Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Tate–Shafarevich group
open-in-new
<h2 id="elements-of-the-tateshafarevich-group">Elements of the Tate–Shafarevich group</h2> <p>Geometrically, the non-trivial elements of the Tate–Shafarevich group can be thought of as the homogeneous spaces of <i>A</i> that have <i>Kv</i>-<a href="/facts/Rational_point/j5v4cxuE">rational points</a> for every <a href="/facts/Place_(mathematics)/LPzDT5CA">place</a> <i>v</i> of <i>K</i>, but no <i>K</i>-rational point. Thus, the group measures the extent to which the <a href="/facts/Hasse_principle/B6el223t">Hasse principle</a> fails to hold for rational equations with coefficients in the field <i>K</i>. Carl-Erik Lind gave an example of such a homogeneous space, by showing that the <a href="/facts/Genus_of_an_algebraic_curve/o1Yhf8ty">genus</a> 1 curve <i>x</i>4 − 17 = 2<i>y</i>2 has solutions over the reals and over all <i>p</i>-adic fields, but has no rational points.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> <a href="/facts/Ernst_S._Selmer/xPrjJLnM">Ernst S. Selmer</a> gave many more examples, such as 3<i>x</i>3 + 4<i>y</i>3 + 5<i>z</i>3 = 0.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p><p>The special case of the Tate–Shafarevich group for the finite group scheme consisting of points of some given finite order <i>n</i> of an abelian variety is closely related to the <a href="/facts/Selmer_group/jnT5RIqE">Selmer group</a>. </p> <h2 id="tateshafarevich-conjecture">Tate–Shafarevich conjecture</h2> <p>The Tate–Shafarevich conjecture states that the Tate–Shafarevich group is finite. <a href="/facts/Karl_Rubin/3xhVe6yE">Karl Rubin</a> proved this for some <a href="/facts/Elliptic_curve/60jvaHSA">elliptic curves</a> of rank at most 1 with <a href="/facts/Complex_multiplication/YQNcbNXP">complex multiplication</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> <a href="/facts/Victor_Kolyvagin/7H6wd82H">Victor A. Kolyvagin</a> extended this to modular elliptic curves over the rationals of analytic rank at most 1.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> (The <a href="/facts/Modularity_theorem/aI3HRxwj">modularity theorem</a> later showed that the modularity assumption always holds.) </p><p>It is known that the Tate–Shafarevich group is a <a href="/facts/Torsion_group/TEvLhDtm">torsion group</a>,<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> thus the conjecture is equivalent to stating that the group is <a href="/facts/Finitely_generated_abelian_group/dwtl7mp4">finitely generated</a>. </p> <h2 id="casselstate-pairing">Cassels–Tate pairing</h2> <p>The Cassels–Tate pairing is a <a href="/facts/Bilinear_pairing/G0qgKRD8">bilinear pairing</a> Ш(<i>A</i>) × Ш(<i>Â</i>) → Q/Z, where <i>A</i> is an abelian variety and <i>Â</i> is its dual. Cassels introduced this for <a href="/facts/Elliptic_curve/60jvaHSA">elliptic curves</a>, when <i>A</i> can be identified with <i>Â</i> and the pairing is an alternating form.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> The kernel of this form is the subgroup of divisible elements, which is trivial if the Tate–Shafarevich conjecture is true. Tate extended the pairing to general abelian varieties, as a variation of <a href="/facts/Tate_duality/hQwrbFm1">Tate duality</a>.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> A choice of polarization on <i>A</i> gives a map from <i>A</i> to <i>Â</i>, which induces a bilinear pairing on Ш(<i>A</i>) with values in Q/Z, but unlike the case of elliptic curves this need not be alternating or even skew symmetric. </p><p>For an elliptic curve, Cassels showed that the pairing is alternating, and a consequence is that if the order of Ш is finite then it is a square. For more general abelian varieties it was sometimes incorrectly believed for many years that the order of Ш is a square whenever it is finite; this mistake originated in a paper by <a href="/facts/Peter_Swinnerton-Dyer/D66SILlt">Swinnerton-Dyer</a>,<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> who misquoted one of the results of Tate.