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Torsion points on CM elliptic curves over real number fields

Authors: Abbey Bourdon, Pete L. Clark and James Stankewicz
Journal: Trans. Amer. Math. Soc. 369 (2017), 8457-8496
MSC (2010): Primary 11G05; Secondary 11G15
Published electronically: May 1, 2017
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Abstract: We study torsion subgroups of elliptic curves with complex multiplication (CM) defined over number fields which admit a real embedding. We give a complete classification of the groups which arise up to isomorphism as the torsion subgroup of a CM elliptic curve defined over a number field of odd degree: there are infinitely many. However, if we fix an odd integer $ d$ and consider number fields of degree $ dp$ as $ p$ ranges over all prime numbers, all but finitely many torsion subgroups that appear for CM elliptic curves actually occur in a degree dividing $ d$. This implies an absolute bound on the size of torsion subgroups of CM elliptic curves defined over number fields of degree $ dp$. In the case where $ d=1$, there are six ``Olson groups'' which arise as torsion subgroups of CM elliptic curves over $ \mathbb{Q}$, and there are precisely $ 17$ ``non-Olson'' CM elliptic curves defined over a number field of (variable) prime degree.

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  • [Ao95] Noboru Aoki, Torsion points on abelian varieties with complex multiplication, Algebraic cycles and related topics (Kitasakado, 1994) World Sci. Publ., River Edge, NJ, 1995, pp. 1-22. MR 1414432
  • [Ao06] Noboru Aoki, Torsion points on CM abelian varieties, Comment. Math. Univ. St. Pauli 55 (2006), no. 2, 207-229. MR 2294929
  • [CCRS13] Pete L. Clark, Brian Cook, and James Stankewicz, Torsion points on elliptic curves with complex multiplication (with an appendix by Alex Rice), Int. J. Number Theory 9 (2013), no. 2, 447-479. MR 3005559,
  • [CCRS14] Pete L. Clark, Patrick Corn, Alex Rice, and James Stankewicz, Computation on elliptic curves with complex multiplication, LMS J. Comput. Math. 17 (2014), no. 1, 509-535. MR 3356044,
  • [Cl04] P.L. Clark, Bounds for torsion on abelian varieties with integral moduli, $ 2004$ preprint.
  • [Cl07] Pete L. Clark, On the indices of curves over local fields, Manuscripta Math. 124 (2007), no. 4, 411-426. MR 2357791,
  • [Co89] David A. Cox, Primes of the form $ x^2 + ny^2$: Fermat, class field theory and complex multiplication, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989. MR 1028322
  • [Co00] Henri Cohen, Advanced topics in computational number theory, Graduate Texts in Mathematics, vol. 193, Springer-Verlag, New York, 2000. MR 1728313
  • [CX08] Pete L. Clark and Xavier Xarles, Local bounds for torsion points on abelian varieties, Canad. J. Math. 60 (2008), no. 3, 532-555. MR 2414956,
  • [DR73] 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
  • [Ei95] David Eisenbud, Commutative algebra: With a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. MR 1322960
  • [GM91] F. Gouvêa and B. Mazur, The square-free sieve and the rank of elliptic curves, J. Amer. Math. Soc. 4 (1991), no. 1, 1-23. MR 1080648,
  • [HK13] Franz Halter-Koch, Quadratic irrationals: An introduction to classical number theory, Pure and Applied Mathematics (Boca Raton), CRC Press, Boca Raton, FL, 2013. MR 3086176
  • [HS99] Marc Hindry and Joseph Silverman, Sur le nombre de points de torsion rationnels sur une courbe elliptique, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 2, 97-100 (French, with English and French summaries). MR 1710502,
  • [Ja89] Nathan Jacobson, Basic algebra. II, 2nd ed., W. H. Freeman and Company, New York, 1989. MR 1009787
  • [Ku76] Daniel Sion Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. (3) 33 (1976), no. 2, 193-237. MR 0434947
  • [Kw99] Soonhak Kwon, Degree of isogenies of elliptic curves with complex multiplication, J. Korean Math. Soc. 36 (1999), no. 5, 945-958. MR 1724020
  • [La87] Serge Lang, Elliptic functions, 2nd ed., Graduate Texts in Mathematics, vol. 112, Springer-Verlag, New York, 1987. With an appendix by J. Tate. MR 890960
