Quadratic points on modular curves with infinite Mordell–Weil group

Box, Josha

doi:10.1090/mcom/3547

Quadratic points on modular curves with infinite Mordell–Weil group
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Math. Comp. 90 (2021), 321-343 Request permission

Abstract:

Bruin and Najman [LMS J. Comput. Math. 18 (2015), no. 1, 578–602] and Ozman and Siksek [Math. Comp. 88 (2019), no. 319, 2461–2484] have recently determined the quadratic points on each modular curve $X_0(N)$ of genus 2, 3, 4, or 5 whose Mordell–Weil group has rank 0. In this paper we do the same for the $X_0(N)$ of genus 2, 3, 4, and 5 and positive Mordell–Weil rank. The values of $N$ are 37, 43, 53, 61, 57, 65, 67, and 73.

The main tool used is a relative symmetric Chabauty method, in combination with the Mordell–Weil sieve. Often the quadratic points are not finite, as the degree 2 map $X_0(N)\to X_0(N)^+$ can be a source of infinitely many such points. In such cases, we describe this map and the rational points on $X_0(N)^+$, and we specify the exceptional quadratic points on $X_0(N)$ not coming from $X_0(N)^+$. In particular, we determine the $j$-invariants of the corresponding elliptic curves and whether they are ${\mathbb {Q}}$-curves or have complex multiplication.