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> Poonen and Stoll gave some examples where the order is twice a square, such as the Jacobian of a certain genus 2 curve over the rationals whose Tate–Shafarevich group has order 2,<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> and <a href="/facts/William_A._Stein/tBqUDE0P">Stein</a> gave some examples where the power of an odd prime dividing the order is odd.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> If the abelian variety has a principal polarization then the form on Ш is skew symmetric which implies that the order of Ш is a square or twice a square (if it is finite), and if in addition the principal polarization comes from a rational divisor (as is the case for elliptic curves) then the form is alternating and the order of Ш is a square (if it is finite). On the other hand building on the results just presented Konstantinou showed that for any <a href="/facts/Squarefree/R3qR9hP1">squarefree</a> number <i>n</i> there is an abelian variety <i>A</i> defined over Q and an integer <i>m</i> with |Ш| = <i>n</i> ⋅ <i>m</i>2.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> In particular Ш is finite in Konstantinou's examples and these examples confirm a conjecture of Stein. Thus modulo squares any integer can be the order of Ш. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Birch_and_Swinnerton-Dyer_conjecture/Issh7Yfr">Birch and Swinnerton-Dyer conjecture</a></li></ul> <h2 id="citations">Citations</h2> <ul><li>Cassels, John William Scott (1962), "Arithmetic on curves of genus 1. III. The Tate–Šafarevič and Selmer groups", <i>Proceedings of the London Mathematical Society</i>, Third Series, 12: 259–296, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2Fplms%2Fs3-12.1.259">10.1112/plms/s3-12.1.259</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0024-6115">0024-6115</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0163913">0163913</a></li> <li>Cassels, John William Scott (1962b), <a href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002179873">"Arithmetic on curves of genus 1. IV. Proof of the Hauptvermutung"</a>, <i><a href="/facts/Journal_f%25C3%25BCr_die_reine_und_angewandte_Mathematik/YCcRSaAQ">Journal für die reine und angewandte Mathematik</a></i>, 211 (211): 95–112, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1515%2Fcrll.1962.211.95">10.1515/crll.1962.211.95</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0075-4102">0075-4102</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0163915">0163915</a></li> <li>Cassels, John William Scott (1991), <a href="https://books.google.com/books?id=zgqUAuEJNJ4C"><i>Lectures on elliptic curves</i></a>, London Mathematical Society Student Texts, vol. 24, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FCBO9781139172530">10.1017/CBO9781139172530</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-41517-0, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1144763">1144763</a></li> <li>Hindry, Marc; <a href="/facts/Joseph_H._Silverman/acyPld7K">Silverman, Joseph H.</a> (2000), <i>Diophantine geometry: an introduction</i>, Graduate Texts in Mathematics, vol. 201, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-98981-5</li> <li><a href="/facts/Ralph_Greenberg/00Wa5D6T">Greenberg, Ralph</a> (1994), "Iwasawa Theory and p-adic Deformation of Motives", in <a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Serre, Jean-Pierre</a>; Jannsen, Uwe; Kleiman, Steven L. (eds.), <i>Motives</i>, Providence, R.I.: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-1637-0</li> <li><a href="/facts/Victor_Kolyvagin/7H6wd82H">Kolyvagin, V. A.