  • [LP92] H. W. Lenstra Jr. and Carl Pomerance, A rigorous time bound for factoring integers, J. Amer. Math. Soc. 5 (1992), no. 3, 483-516. MR 1137100,
  • [LR13] Álvaro Lozano-Robledo, On the field of definition of $ p$-torsion points on elliptic curves over the rationals, Math. Ann. 357 (2013), no. 1, 279-305. MR 3084348,
  • [Ma87] D. W. Masser, Small values of heights on families of abelian varieties, Diophantine approximation and transcendence theory (Bonn, 1985) Lecture Notes in Math., vol. 1290, Springer, Berlin, 1987, pp. 109-148. MR 927559,
  • [Mi86] J. S. Milne, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 103-150. MR 861974
  • [MR10] B. Mazur and K. Rubin, Ranks of twists of elliptic curves and Hilbert's tenth problem, Invent. Math. 181 (2010), no. 3, 541-575. MR 2660452,
  • [Me96] 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,
  • [Na13] Filip Najman, The number of twists with large torsion of an elliptic curve, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 109 (2015), no. 2, 535-547. MR 3383431,
  • [Ol74] Loren D. Olson, Points of finite order on elliptic curves with complex multiplication, Manuscripta Math. 14 (1974), 195-205. MR 0352104
  • [Pa89] James L. Parish, Rational torsion in complex-multiplication elliptic curves, J. Number Theory 33 (1989), no. 2, 257-265. MR 1034205,
  • [Pi82] Richard S. Pierce, Associative algebras, Studies in the History of Modern Science, 9, Graduate Texts in Mathematics, vol. 88, Springer-Verlag, New York-Berlin, 1982. MR 674652
  • [Po09] Paul Pollack, Not always buried deep: A second course in elementary number theory, American Mathematical Society, Providence, RI, 2009. MR 2555430
  • [PY01] Dipendra Prasad and C. S. Yogananda, Bounding the torsion in CM elliptic curves, C. R. Math. Acad. Sci. Soc. R. Can. 23 (2001), no. 1, 1-5 (English, with French summary). MR 1816457
  • [QZ01] Derong Qiu and Xianke Zhang, Elliptic curves and their torsion subgroups over number fields of type $ (2,2,\dots ,2)$, Sci. China Ser. A 44 (2001), no. 2, 159-167. MR 1824316,
  • [Se66] Jean-Pierre Serre, Groupes de Lie $ l$-adiques attachés aux courbes elliptiques, Les Tendances Géom. en Algébre et Théorie des Nombres, Éditions du Centre National de la Recherche Scientifique, Paris, 1966, pp. 239-256 (French). MR 0218366
  • [Se67] J.-P. Serre, Complex multiplication, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) Thompson, Washington, D.C., 1967, pp. 292-296. MR 0244199
  • [Sh94] Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Kanô Memorial Lectures, 1, Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original. MR 1291394
  • [Si86] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210
  • [Si88] Alice Silverberg, Torsion points on abelian varieties of CM-type, Compositio Math. 68 (1988), no. 3, 241-249. MR 971328
  • [Si92] A. Silverberg, Points of finite order on abelian varieties, $ p$-adic Methods in Number Theory and Algebraic Geometry, Contemp. Math., vol. 133, Amer. Math. Soc., Providence, RI, 1992, pp. 175-193. MR 1183978,
  • [Si94] Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368
  • [SS58] A. Schinzel and W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arith. 4 (1958), 185-208; erratum 5 (1958), 259 (French). MR 0106202
  • [ST68] Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492-517. MR 0236190

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Additional Information

Abbey Bourdon
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30605

Pete L. Clark
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30605

James Stankewicz
Affiliation: Heilbronn Institute for Mathematical Research, University of Bristol, Bristol BS8 1TW, United Kingdom

Received by editor(s): March 24, 2015
Received by editor(s) in revised form: October 4, 2015, and January 21, 2016
Published electronically: May 1, 2017
Additional Notes: The first author was supported in part by NSF grant DMS-1344994 (RTG in Algebra, Algebraic Geometry, and Number Theory, at the University of Georgia). The third author was supported by the Villum Fonden through the network for Experimental Mathematics in Number Theory, Operator Algebras, and Topology.
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