Typically bounding torsion on elliptic curves isogenous to rational $j$-invariant
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- by Tyler Genao
- Proc. Amer. Math. Soc. 151 (2023), 1907-1914
- DOI: https://doi.org/10.1090/proc/16298
- Published electronically: February 10, 2023
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Abstract:
We prove that the family $\mathcal {I}_{F_0}$ of elliptic curves over number fields that are geometrically isogenous to an elliptic curve with $F_0$-rational $j$-invariant is typically bounded in torsion. Under an additional uniformity assumption, we also prove that the family $\mathcal {I}_{d_0}$ of elliptic curves over number fields that are geometrically isogenous to an elliptic curve with degree $d_0$ $j$-invariant is typically bounded in torsion.References
- Keisuke Arai, On uniform lower bound of the Galois images associated to elliptic curves, J. ThĆ©or. Nombres Bordeaux 20 (2008), no.Ā 1, 23ā43 (English, with English and French summaries). MR 2434156, DOI 10.5802/jtnb.614
- Abbey Bourdon, Pete L. Clark, and Paul Pollack, Anatomy of torsion in the CM case, Math. Z. 285 (2017), no.Ā 3-4, 795ā820. MR 3623731, DOI 10.1007/s00209-016-1727-5
- A. Bourdon and F. Najman, Sporadic points of odd degree on $X_1(N)$ coming from $\mathbb {Q}$-curves, preprint, arXiv:2107.10909, 2021.
- Pete L. Clark, CM elliptic curves: Volcanoes, reality and applications, Part I, preprint, arXiv:2212.13316, 2023.
- Pete L. Clark, Marko Milosevic, and Paul Pollack, Typically bounding torsion, J. Number Theory 192 (2018), 150ā167. MR 3841549, DOI 10.1016/j.jnt.2018.04.005
- J. E. Cremona and Filip Najman, $\Bbb Q$-curves over odd degree number fields, Res. Number Theory 7 (2021), no.Ā 4, Paper No. 62, 30. MR 4314224, DOI 10.1007/s40993-021-00270-0
- Pete L. Clark and Paul Pollack, Pursuing polynomial bounds on torsion, Israel J. Math. 227 (2018), no.Ā 2, 889ā909. MR 3846346, DOI 10.1007/s11856-018-1751-8
- Noam D. Elkies, On elliptic $K$-curves, Modular curves and abelian varieties, Progr. Math., vol. 224, BirkhĆ¤user, Basel, 2004, pp.Ā 81ā91. MR 2058644
- Jordan S. Ellenberg, $\Bbb Q$-curves and Galois representations, Modular curves and abelian varieties, Progr. Math., vol. 224, BirkhĆ¤user, Basel, 2004, pp.Ā 93ā103. MR 2058645
- Paul ErdÅs and Samuel S. Wagstaff Jr., The fractional parts of the Bernoulli numbers, Illinois J. Math. 24 (1980), no.Ā 1, 104ā112. MR 550654
- Tyler Genao, Typically bounding torsion on elliptic curves with rational $j$-invariant, J. Number Theory 238 (2022), 823ā841. MR 4430120, DOI 10.1016/j.jnt.2021.10.003
- H. Heilbronn, On the class number in imaginary quadratic fields. Quart. J. Math. Oxford Ser. 25 (1934), 150ā160.
- Chandrashekhar Khare and Jean-Pierre Wintenberger, Serreās modularity conjecture. I, Invent. Math. 178 (2009), no.Ā 3, 485ā504. MR 2551763, DOI 10.1007/s00222-009-0205-7
- Chandrashekhar Khare and Jean-Pierre Wintenberger, Serreās modularity conjecture. II, Invent. Math. 178 (2009), no.Ā 3, 505ā586. MR 2551764, DOI 10.1007/s00222-009-0206-6
- Eric Larson and Dmitry Vaintrob, Determinants of subquotients of Galois representations associated with abelian varieties, J. Inst. Math. Jussieu 13 (2014), no.Ā 3, 517ā559. With an appendix by Brian Conrad. MR 3211798, DOI 10.1017/S1474748013000182
- LoĆÆc Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), no.Ā 1-3, 437ā449 (French). MR 1369424, DOI 10.1007/s002220050059
- Kenneth A. Ribet, Abelian varieties over $\textbf {Q}$ and modular forms, Algebra and topology 1992 (TaejÅn), Korea Adv. Inst. Sci. Tech., TaejÅn, 1992, pp.Ā 53ā79. MR 1212980
- Jean-Pierre Serre, PropriĆ©tĆ©s galoisiennes des points dāordre fini des courbes elliptiques, Invent. Math. 15 (1972), no.Ā 4, 259ā331 (French). MR 387283, DOI 10.1007/BF01405086
- Jean-Pierre Serre, Abelian $l$-adic representations and elliptic curves, Research Notes in Mathematics, vol. 7, A K Peters, Ltd., Wellesley, MA, 1998. With the collaboration of Willem Kuyk and John Labute; Revised reprint of the 1968 original. MR 1484415
- Joseph H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094, DOI 10.1007/978-0-387-09494-6
Bibliographic Information
- Tyler Genao
- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- MR Author ID: 1179563
- ORCID: 0000-0003-4242-3784
- Email: tylergenao@uga.edu
- Received by editor(s): January 1, 2022
- Received by editor(s) in revised form: June 17, 2022, and September 16, 2022
- Published electronically: February 10, 2023
- Additional Notes: This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1842396. Partial support was also provided by the Research and Training Group grant DMS-1344994 funded by the National Science Foundation.
- Communicated by: Romyar T. Sharifi
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 1907-1914
- MSC (2020): Primary 11G05
- DOI: https://doi.org/10.1090/proc/16298
- MathSciNet review: 4556187