</a> (1988), "Finiteness of E(Q) and SH(E,Q) for a subclass of Weil curves", <i>Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya</i>, 52 (3): 522–540, 670–671, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0373-2436">0373-2436</a>, 954295</li> <li><a href="/facts/Serge_Lang/I7MW2rUu">Lang, Serge</a>; <a href="/facts/John_Tate_(mathematician)/nBntLCG0">Tate, John</a> (1958), "Principal homogeneous spaces over abelian varieties", <i><a href="/facts/American_Journal_of_Mathematics/lkgftcD7">American Journal of Mathematics</a></i>, 80 (3): 659–684, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2372778">10.2307/2372778</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9327">0002-9327</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2372778">2372778</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0106226">0106226</a></li> <li>Lind, Carl-Erik (1940). <a href="https://books.google.com/books?id=ZggUAQAAIAAJ"><i>Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins</i></a> (Thesis). Vol. 1940. University of Uppsala. 97 pp. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0022563">0022563</a>.</li> <li>Poonen, Bjorn; Stoll, Michael (1999), "The Cassels-Tate pairing on polarized abelian varieties", <i><a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a></i>, Second Series, 150 (3): 1109–1149, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/9911267">math/9911267</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F121064">10.2307/121064</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/121064">121064</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1740984">1740984</a></li> <li>Rubin, Karl (1987), "Tate–Shafarevich groups and L-functions of elliptic curves with complex multiplication", <i><a href="/facts/Inventiones_Mathematicae/bARcL95Y">Inventiones Mathematicae</a></i>, 89 (3): 527–559, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1987InMat..89..527R">1987InMat..89..527R</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01388984">10.1007/BF01388984</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0020-9910">0020-9910</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0903383">0903383</a></li> <li>Selmer, Ernst S. (1951), "The Diophantine equation ax³+by³+cz³=0", <i><a href="/facts/Acta_Mathematica/11kbuhDT">Acta Mathematica</a></i>, 85: 203–362, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF02395746">10.1007/BF02395746</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0001-5962">0001-5962</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0041871">0041871</a></li> <li>Shafarevich, I. R. (1959), "The group of principal homogeneous algebraic manifolds", <i>Doklady Akademii Nauk SSSR</i> (in Russian), 124: 42–43, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-3264">0002-3264</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0106227">0106227</a> English translation in his collected mathematical papers</li> <li><a href="/facts/William_A._Stein/tBqUDE0P">Stein, William A.</a> (2004), <a href="https://wstein.org/papers/nonsquaresha/final2.pdf">"Shafarevich–Tate groups of nonsquare order"</a> (PDF), <i>Modular curves and abelian varieties</i>, Progr. Math., vol. 224, Basel, Boston, Berlin: Birkhäuser, pp. 277–289, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2058655">2058655</a></li> <li><a href="/facts/Peter_Swinnerton-Dyer/D66SILlt">Swinnerton-Dyer, P.</a> (1967), <a href="https://books.google.com/books?id=I983HAAACAAJ">"The conjectures of Birch and Swinnerton-Dyer, and of Tate"</a>, in Springer, Tonny A. (ed.), <i>Proceedings of a Conference on Local Fields (Driebergen, 1966)</i>, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, pp. 132–157, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0230727">0230727</a></li> <li><a href="/facts/John_Tate_(mathematician)/nBntLCG0">Tate, John</a> (1958), <a href="http://www.numdam.org/item?id=SB_1956-1958__4__265_0"><i>WC-groups over p-adic fields</i></a>, Séminaire Bourbaki; 10e année: 1957/1958, vol. 13, Paris: Secrétariat Mathématique, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0105420">0105420</a></li> <li><a href="/facts/John_Tate_(mathematician)/nBntLCG0">Tate, John</a> (1963), <a