Cyclic isogenies of elliptic curves over fixed quadratic fields

Banwait, Barinder; Najman, Filip; Padurariu, Oana

doi:10.1090/mcom/3894

Cyclic isogenies of elliptic curves over fixed quadratic fields
HTML articles powered by AMS MathViewer

by Barinder S. Banwait, Filip Najman and Oana Padurariu

Math. Comp. 93 (2024), 841-862

DOI: https://doi.org/10.1090/mcom/3894

Published electronically: August 30, 2023

HTML | PDF | Request permission

Abstract:

Building on Mazur’s 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over $\mathbb {Q}$. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised.

In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields $\mathbb {Q}(\sqrt {d})$ with $|d| < 10^4$ we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over $19$ quadratic fields, including $\mathbb {Q}(\sqrt {213})$ and $\mathbb {Q}(\sqrt {-2289})$. To make this procedure work, we determine all of the finitely many quadratic points on the modular curves $X_0(125)$ and $X_0(169)$, which may be of independent interest.

References

Nikola Adžaga, Vishal Arul, Lea Beneish, Mingjie Chen, Shiva Chidambaram, Timo Keller, and Boya Wen, Quadratic Chabauty for Atkin-Lehner quotients of modular curves of prime level and genus 4, 5, 6, Acta Arith. 208 (2023), no. 1, 15–49. MR 4616870, DOI 10.4064/aa220110-7-3
V. Arul and J. Steffen Müller, Rational points on $X_0^+(125)$, preprint, available at arXiv:2205.14744, 2022.
Barinder S. Banwait, Explicit isogenies of prime degree over quadratic fields, Int. Math. Res. Not. IMRN 14 (2023), 11829–11876. MR 4615219, DOI 10.1093/imrn/rnac134
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Barinder S. Banwait, Explicit isogenies of prime degree over quadratic fields, Int. Math. Res. Not. IMRN 14 (2023), 11829–11876. MR 4615219, DOI 10.1093/imrn/rnac134
Jennifer Balakrishnan, Netan Dogra, J. Steffen Müller, Jan Tuitman, and Jan Vonk, Explicit Chabauty-Kim for the split Cartan modular curve of level 13, Ann. of Math. (2) 189 (2019), no. 3, 885–944. MR 3961086, DOI 10.4007/annals.2019.189.3.6
Josha Box, Stevan Gajović, and Pip Goodman, Cubic and quartic points on modular curves using generalised symmetric Chabauty, Int. Math. Res. Not. IMRN 7 (2023), 5604–5659. MR 4565697, DOI 10.1093/imrn/rnab358
Nicolas Billerey, Critères d’irréductibilité pour les représentations des courbes elliptiques, Int. J. Number Theory 7 (2011), no. 4, 1001–1032 (French, with English and French summaries). MR 2812649, DOI 10.1142/S1793042111004538
Peter Bruin and Filip Najman, A criterion to rule out torsion groups for elliptic curves over number fields, Res. Number Theory 2 (2016), Paper No. 3, 13. MR 3501016, DOI 10.1007/s40993-015-0031-5
Josha Box, Quadratic points on modular curves with infinite Mordell-Weil group, Math. Comp. 90 (2021), no. 327, 321–343. MR 4166463, DOI 10.1090/mcom/3547
Pete L. Clark, Tyler Genao, Paul Pollack, and Frederick Saia, The least degree of a CM point on a modular curve, J. Lond. Math. Soc. (2) 105 (2022), no. 2, 825–883. MR 4400938, DOI 10.1112/jlms.12518
J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193
Maarten Derickx, Anastassia Etropolski, Mark van Hoeij, Jackson S. Morrow, and David Zureick-Brown, Sporadic cubic torsion, Algebra Number Theory 15 (2021), no. 7, 1837–1864. MR 4333666, DOI 10.2140/ant.2021.15.1837
Maarten Derickx, Sheldon Kamienny, William Stein, and Michael Stoll, Torsion points on elliptic curves over number fields of small degree, Algebra Number Theory 17 (2023), no. 2, 267–308. MR 4564759, DOI 10.2140/ant.2023.17.267
Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), no. 2, 225–320. MR 833192, DOI 10.1007/BF01388809
S. Kamienny, Torsion points on elliptic curves and $q$-coefficients of modular forms, Invent. Math. 109 (1992), no. 2, 221–229. MR 1172689, DOI 10.1007/BF01232025
M. A. Kenku, The modular curve $X_{0}(39)$ and rational isogeny, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 1, 21–23. MR 510395, DOI 10.1017/S0305004100055444
