Torsion points on CM elliptic curves over real number fields

Bourdon, Abbey; Clark, Pete; Stankewicz, James

doi:10.1090/tran/6905

Torsion points on CM elliptic curves over real number fields
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by Abbey Bourdon, Pete L. Clark and James Stankewicz PDF

Trans. Amer. Math. Soc. 369 (2017), 8457-8496 Request permission

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.

References

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
Noboru Aoki, Torsion points on CM abelian varieties, Comment. Math. Univ. St. Pauli 55 (2006), no. 2, 207–229. MR 2294929
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, DOI 10.1142/S1793042112501436
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, DOI 10.1112/S1461157014000072
P.L. Clark, Bounds for torsion on abelian varieties with integral moduli, $2004$ preprint.
Pete L. Clark, On the indices of curves over local fields, Manuscripta Math. 124 (2007), no. 4, 411–426. MR 2357791, DOI 10.1007/s00229-007-0126-y
David A. Cox, Primes of the form $x^2 + ny^2$, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989. Fermat, class field theory and complex multiplication. MR 1028322
Henri Cohen, Advanced topics in computational number theory, Graduate Texts in Mathematics, vol. 193, Springer-Verlag, New York, 2000. MR 1728313, DOI 10.1007/978-1-4419-8489-0
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, DOI 10.4153/CJM-2008-026-x
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
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
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, DOI 10.1090/S0894-0347-1991-1080648-7
Franz Halter-Koch, Quadratic irrationals, Pure and Applied Mathematics (Boca Raton), CRC Press, Boca Raton, FL, 2013. An introduction to classical number theory. MR 3086176, DOI 10.1201/b14968
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, DOI 10.1016/S0764-4442(99)80469-8
Nathan Jacobson, Basic algebra. II, 2nd ed., W. H. Freeman and Company, New York, 1989. MR 1009787
Daniel Sion Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. (3) 33 (1976), no. 2, 193–237. MR 434947, DOI 10.1112/plms/s3-33.2.193
Soonhak Kwon, Degree of isogenies of elliptic curves with complex multiplication, J. Korean Math. Soc. 36 (1999), no. 5, 945–958. MR 1724020
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, DOI 10.1007/978-1-4612-4752-4
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, DOI 10.1090/S0894-0347-1992-1137100-0
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, DOI 10.1007/s00208-013-0906-5
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, DOI 10.1007/BFb0078706
J. S. Milne, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 103–150. MR 861974
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, DOI 10.1007/s00222-010-0252-0
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
Filip Najman, The number of twists with large torsion of an elliptic curve, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 109 (2015), no. 2, 535–547. MR 3383431, DOI 10.1007/s13398-014-0199-x
Loren D. Olson, Points of finite order on elliptic curves with complex multiplication, Manuscripta Math. 14 (1974), 195–205. MR 352104, DOI 10.1007/BF01171442
James L. Parish, Rational torsion in complex-multiplication elliptic curves, J. Number Theory 33 (1989), no. 2, 257–265. MR 1034205, DOI 10.1016/0022-314X(89)90012-7
Richard S. Pierce, Associative algebras, Studies in the History of Modern Science, vol. 9, Springer-Verlag, New York-Berlin, 1982. MR 674652
Paul Pollack, Not always buried deep, American Mathematical Society, Providence, RI, 2009. A second course in elementary number theory. MR 2555430, DOI 10.1090/mbk/068
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
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, DOI 10.1007/BF02874418
Jean-Pierre Serre, Groupes de Lie $l$-adiques attachés aux courbes elliptiques, Les Tendances Géom. en Algèbre et Théorie des Nombres, Éditions du Centre National de la Recherche Scientifique (CNRS), Paris, 1966, pp. 239–256 (French). MR 0218366
J.-P. Serre, Complex multiplication, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) Thompson, Washington, D.C., 1967, pp. 292–296. MR 0244199
Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original; Kanô Memorial Lectures, 1. MR 1291394
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
Alice Silverberg, Torsion points on abelian varieties of CM-type, Compositio Math. 68 (1988), no. 3, 241–249. MR 971328
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, DOI 10.1090/conm/133/1183978
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
A. Schinzel and W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arith. 4 (1958), 185–208; erratum 5 (1958), 259 (French). MR 106202, DOI 10.4064/aa-4-3-185-208
Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. MR 236190, DOI 10.2307/1970722