href="https://web.archive.org/web/20110717144510/http://mathunion.org/ICM/ICM1962.1/">"Duality theorems in Galois cohomology over number fields"</a>, <i>Proceedings of the International Congress of Mathematicians (Stockholm, 1962)</i>, Djursholm: Inst. Mittag-Leffler, pp. 288–295, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0175892">0175892</a>, archived from <a href="http://mathunion.org/ICM/ICM1962.1/">the original</a> on 2011-07-17</li> <li><a href="/facts/Andr%25C3%25A9_Weil/Q4Dd0Gr4">Weil, André</a> (1955), "On algebraic groups and homogeneous spaces", <i><a href="/facts/American_Journal_of_Mathematics/lkgftcD7">American Journal of Mathematics</a></i>, 77 (3): 493–512, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2372637">10.2307/2372637</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9327">0002-9327</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2372637">2372637</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0074084">0074084</a></li> <li>Konstantinou, Alexandros (2024-04-25). "A note on the order of the Tate-Shafarevich group modulo squares". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/2404.16785">2404.16785</a> [<a href="https://arxiv.org/archive/math.NT">math.NT</a>].</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Lang & Tate 1958. - Lang, Serge; Tate, John (1958), "Principal homogeneous spaces over abelian varieties", American Journal of Mathematics, 80 (3): 659–684, doi:10.2307/2372778, ISSN 0002-9327, JSTOR 2372778, MR 0106226 <a href="https://doi.org/10.2307%2F2372778" target="_blank">https://doi.org/10.2307%2F2372778</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Shafarevich 1959. - Shafarevich, I. R. (1959), "The group of principal homogeneous algebraic manifolds", Doklady Akademii Nauk SSSR (in Russian), 124: 42–43, ISSN 0002-3264, MR 0106227 <a href="https://search.worldcat.org/issn/0002-3264" target="_blank">https://search.worldcat.org/issn/0002-3264</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Cassels 1962b. - Cassels, John William Scott (1962b), "Arithmetic on curves of genus 1. IV. Proof of the Hauptvermutung", Journal für die reine und angewandte Mathematik, 211 (211): 95–112, doi:10.1515/crll.1962.211.95, ISSN 0075-4102, MR 0163915 <a href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002179873" target="_blank">http://resolver.sub.uni-goettingen.de/purl?GDZPPN002179873</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Cassels 1962. - Cassels, John William Scott (1962), "Arithmetic on curves of genus 1. III. The Tate–Šafarevič and Selmer groups", Proceedings of the London Mathematical Society, Third Series, 12: 259–296, doi:10.1112/plms/s3-12.1.259, ISSN 0024-6115, MR 0163913 <a href="https://doi.org/10.1112%2Fplms%2Fs3-12.1.259" target="_blank">https://doi.org/10.1112%2Fplms%2Fs3-12.1.259</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Lind 1940. - Lind, Carl-Erik (1940). Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins (Thesis). Vol. 1940. University of Uppsala. 97 pp. MR 0022563. <a href="https://books.google.com/books?id=ZggUAQAAIAAJ" target="_blank">https://books.google.com/books?id=ZggUAQAAIAAJ</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Selmer 1951. - Selmer, Ernst S. (1951), "The Diophantine equation ax³+by³+cz³=0", Acta Mathematica, 85: 203–362, doi:10.1007/BF02395746, ISSN 0001-5962, MR 0041871 <a href="https://doi.org/10.1007%2FBF02395746" target="_blank">https://doi.org/10.1007%2FBF02395746</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Rubin 1987. - Rubin, Karl (1987), "Tate–Shafarevich groups and L-functions of elliptic curves with complex multiplication", Inventiones Mathematicae, 89 (3): 527–559, Bibcode:1987InMat..89..527R, doi:10.1007/BF01388984, ISSN 0020-9910, MR 0903383 <a href="https://ui.adsabs.harvard.edu/abs/1987InMat..89..527R" target="_blank">https://ui.adsabs.harvard.edu/abs/1987InMat..89..527R</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Kolyvagin 