{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:12:07Z","timestamp":1776845527523,"version":"3.51.2"},"reference-count":46,"publisher":"American Mathematical Society (AMS)","issue":"298","license":[{"start":{"date-parts":[[2016,8,12]],"date-time":"2016-08-12T00:00:00Z","timestamp":1470960000000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1103831"],"award-info":[{"award-number":["1103831"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["STO 299\/5-1"],"award-info":[{"award-number":["STO 299\/5-1"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["KU 2359\/2-1"],"award-info":[{"award-number":["KU 2359\/2-1"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1161226"],"award-info":[{"award-number":["DMS-1161226"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1147802"],"award-info":[{"award-number":["DMS-1147802"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["1103831"],"award-info":[{"award-number":["1103831"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["STO 299\/5-1"],"award-info":[{"award-number":["STO 299\/5-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["KU 2359\/2-1"],"award-info":[{"award-number":["KU 2359\/2-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["DMS-1161226"],"award-info":[{"award-number":["DMS-1161226"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["DMS-1147802"],"award-info":[{"award-number":["DMS-1147802"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1103831"],"award-info":[{"award-number":["1103831"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["STO 299\/5-1"],"award-info":[{"award-number":["STO 299\/5-1"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["KU 2359\/2-1"],"award-info":[{"award-number":["KU 2359\/2-1"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1161226"],"award-info":[{"award-number":["DMS-1161226"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1147802"],"award-info":[{"award-number":["DMS-1147802"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Mazur, Tate, and Teitelbaum gave a\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -adic analogue of the Birch and Swinnerton-Dyer conjecture for elliptic curves. We provide a generalization of their conjecture in the good ordinary case to higher dimensional modular abelian varieties over the rationals by constructing the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -adic\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L\">\n                        <mml:semantics>\n                          <mml:mi>L<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">L<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -function of a modular abelian variety and showing that it satisfies the appropriate interpolation property. This relies on a careful normalization of the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -adic\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L\">\n                        <mml:semantics>\n                          <mml:mi>L<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">L<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -function, which we achieve by a comparison of periods. Our generalization agrees with the conjecture of Mazur, Tate, and Teitelbaum in dimension 1 and the classical Birch and Swinnerton-Dyer conjecture formulated by Tate in rank 0. We describe the theoretical techniques used to formulate the conjecture and give numerical evidence supporting the conjecture in the case when the modular abelian variety is of dimension 2.\n                  <\/p>","DOI":"10.1090\/mcom\/3029","type":"journal-article","created":{"date-parts":[[2015,8,12]],"date-time":"2015-08-12T11:20:06Z","timestamp":1439378406000},"page":"983-1016","source":"Crossref","is-referenced-by-count":10,"title":["A \ud835\udc5d-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties"],"prefix":"10.1090","volume":"85","author":[{"given":"Jennifer","family":"Balakrishnan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J.","family":"M\u00fcller","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"William","family":"Stein","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2015,8,12]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"617","DOI":"10.4310\/PAMQ.2006.v2.n2.a11","article-title":"The Manin constant","volume":"2","author":"Agashe, Amod","year":"2006","journal-title":"Pure Appl. Math. Q.","ISSN":"https:\/\/id.crossref.org\/issn\/1558-8599","issn-type":"print"},{"issue":"249","key":"2","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1090\/S0025-5718-04-01644-8","article-title":"Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero","volume":"74","author":"Agashe, Amod","year":"2005","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"11","key":"3","doi-asserted-by":"publisher","first-page":"2405","DOI":"10.1093\/imrn\/rnr111","article-title":"Computing local \ud835\udc5d-adic height pairings on hyperelliptic curves","author":"Balakrishnan, Jennifer S.","year":"2012","journal-title":"Int. Math. Res. Not. IMRN","ISSN":"https:\/\/id.crossref.org\/issn\/1073-7928","issn-type":"print"},{"key":"4","isbn-type":"print","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1007\/978-3-642-14518-6_6","article-title":"Explicit Coleman integration for hyperelliptic curves","author":"Balakrishnan, Jennifer S.","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9783642145179"},{"issue":"3","key":"5","first-page":"227","article-title":"Variante \ud835\udc5d-adique de la conjecture de Birch et Swinnerton-Dyer (le cas supersingulier)","volume":"317","author":"Bernardi, Dominique","year":"1993","journal-title":"C. R. Acad. Sci. Paris S\\'{e}r. I Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-4442","issn-type":"print"},{"key":"6","isbn-type":"print","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1090\/crmp\/036\/02","article-title":"The \ud835\udc5d-adic height pairings of Coleman-Gross and of Nekov\u00e1\u0159","author":"Besser, Amnon","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0821833316"},{"issue":"3","key":"7","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1007\/s002290050140","article-title":"Rational points of the group of components of a N\u00e9ron model","volume":"98","author":"Bosch, Siegfried","year":"1999","journal-title":"Manuscripta Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-2611","issn-type":"print"},{"issue":"3-4","key":"8","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1006\/jsco.1996.0125","article-title":"The Magma algebra system. I. The user language","volume":"24","author":"Bosma, Wieb","year":"1997","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"53","DOI":"10.2307\/1971508","article-title":"Eisenstein series on the metaplectic group and nonvanishing theorems for automorphic \ud835\udc3f-functions and their derivatives","volume":"131","author":"Bump, Daniel","year":"1990","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"177-178","key":"10","first-page":"Exp. No. 701, 33--59","article-title":"On \ud835\udc5d-adic \ud835\udc3f-functions","author":"Coates, John","year":"1989","journal-title":"Ast\\'{e}risque","ISSN":"https:\/\/id.crossref.org\/issn\/0303-1179","issn-type":"print"},{"issue":"3","key":"11","doi-asserted-by":"publisher","first-page":"631","DOI":"10.1007\/BF01239529","article-title":"The universal vectorial bi-extension and \ud835\udc5d-adic heights","volume":"103","author":"Coleman, Robert F.","year":"1991","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"key":"12","isbn-type":"print","doi-asserted-by":"publisher","first-page":"73","DOI":"10.2969\/aspm\/01710073","article-title":"\ud835\udc5d-adic heights on curves","author":"Coleman, Robert F.","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/0121773701"},{"issue":"236","key":"13","doi-asserted-by":"publisher","first-page":"1675","DOI":"10.1090\/S0025-5718-01-01320-5","article-title":"Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves","volume":"70","author":"Flynn, E. Victor","year":"2001","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"14","unstructured":"S. D. Galbraith, Equations for modular curves, Oxford Ph.D. thesis, 1996."},{"issue":"268","key":"15","doi-asserted-by":"publisher","first-page":"2397","DOI":"10.1090\/S0025-5718-09-02253-4","article-title":"Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves","volume":"78","author":"Grigorov, Grigor","year":"2009","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"16","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":"1","key":"17","doi-asserted-by":"publisher","first-page":"23","DOI":"10.2307\/2374455","article-title":"Heights and Arakelov\u2019s intersection theory","volume":"107","author":"Hriljac, Paul","year":"1985","journal-title":"Amer. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9327","issn-type":"print"},{"issue":"295","key":"18","first-page":"ix, 117--290","article-title":"\ud835\udc5d-adic Hodge theory and values of zeta functions of modular forms","author":"Kato, Kazuya","year":"2004","journal-title":"Ast\\'{e}risque","ISSN":"https:\/\/id.crossref.org\/issn\/0303-1179","issn-type":"print"},{"issue":"1","key":"19","first-page":"51","article-title":"Conducteur et discriminant minimal de courbes de genre 2","volume":"94","author":"Liu, Qing","year":"1994","journal-title":"Compositio Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-437X","issn-type":"print"},{"issue":"11","key":"20","doi-asserted-by":"publisher","first-page":"4577","DOI":"10.1090\/S0002-9947-96-01684-4","article-title":"Mod\u00e8les entiers des courbes hyperelliptiques sur un corps de valuation discr\u00e8te","volume":"348","author":"Liu, Qing","year":"1996","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"21","series-title":"Oxford Graduate Texts in Mathematics","isbn-type":"print","volume-title":"Algebraic geometry and arithmetic curves","volume":"6","author":"Liu, Qing","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/0198502842"},{"key":"22","first-page":"577","article-title":"Computation of \ud835\udc5d-adic heights and log convergence","author":"Mazur, Barry","year":"2006","journal-title":"Doc. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1431-0635","issn-type":"print"},{"key":"23","isbn-type":"print","first-page":"195","article-title":"Canonical height pairings via biextensions","author":"Mazur, B.","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/3764331321"},{"issue":"1","key":"24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF01388731","article-title":"On \ud835\udc5d-adic analogues of the conjectures of Birch and Swinnerton-Dyer","volume":"84","author":"Mazur, B.","year":"1986","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"key":"25","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1112\/S1461157011000180","article-title":"Proving the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and one","volume":"14","author":"Miller, Robert L.","year":"2011","journal-title":"LMS J. Comput. Math."