{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:41:32Z","timestamp":1776836492886,"version":"3.51.2"},"reference-count":44,"publisher":"American Mathematical Society (AMS)","issue":"339","license":[{"start":{"date-parts":[[2023,8,25]],"date-time":"2023-08-25T00:00:00Z","timestamp":1692921600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100003343","name":"Cambridge Trust","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100003343","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We study\n \n \n \n N<\/mml:mi>\n N<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -congruences between quadratic twists of elliptic curves. If\n \n \n \n N<\/mml:mi>\n N<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In many, but not all cases, the modular curves in question correspond to the normaliser of a Cartan subgroup of\n \n \n \n \n \n GL<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \n \u2061\n \n <\/mml:mo>\n (<\/mml:mo>\n \n Z<\/mml:mi>\n <\/mml:mrow>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n N<\/mml:mi>\n \n Z<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n \\operatorname {GL}_2(\\mathbb {Z}\/N\\mathbb {Z})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . By computing explicit models for these double covers we find all pairs\n \n \n \n \n (<\/mml:mo>\n N<\/mml:mi>\n ,<\/mml:mo>\n r<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n (N, r)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n such that there exist infinitely many\n \n \n \n j<\/mml:mi>\n j<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -invariants of elliptic curves\n \n \n \n \n E<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n Q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n E\/\\mathbb {Q}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n which are\n \n \n \n N<\/mml:mi>\n N<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -congruent with power\n \n \n \n r<\/mml:mi>\n r<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n to a quadratic twist of\n \n \n \n E<\/mml:mi>\n E<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We also find an example of a\n \n \n \n 48<\/mml:mn>\n 48<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -congruence over\n \n \n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {Q}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We make a conjecture classifying nontrivial\n \n \n \n \n (<\/mml:mo>\n N<\/mml:mi>\n ,<\/mml:mo>\n r<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n (N,r)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -congruences between quadratic twists of elliptic curves over\n \n \n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {Q}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>\n

\n Finally, we give a more detailed analysis of the level\n \n \n \n 15<\/mml:mn>\n 15<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n case. We use elliptic Chabauty to determine the rational points on a modular curve of genus\n \n \n \n 2<\/mml:mn>\n 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n whose Jacobian has rank\n \n \n \n 2<\/mml:mn>\n 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and which arises as a double cover of the modular curve\n \n \n \n \n X<\/mml:mi>\n (<\/mml:mo>\n \n n<\/mml:mi>\n s<\/mml:mi>\n \n <\/mml:mrow>\n \n 3<\/mml:mn>\n +<\/mml:mo>\n <\/mml:msup>\n ,<\/mml:mo>\n \n n<\/mml:mi>\n s<\/mml:mi>\n \n <\/mml:mrow>\n \n 5<\/mml:mn>\n +<\/mml:mo>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n X(\\mathrm {ns\\,} 3^+, \\mathrm {ns\\,} 5^+)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . As a consequence we obtain a new proof of the class number\n \n \n \n 1<\/mml:mn>\n 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n problem.\n <\/p>","DOI":"10.1090\/mcom\/3770","type":"journal-article","created":{"date-parts":[[2022,6,22]],"date-time":"2022-06-22T09:19:01Z","timestamp":1655889541000},"page":"409-450","source":"Crossref","is-referenced-by-count":3,"title":["Congruences of elliptic curves arising from nonsurjective mod \ud835\udc41 Galois representations"],"prefix":"10.1090","volume":"92","author":[{"given":"Sam","family":"Frengley","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2022,8,25]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"715","DOI":"10.1016\/j.jnt.2008.09.013","article-title":"A modular curve of level 9 and the class number one problem","volume":"129","author":"Baran, Burcu","year":"2009","journal-title":"J. 