M. A. Kenku, The modular curve $X_{0}(169)$ and rational isogeny, J. London Math. Soc. (2) 22 (1980), no. 2, 239–244. MR 588271, DOI 10.1112/jlms/s2-22.2.239
M. A. Kenku, The modular curves $X_{0}(65)$ and $X_{0}(91)$ and rational isogeny, Math. Proc. Cambridge Philos. Soc. 87 (1980), no. 1, 15–20. MR 549292, DOI 10.1017/S0305004100056462
M. A. Kenku, On the modular curves $X_{0}(125)$, $X_{1}(25)$ and $X_{1}(49)$, J. London Math. Soc. (2) 23 (1981), no. 3, 415–427. MR 616546, DOI 10.1112/jlms/s2-23.3.415
V. A. Kolyvagin and D. Yu. Logachëv, Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties, Algebra i Analiz 1 (1989), no. 5, 171–196 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 5, 1229–1253. MR 1036843
M. A. Kenku and F. Momose, Torsion points on elliptic curves defined over quadratic fields, Nagoya Math. J. 109 (1988), 125–149. MR 931956, DOI 10.1017/S0027763000002816
V. A. Kolyvagin, Euler systems, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 435–483. MR 1106906
Samuel Le Fourn and Pedro Lemos, Residual Galois representations of elliptic curves with image contained in the normaliser of a nonsplit Cartan, Algebra Number Theory 15 (2021), no. 3, 747–771. MR 4261100, DOI 10.2140/ant.2021.15.747
Eric Larson and Dmitry Vaintrob, Determinants of subquotients of Galois representations associated with abelian varieties, J. Inst. Math. Jussieu 13 (2014), no. 3, 517–559. With an appendix by Brian Conrad. MR 3211798, DOI 10.1017/S1474748013000182
B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186 (1978). With an appendix by Mazur and M. Rapoport. MR 488287, DOI 10.1007/BF02684339
B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162. MR 482230, DOI 10.1007/BF01390348
Loïc Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), no. 1-3, 437–449 (French). MR 1369424, DOI 10.1007/s002220050059
Philippe Michaud-Jacobs, Fermat’s last theorem and modular curves over real quadratic fields, Acta Arith. 203 (2022), no. 4, 319–351. MR 4444259, DOI 10.4064/aa210812-2-4
B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves, Invent. Math. 25 (1974), 1–61. MR 354674, DOI 10.1007/BF01389997
Filip Najman, Complete classification of torsion of elliptic curves over quadratic cyclotomic fields, J. Number Theory 130 (2010), no. 9, 1964–1968. MR 2653208, DOI 10.1016/j.jnt.2009.12.008
Filip Najman, Isogenies of non-CM elliptic curves with rational $j$-invariants over number fields, Math. Proc. Cambridge Philos. Soc. 164 (2018), no. 1, 179–184. MR 3733245, DOI 10.1017/S0305004117000160
Filip Najman and Antonela Trbović, Splitting of primes in number fields generated by points on some modular curves, Res. Number Theory 8 (2022), no. 2, Paper No. 28, 18. MR 4412569, DOI 10.1007/s40993-022-00325-w
Filip Najman and Borna Vukorepa, Quadratic points on bielliptic modular curves, Math. Comp. 92 (2023), no. 342, 1791–1816. MR 4570342, DOI 10.1090/mcom/3805
Ekin Ozman and Samir Siksek, Quadratic points on modular curves, Math. Comp. 88 (2019), no. 319, 2461–2484. MR 3957901, DOI 10.1090/mcom/3407
Ekin Ozman, Points on quadratic twists of $X_0(N)$, Acta Arith. 152 (2012), no. 4, 323–348. MR 2890545, DOI 10.4064/aa152-4-1
Samir Siksek, Chabauty for symmetric powers of curves, Algebra Number Theory 3 (2009), no. 2, 209–236. MR 2491943, DOI 10.2140/ant.2009.3.209
The Sage Developers, SageMath, the Sage Mathematics Software System (Version 9.2), 2020. https://www.sagemath.org.
The PARI Group, Univ. Bordeaux. PARI/GP version 2.14.0, 2021. available from http://pari.math.u-bordeaux.fr/.
Antonela Trbović, Torsion groups of elliptic curves over quadratic fields $\Bbb Q(\sqrt {d})$, $0<d<100$, Acta Arith. 192 (2020), no. 2, 141–153. MR 4042408, DOI 10.4064/aa180725-8-1
Borna Vukorepa, Isogenies over quadratic fields of elliptic curves with rational j-invariant, preprint, available at arXiv:2203.10672, 2022.