},{"issue":"285","key":"26","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1090\/S0025-5718-2013-02719-6","article-title":"Computing canonical heights using arithmetic intersection theory","volume":"83","author":"M\u00fcller, Jan Steffen","year":"2014","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"27","series-title":"Modern Birkh\\\"{a}user Classics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-4578-6","volume-title":"Tata lectures on theta. I","author":"Mumford, David","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9780817645724"},{"key":"28","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1007\/BF01297652","article-title":"The complete classification of fibres in pencils of curves of genus two","volume":"9","author":"Namikawa, Yukihiko","year":"1973","journal-title":"Manuscripta Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-2611","issn-type":"print"},{"key":"29","isbn-type":"print","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1007\/s10107-005-0696-y","article-title":"On \ud835\udc5d-adic height pairings","author":"Nekov\u00e1\u0159, Jan","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/0817636846"},{"issue":"5","key":"30","doi-asserted-by":"publisher","first-page":"1513","DOI":"10.1090\/S0002-9939-2011-11014-1","article-title":"Periods of quadratic twists of elliptic curves","volume":"140","author":"Pal, Vivek","year":"2012","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"issue":"17","key":"31","first-page":"130","article-title":"Arithm\u00e9tique des courbes elliptiques et th\u00e9orie d\u2019Iwasawa","author":"Perrin-Riou, Bernadette","year":"1984","journal-title":"M\\'{e}m. Soc. Math. France (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0037-9484","issn-type":"print"},{"issue":"4","key":"32","doi-asserted-by":"crossref","first-page":"399","DOI":"10.24033\/bsmf.2085","article-title":"Fonctions \ud835\udc3f \ud835\udc5d-adiques, th\u00e9orie d\u2019Iwasawa et points de Heegner","volume":"115","author":"Perrin-Riou, Bernadette","year":"1987","journal-title":"Bull. Soc. Math. France","ISSN":"https:\/\/id.crossref.org\/issn\/0037-9484","issn-type":"print"},{"issue":"3","key":"33","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1215\/S0012-7094-03-11835-9","article-title":"On the \ud835\udc5d-adic \ud835\udc3f-function of a modular form at a supersingular prime","volume":"118","author":"Pollack, Robert","year":"2003","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"},{"issue":"1","key":"34","doi-asserted-by":"publisher","first-page":"1","DOI":"10.24033\/asens.2139","article-title":"Overconvergent modular symbols and \ud835\udc5d-adic \ud835\udc3f-functions","volume":"44","author":"Pollack, Robert","year":"2011","journal-title":"Ann. Sci. \\'{E}c. Norm. Sup\\'{e}r. (4)","ISSN":"https:\/\/id.crossref.org\/issn\/0012-9593","issn-type":"print"},{"issue":"1","key":"35","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1007\/BF01239508","article-title":"The \u201cmain conjectures\u201d of Iwasawa theory for imaginary quadratic fields","volume":"103","author":"Rubin, Karl","year":"1991","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"issue":"3","key":"36","doi-asserted-by":"publisher","first-page":"401","DOI":"10.1007\/BF01389362","article-title":"\ud835\udc5d-adic height pairings. I","volume":"69","author":"Schneider, Peter","year":"1982","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"issue":"2","key":"37","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1007\/BF01388978","article-title":"\ud835\udc5d-adic height pairings. II","volume":"79","author":"Schneider, Peter","year":"1985","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"key":"38","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"},{"issue":"3","key":"39","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1007\/BF01391466","article-title":"On the periods of modular forms","volume":"229","author":"Shimura, Goro","year":"1977","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"issue":"1","key":"40","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00222-013-0448-1","article-title":"The Iwasawa main conjectures for \ud835\udc3a\ud835\udc3f\u2082","volume":"195","author":"Skinner, Christopher","year":"2014","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"key":"41","series-title":"Graduate Studies in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/079","volume-title":"Modular forms, a computational approach","volume":"79","author":"Stein, William","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9780821839607"},{"key":"42","unstructured":"W. A. Stein et al, Sage Mathematics Software (Version 6.2). The Sage Development Team, 2014. \\url{http:\/\/www.sagemath.org}."},{"issue":"283","key":"43","doi-asserted-by":"publisher","first-page":"1757","DOI":"10.1090\/S0025-5718-2012-02649-4","article-title":"Algorithms for the arithmetic of elliptic curves using Iwasawa theory","volume":"82","author":"Stein, William","year":"2013","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"44","doi-asserted-by":"publisher","first-page":"165","DOI":"10.4064\/aa104-2-6","article-title":"On the height constant for curves of genus two. II","volume":"104","author":"Stoll, Michael","year":"2002","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"45","isbn-type":"print","first-page":"Exp. No. 306, 415--440","article-title":"On the conjectures of Birch and Swinnerton-Dyer and a geometric analog","author":"Tate, John","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/2856290426"},{"issue":"2","key":"46","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1215\/S0012-7094-99-09811-3","article-title":"Canonical periods and congruence formulae","volume":"98","author":"Vatsal, V.","year":"1999","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2016-85-298\/S0025-5718-2015-03029-4\/S0025-5718-2015-03029-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-298\/S0025-5718-2015-03029-4\/S0025-5718-2015-03029-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:42:39Z","timestamp":1776796959000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-298\/S0025-5718-2015-03029-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,8,12]]},"references-count":46,"journal-issue":{"issue":"298","published-print":{"date-parts":[[2016,3]]}},"alternative-id":["S0025-5718-2015-03029-4"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3029","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":[[2015,8,12]]}}}