1988. - Kolyvagin, V. A. (1988), "Finiteness of E(Q) and SH(E,Q) for a subclass of Weil curves", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 52 (3): 522–540, 670–671, ISSN 0373-2436, 954295 <a href="https://search.worldcat.org/issn/0373-2436" target="_blank">https://search.worldcat.org/issn/0373-2436</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Kolyvagin, V. A. (1991), "On the structure of shafarevich-tate groups", Algebraic Geometry, Lecture Notes in Mathematics, vol. 1479, Springer Berlin Heidelberg, pp. 94–121, doi:10.1007/bfb0086267, ISBN 978-3-540-54456-2 <a href="978-3-540-54456-2" target="_blank">978-3-540-54456-2</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Poonen, Bjorn (2024-09-01). "THE SELMER GROUP, THE SHAFAREVICH-TATE GROUP, AND THE WEAK MORDELL-WEIL THEOREM" (PDF). <a href="https://math.mit.edu/~poonen/f01/weakmw.pdf" target="_blank">https://math.mit.edu/~poonen/f01/weakmw.pdf</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Cassels 1962. - Cassels, John William Scott (1962), "Arithmetic on curves of genus 1. III. The Tate–Šafarevič and Selmer groups", Proceedings of the London Mathematical Society, Third Series, 12: 259–296, doi:10.1112/plms/s3-12.1.259, ISSN 0024-6115, MR 0163913 <a href="https://doi.org/10.1112%2Fplms%2Fs3-12.1.259" target="_blank">https://doi.org/10.1112%2Fplms%2Fs3-12.1.259</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Tate 1963. - Tate, John (1963), "Duality theorems in Galois cohomology over number fields", Proceedings of the International Congress of Mathematicians (Stockholm, 1962), Djursholm: Inst. Mittag-Leffler, pp. 288–295, MR 0175892, archived from the original on 2011-07-17 <a href="https://web.archive.org/web/20110717144510/http://mathunion.org/ICM/ICM1962.1/" target="_blank">https://web.archive.org/web/20110717144510/http://mathunion.org/ICM/ICM1962.1/</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Swinnerton-Dyer 1967. - Swinnerton-Dyer, P. (1967), "The conjectures of Birch and Swinnerton-Dyer, and of Tate", in Springer, Tonny A. (ed.), Proceedings of a Conference on Local Fields (Driebergen, 1966), Berlin, New York: Springer-Verlag, pp. 132–157, MR 0230727 <a href="https://books.google.com/books?id=I983HAAACAAJ" target="_blank">https://books.google.com/books?id=I983HAAACAAJ</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Tate 1963. - Tate, John (1963), "Duality theorems in Galois cohomology over number fields", Proceedings of the International Congress of Mathematicians (Stockholm, 1962), Djursholm: Inst. Mittag-Leffler, pp. 288–295, MR 0175892, archived from the original on 2011-07-17 <a href="https://web.archive.org/web/20110717144510/http://mathunion.org/ICM/ICM1962.1/" target="_blank">https://web.archive.org/web/20110717144510/http://mathunion.org/ICM/ICM1962.1/</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Poonen & Stoll 1999. - Poonen, Bjorn; Stoll, Michael (1999), "The Cassels-Tate pairing on polarized abelian varieties", Annals of Mathematics, Second Series, 150 (3): 1109–1149, arXiv:math/9911267, doi:10.2307/121064, ISSN 0003-486X, JSTOR 121064, MR 1740984 <a href="https://arxiv.org/abs/math/9911267" target="_blank">https://arxiv.org/abs/math/9911267</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Stein 2004. - Stein, William A. (2004), "Shafarevich–Tate groups of nonsquare order" (PDF), Modular curves and abelian varieties, Progr. Math., vol. 224, Basel, Boston, Berlin: Birkhäuser, pp. 277–289, MR 2058655 <a href="https://wstein.org/papers/nonsquaresha/final2.pdf" target="_blank">https://wstein.org/papers/nonsquaresha/final2.pdf</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Konstantinou 2024. - Konstantinou, Alexandros (2024-04-25). "A note on the order of the Tate-Shafarevich group modulo squares". arXiv:2404.16785 [math.NT]. <a href="https://arxiv.org/abs/2404.16785" target="_blank">https://arxiv.org/abs/2404.16785</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> </ol>