Mazur, Visualizing elements in the Shafarevich-Tate group, Experiment. Math. 9 (2000), no. 1, 13\u201328.","DOI":"10.1080\/10586458.2000.10504633"},{"issue":"5-6","key":"11","doi-asserted-by":"publisher","first-page":"745","DOI":"10.4310\/MRL.2001.v8.n6.a5","article-title":"Component groups of purely toric quotients","volume":"8","author":"Conrad, Brian","year":"2001","journal-title":"Math. Res. Lett.","ISSN":"https:\/\/id.crossref.org\/issn\/1073-2780","issn-type":"print"},{"key":"12","doi-asserted-by":"crossref","unstructured":"[Del01] C. Delaunay, Heuristics on Tate-Shafarevitch groups of elliptic curves defined over \ud835\udc44, Experiment. Math. 10 (2001), no. 2, 191\u2013196.","DOI":"10.1080\/10586458.2001.10504442"},{"key":"13","isbn-type":"print","first-page":"39","article-title":"Modular forms and modular curves","author":"Diamond, Fred","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0821803131"},{"key":"14","isbn-type":"print","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1007\/978-1-4612-0457-2_3","article-title":"On the Manin constants of modular elliptic curves","author":"Edixhoven, Bas","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0817635130"},{"key":"15","unstructured":"[Eme01] M. Emerton, Optimal quotients of modular Jacobians. Preprint."},{"key":"16","doi-asserted-by":"crossref","unstructured":"[FpS:01] E. V. Flynn, F. Lepr\u00e9vost, E. F. Schaefer, W. A. Stein, M. Stoll, and J. L. Wetherell, Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves, Math. Comp. 70 (2001), no. 236, 1675\u20131697.","DOI":"10.1090\/S0025-5718-01-01320-5"},{"key":"17","first-page":"Ai, A1423--A1425","article-title":"Groupes finis commutatifs sur les vecteurs de Witt","volume":"280","author":"Fontaine, Jean-Marc","year":"1975","journal-title":"C. R. Acad. Sci. Paris S\\'{e}r. A-B","ISSN":"https:\/\/id.crossref.org\/issn\/0151-0509","issn-type":"print"},{"key":"18","isbn-type":"print","doi-asserted-by":"publisher","first-page":"527","DOI":"10.1090\/pspum\/055.1\/1265543","article-title":"\ud835\udc3f-functions at the central critical point","author":"Gross, Benedict H.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0821816365"},{"key":"19","first-page":"46","article-title":"Le groupe de Brauer. I. Alg\u00e8bres d\u2019Azumaya et interpr\u00e9tations diverses","author":"Grothendieck, Alexander","year":"1968"},{"key":"20","series-title":"Lecture Notes in Mathematics, Vol. 288","volume-title":"Groupes de monodromie en g\\'{e}om\\'{e}trie alg\\'{e}brique. I","year":"1972"},{"issue":"2","key":"21","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1007\/BF01388809","article-title":"Heegner points and derivatives of \ud835\udc3f-series","volume":"84","author":"Gross, Benedict H.","year":"1986","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"issue":"3","key":"22","doi-asserted-by":"publisher","first-page":"481","DOI":"10.1007\/BF01394256","article-title":"Galois properties of torsion points on abelian varieties","volume":"62","author":"Katz, Nicholas M.","year":"1981","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"issue":"5","key":"23","first-page":"171","article-title":"Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties","volume":"1","author":"Kolyvagin, V. A.","year":"1989","journal-title":"Algebra i Analiz","ISSN":"https:\/\/id.crossref.org\/issn\/0234-0852","issn-type":"print"},{"issue":"4","key":"24","doi-asserted-by":"publisher","first-page":"851","DOI":"10.1070\/IM1992v039n01ABEH002228","article-title":"Finiteness of SH over totally real fields","volume":"55","author":"Kolyvagin, V. A.","year":"1991","journal-title":"Izv. Akad. Nauk SSSR Ser. Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/0373-2436","issn-type":"print"},{"key":"25","unstructured":"[KS00] D. R. Kohel and W. A. Stein, Component Groups of Quotients of \ud835\udc3d\u2080(\ud835\udc41), Proceedings of the 4th International Symposium (ANTS-IV), Leiden, Netherlands, July 2\u20137, 2000 (Berlin), Springer, 2000."},{"key":"26","series-title":"Encyclopaedia of Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-58227-1","volume-title":"Number theory. III","volume":"60","author":"Lang, Serge","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/3540530045"},{"issue":"3","key":"27","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1016\/0022-4049(85)90079-9","article-title":"Abelian varieties having purely additive reduction","volume":"36","author":"Lenstra, H. W., Jr.","year":"1985","journal-title":"J. Pure Appl. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0022-4049","issn-type":"print"},{"issue":"2","key":"28","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1080\/10586458.2002.10504694","article-title":"Some computations with Hecke rings and deformation rings","volume":"11","author":"Lario, Joan-C.","year":"2002","journal-title":"Experiment. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"key":"29","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1007\/BF01389815","article-title":"Rational points of abelian varieties with values in towers of number fields","volume":"18","author":"Mazur, Barry","year":"1972","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"issue":"47","key":"30","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/BF02684339","article-title":"Modular curves and the Eisenstein ideal","author":"Mazur, B.","year":"1977","journal-title":"Inst. Hautes \\'{E}tudes Sci. Publ. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0073-8301","issn-type":"print"},{"issue":"2","key":"31","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1007\/BF01390348","article-title":"Rational isogenies of prime degree (with an appendix by D. Goldfeld)","volume":"44","author":"Mazur, B.","year":"1978","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"key":"32","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1007\/BF01425572","article-title":"Points of order 13 on elliptic curves","volume":"22","author":"Mazur, B.","year":"1973","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"key":"33","doi-asserted-by":"crossref","unstructured":"[Mil86] J. S. Milne, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, pp. 103\u2013150.","DOI":"10.1007\/978-1-4613-8655-1_5"},{"key":"34","first-page":"221","article-title":"Rational points on certain elliptic modular curves","author":"Ogg, A. P.","year":"1973"},{"issue":"3","key":"35","doi-asserted-by":"publisher","first-page":"1109","DOI":"10.2307\/121064","article-title":"The Cassels-Tate pairing on polarized abelian varieties","volume":"150","author":"Poonen, Bjorn","year":"1999","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"36","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. Japan","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5645","issn-type":"print"},{"key":"37","series-title":"Publications of the Mathematical Society of Japan","isbn-type":"print","volume-title":"Introduction to the arithmetic theory of automorphic functions","volume":"11","author":"Shimura, Goro","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0691080925"},{"key":"38","series-title":"Progress in Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4684-9165-4","volume-title":"Arithmetic on modular curves","volume":"20","author":"Stevens, Glenn","year":"1982","ISBN":"https:\/\/id.crossref.org\/isbn\/3764330880"},{"key":"39","unstructured":"[Ste00] W. A. Stein, Explicit approaches to modular abelian varieties, Ph.D. thesis, University of California, Berkeley (2000)."},{"key":"40","unstructured":"[Ste02a] W. A. Stein, An introduction to computing modular forms using modular symbols, to appear in an MSRI Proceedings (2002)."},{"key":"41","unstructured":"[Ste02b] W. A. Stein, Shafarevich-Tate groups of nonsquare order, Proceedings of MCAV 2002, Progress of Mathematics (to appear)."},{"key":"42","isbn-type":"print","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1007\/BFb0072985","article-title":"On the congruence of modular forms","author":"Sturm, Jacob","year":"1987","ISBN":"https:\/\/id.crossref.org\/isbn\/3540176691"},{"key":"43","first-page":"288","article-title":"Duality theorems in Galois cohomology over number fields","author":"Tate, John","year":"1963"},{"key":"44","unstructured":"[Tat66] J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, S\u00e9minaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1966 (reprinted in 1995), Exp. No. 306, 415\u2013440."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2005-74-249\/S0025-5718-04-01644-8\/S0025-5718-04-01644-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2005-74-249\/S0025-5718-04-01644-8\/S0025-5718-04-01644-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T13:58:46Z","timestamp":1776779926000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2005-74-249\/S0025-5718-04-01644-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,5,18]]},"references-count":44,"journal-issue":{"issue":"249","published-print":{"date-parts":[[2005,1]]}},"alternative-id":["S0025-5718-04-01644-8"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-04-01644-8","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":[[2004,5,18]]}}}