References

Dan Abramovich and Joe Harris, Abelian varieties and curves in $W_d(C)$, Compositio Math. 78 (1991), no. 2, 227–238. MR 1104789
J. Balakrishnan, A. Best, F. Bianchi, B. Lawrence, S. Müller, N. Triantafillou, and J. Vonk, Two recent $p$-adic approaches towards the (effective) mordell conjecture, arXiv:1910.12755.
Jennifer S. Balakrishnan and Netan Dogra, Quadratic Chabauty and rational points, I: $p$-adic heights, Duke Math. J. 167 (2018), no. 11, 1981–2038. With an appendix by J. Steffen Müller. MR 3843370, DOI 10.1215/00127094-2018-0013
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
Francesc Bars, Bielliptic modular curves, J. Number Theory 76 (1999), no. 1, 154–165. MR 1688168, DOI 10.1006/jnth.1998.2343
Nils Bruin and Michael Stoll, The Mordell-Weil sieve: proving non-existence of rational points on curves, LMS J. Comput. Math. 13 (2010), 272–306. MR 2685127, DOI 10.1112/S1461157009000187
Peter Bruin and Filip Najman, Hyperelliptic modular curves $X_0(n)$ and isogenies of elliptic curves over quadratic fields, LMS J. Comput. Math. 18 (2015), no. 1, 578–602. MR 3389884, DOI 10.1112/S1461157015000157
Yann Bugeaud, Maurice Mignotte, Samir Siksek, Michael Stoll, and Szabolcs Tengely, Integral points on hyperelliptic curves, Algebra Number Theory 2 (2008), no. 8, 859–885. MR 2457355, DOI 10.2140/ant.2008.2.859
Claude Chabauty, Sur les points rationnels des variétés algébriques dont l’irrégularité est supérieure à la dimension, C. R. Acad. Sci. Paris 212 (1941), 1022–1024 (French). MR 11005
Robert F. Coleman, Effective Chabauty, Duke Math. J. 52 (1985), no. 3, 765–770. MR 808103, DOI 10.1215/S0012-7094-85-05240-8
J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193
P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 143–316 (French). MR 0337993
M. Derickx, S. Kamienny, W. Stein, and M. Stoll, Torsion points on elliptic curves over number fields of small degree, arXiv:1707.00364v1.1.
V. G. Drinfel′d, Two theorems on modular curves, Funkcional. Anal. i Priložen. 7 (1973), no. 2, 83–84 (Russian). MR 0318157
Gerd Faltings, Diophantine approximation on abelian varieties, Ann. of Math. (2) 133 (1991), no. 3, 549–576. MR 1109353, DOI 10.2307/2944319
Nuno Freitas, Bao V. Le Hung, and Samir Siksek, Elliptic curves over real quadratic fields are modular, Invent. Math. 201 (2015), no. 1, 159–206. MR 3359051, DOI 10.1007/s00222-014-0550-z
Joe Harris and Joe Silverman, Bielliptic curves and symmetric products, Proc. Amer. Math. Soc. 112 (1991), no. 2, 347–356. MR 1055774, DOI 10.1090/S0002-9939-1991-1055774-0
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
Marc Hindry and Joseph H. Silverman, Diophantine geometry, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000. An introduction. MR 1745599, DOI 10.1007/978-1-4612-1210-2
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, 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
Minhyong Kim, The motivic fundamental group of $\mathbf P^1\sbs \{0,1,\infty \}$ and the theorem of Siegel, Invent. Math. 161 (2005), no. 3, 629–656. MR 2181717, DOI 10.1007/s00222-004-0433-9
Minhyong Kim, The unipotent Albanese map and Selmer varieties for curves, Publ. Res. Inst. Math. Sci. 45 (2009), no. 1, 89–133. MR 2512779, DOI 10.2977/prims/1234361156
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
J. Lehner and M. Newman, Weierstrass points of $\Gamma _{0}\,(n)$, Ann. of Math. (2) 79 (1964), 360–368. MR 161841, DOI 10.2307/1970550
San Ling, On the $\mathbf Q$-rational cuspidal subgroup and the component group of $J_0(p^r)$, Israel J. Math. 99 (1997), 29–54. MR 1469086, DOI 10.1007/BF02760675
Ju. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19–66 (Russian). MR 0314846
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, 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
B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves, Invent. Math. 25 (1974), 1–61. MR 354674, DOI 10.1007/BF01389997
William McCallum and Bjorn Poonen, The method of Chabauty and Coleman, Explicit methods in number theory, Panor. Synthèses, vol. 36, Soc. Math. France, Paris, 2012, pp. 99–117 (English, with English and French summaries). MR 3098132
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
Andrew P. Ogg, Hyperelliptic modular curves, Bull. Soc. Math. France 102 (1974), 449–462. MR 364259
Masami Ohta, Eisenstein ideals and the rational torsion subgroups of modular Jacobian varieties II, Tokyo J. Math. 37 (2014), no. 2, 273–318. MR 3304683, DOI 10.3836/tjm/1422452795
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
Pierre Parent, Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres, J. Reine Angew. Math. 506 (1999), 85–116 (French, with French summary). MR 1665681, DOI 10.1515/crll.1999.009
Yuan Ren, Rational torsion subgroups of modular Jacobian varieties, J. Number Theory 190 (2018), 169–186. MR 3805452, DOI 10.1016/j.jnt.2018.02.009
Kenneth A. Ribet, Abelian varieties over $\bf Q$ and modular forms, Modular curves and abelian varieties, Progr. Math., vol. 224, Birkhäuser, Basel, 2004, pp. 241–261. MR 2058653, DOI 10.1007/978-3-0348-7919-4_{1}5
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
Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368, DOI 10.1007/978-1-4612-0851-8
William Stein, Modular forms, a computational approach, Graduate Studies in Mathematics, vol. 79, American Mathematical Society, Providence, RI, 2007. With an appendix by Paul E. Gunnells. MR 2289048, DOI 10.1090/gsm/079
William Arthur Stein, Explicit approaches to modular abelian varieties, ProQuest LLC, Ann Arbor, MI, 2000. Thesis (Ph.D.)–University of California, Berkeley. MR 2701042
Michael Stoll, On the height constant for curves of genus two. II, Acta Arith. 104 (2002), no. 2, 165–182. MR 1914251, DOI 10.4064/aa104-2-6
Hwajong Yoo, The index of an Eisenstein ideal and multiplicity one, Math. Z. 282 (2016), no. 3-4, 1097–1116. MR 3473658, DOI 10.1007/s00209-015-1579-4