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Frengley, Github repository, \\url{https:\/\/github.com\/SamFrengley\/quadratic-twist-type-congruences.git}, 2021. \u2191"},{"issue":"2","key":"23","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1080\/10586458.1998.10504366","article-title":"Sur la courbe modulaire \ud835\udc4b_{\\ssize\ud835\udc5b\ud835\udc51\ud835\udc52\u0301\ud835\udc5d}(11)","volume":"7","author":"Halberstadt, Emmanuel","year":"1998","journal-title":"Experiment. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"key":"24","unstructured":"[Hal21] E. Halberstadt, Calculs explicites sur les courbes modulaires, \\url{https:\/\/hal.archives-ouvertes.fr\/hal-03149641}, February 2021, Les pr\u00e9sentes notes constituent une introduction \u00e9l\u00e9mentaire aux courbes modulaires. \u2191"},{"key":"25","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1007\/BF01174749","article-title":"Diophantische Analysis und Modulfunktionen","volume":"56","author":"Heegner, Kurt","year":"1952","journal-title":"Math. Z.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5874","issn-type":"print"},{"issue":"1","key":"26","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1080\/10586458.2003.10504710","article-title":"Sur la courbe modulaire \ud835\udc4b_{\ud835\udc38}(7)","volume":"12","author":"Halberstadt, Emmanuel","year":"2003","journal-title":"Experiment. 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Lozano-Robledo, Galois representations attached to elliptic curves with complex multiplication, Algebra Number Theory, 16 (2022) 777\u2013837, arXiv:1809.02584. \u2191","DOI":"10.2140\/ant.2022.16.777"},{"issue":"1","key":"31","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1215\/S0012-7094-07-13714-1","article-title":"Twists of \ud835\udc4b(7) and primitive solutions to \ud835\udc65\u00b2+\ud835\udc66\u00b3=\ud835\udc67\u2077","volume":"137","author":"Poonen, Bjorn","year":"2007","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"},{"key":"32","isbn-type":"print","first-page":"148","article-title":"Families of elliptic curves with constant mod \ud835\udc5d representations","author":"Rubin, K.","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/1571460268"},{"key":"33","unstructured":"[RSZ21a] J. Rouse, A. V. Sutherland, and D. Zureick-Brown, \u2113-Adic images of Galois for elliptic curves over \u211a, Forum Math. 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Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"key":"39","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0851-8","volume-title":"Advanced topics in the arithmetic of elliptic curves","volume":"151","author":"Silverman, Joseph H.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0387943285"},{"key":"40","isbn-type":"print","first-page":"447","article-title":"Explicit families of elliptic curves with prescribed mod \ud835\udc41 representations","author":"Silverberg, Alice","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/0387946098"},{"key":"41","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-09494-6","volume-title":"The arithmetic of elliptic curves","volume":"106","author":"Silverman, Joseph H.","year":"2009","ISBN":"https:\/\/id.crossref.org\/isbn\/9780387094939","edition":"2"},{"issue":"5","key":"42","doi-asserted-by":"publisher","first-page":"1199","DOI":"10.2140\/ant.2017.11.1199","article-title":"Modular curves of prime-power level with infinitely many rational points","volume":"11","author":"Sutherland, Andrew V.","year":"2017","journal-title":"Algebra Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/1937-0652","issn-type":"print"},{"key":"43","unstructured":"[{Zyw}15a] D. Zywina, On the possible images of the mod \u2113 representations associated to elliptic curves over \u211a, E-prints, arXiv:1508.07660, 2015. \u2191"},{"key":"44","unstructured":"[{Zyw}15b] D. Zywina, Possible indices for the Galois image of elliptic curves over \u211a, E-prints, arXiv:1508.07663, 2015. \u2191"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-339\/S0025-5718-2022-03770-4\/mcom3770_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-339\/S0025-5718-2022-03770-4\/S0025-5718-2022-03770-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:47:39Z","timestamp":1776833259000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-339\/S0025-5718-2022-03770-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,25]]},"references-count":44,"journal-issue":{"issue":"339","published-print":{"date-parts":[[2023,1]]}},"alternative-id":["S0025-5718-2022-03770-4"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3770","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":[[2022,8,25]]}}}