On a probabilistic local-global principle for torsion on elliptic curves

John Cullinan; Meagan Kenney; John Voight

doi:10.5802/jtnb.1193

John Cullinan ¹; Meagan Kenney ²; John Voight ³

¹ Department of Mathematics Bard College Annandale-On-Hudson, NY 12504, USA
² Department of Mathematics University of Minnesota Minneapolis, MN 55455, USA
³ Department of Mathematics Dartmouth College 6188 Kemeny Hall, Hanover, NH 03755, USA

Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 41-90.

Abstract
Résumé

Let $m$ be a positive integer and let $E$ be an elliptic curve over $ℚ$ with the property that $m ∣ # E (F_{p})$ for a density $1$ set of primes $p$ . Building upon work of Katz and Harron–Snowden, we study the probability that $m ∣ # E {(ℚ)}_{t o r}$ : we find it is nonzero for all $m \in {1, 2, \dots, 10} \cup {12, 16}$ and we compute it exactly when $m \in {1, 2, 3, 4, 5, 7}$ . As a supplement, we give an asymptotic count of elliptic curves with extra level structure when the parametrizing modular curve arises from the quotient by a torsion-free group of genus zero.

Soit $m$ un entier positif et soit $E$ une courbe elliptique sur $ℚ$ telle que $m ∣ # E (F_{p})$ pour un ensemble de densité $1$ de nombres premiers $p$ . En nous appuyant sur les travaux de Katz et Harron–Snowden, nous étudions la probabilité que $m ∣ # E {(ℚ)}_{t o r}$ : nous trouvons qu’elle est non nulle pour tout $m \in {1, 2, \dots, 10} \cup {12, 16}$ et nous la calculons explicitement pour $m \in {1, 2, 3, 4, 5, 7}$ . En complément, nous donnons un comptage asymptotique des courbes elliptiques avec une structure de niveau supplémentaire lorsque la courbe modulaire paramétrant ces structures provient du quotient par un groupe sans torsion de genre zéro.

Received: 2020-05-15
Revised: 2021-07-17
Accepted: 2021-09-20
Published online: 2022-07-07

DOI: 10.5802/jtnb.1193

Classification: 11G05, 14H52
Keywords: Elliptic curves, torsion subgroups, arithmetic statistics

Author's affiliations:

John Cullinan ¹; Meagan Kenney ²; John Voight ³

¹ Department of Mathematics Bard College Annandale-On-Hudson, NY 12504, USA
² Department of Mathematics University of Minnesota Minneapolis, MN 55455, USA
³ Department of Mathematics Dartmouth College 6188 Kemeny Hall, Hanover, NH 03755, USA

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{JTNB_2022__34_1_41_0,
     author = {John Cullinan and Meagan Kenney and John Voight},
     title = {On a probabilistic local-global principle for torsion on elliptic curves},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {41--90},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {34},
     number = {1},
     year = {2022},
     doi = {10.5802/jtnb.1193},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1193/}
}

TY  - JOUR
AU  - John Cullinan
AU  - Meagan Kenney
AU  - John Voight
TI  - On a probabilistic local-global principle for torsion on elliptic curves
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2022
SP  - 41
EP  - 90
VL  - 34
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1193/
DO  - 10.5802/jtnb.1193
LA  - en
ID  - JTNB_2022__34_1_41_0
ER  -

%0 Journal Article
%A John Cullinan
%A Meagan Kenney
%A John Voight
%T On a probabilistic local-global principle for torsion on elliptic curves
%J Journal de théorie des nombres de Bordeaux
%D 2022
%P 41-90
%V 34
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1193/
%R 10.5802/jtnb.1193
%G en
%F JTNB_2022__34_1_41_0

John Cullinan; Meagan Kenney; John Voight. On a probabilistic local-global principle for torsion on elliptic curves. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 41-90. doi : 10.5802/jtnb.1193. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1193/

References
Cited by

[1] Eran Assaf Computing classical modular forms for arbitrary congruence subgroups, Arithmetic Geometry, Number Theory, and Computation (Simons Symposia), 2021, pp. 43-104 | DOI

[2] Burcu Baran Normalizers of non-split Cartan subgroups, modular curves, and the class number one problem, J. Number Theory, Volume 130 (2010) no. 12, pp. 2753-2772 | DOI | MR | Zbl

[3] Manjul Bhargava; Arul Shankar Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves, Ann. Math., Volume 181 (2015) no. 1, pp. 191-242 | DOI | MR | Zbl

[4] Brandon Boggess; Soumya Sankar Counting elliptic curves with a rational $N$ -isogeny for small $N$ (2020) (https://arxiv.org/abs/2009.05223)

[5] Wieb Bosma; John Cannon; Catherine Playoust The Magma algebra system. I. The user language, J. Symb. Comput., Volume 24 (1997) no. 3-4, pp. 235-265 | DOI | MR | Zbl

[6] Peter Bruin; Filip Najman Counting elliptic curves with prescribed level structures over number fields (2021) (https://arxiv.org/abs/2008.05280)

[7] Garen Chiloyan; Álvaro Lozano-Robledo A classification of isogeny-torsion graphs of elliptic curves over $ℚ$ , Trans. Lond. Math. Soc., Volume 8 (2021) no. 1, pp. 1-34 | DOI

[8] Peter J. Cho; Keunyoung Jeong Probabilistic behaviors of elliptic curves with torsion points (2020) (https://arxiv.org/abs/2005.06862)

[9] David A. Cox; John Little; Donal O’Shea Using algebraic geometry, Graduate Texts in Mathematics, 185, Springer, 2005

[10] John Cullinan; John Voight Universal polynomials for $m$ -full torsion groups, 2020 (available at http://math.dartmouth.edu/~jvoight/code/compute_universal.m)

[11] Harold Davenport On a principle of Lipschitz, J. Lond. Math. Soc., Volume 26 (1951), pp. 179-183 | DOI | MR | Zbl

[12] Pierre Deligne; Michael 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 Mathematics), Volume 349, Springer, 1972, pp. 143-316 | Zbl

[13] Fred Diamond; Jerry Shurman A first course in modular forms, Graduate Texts in Mathematics, 228, Springer, 2005

[14] Jordan S. Ellenberg; Matthew Satriano; David Zureick-Brown Heights on stacks and a generalized Batyrev–Manin–Malle conjecture (2021) (https://arxiv.org/abs/2106.11340v1)

[15] Irene García-Selfa; José M. Tornero A complete Diophantine characterization of the rational torsion of an elliptic curve, Acta Math. Sin., Engl. Ser., Volume 28 (2012) no. 1, pp. 83-96 | DOI | MR | Zbl

[16] Ralph Greenberg The image of Galois representations attached to elliptic curves with an isogeny, Am. J. Math., Volume 134 (2012) no. 5, pp. 1167-1196 | DOI | MR | Zbl

[17] Robert Harron; Andrew Snowden Counting elliptic curves with prescribed torsion, J. Reine Angew. Math., Volume 729 (2017), pp. 151-170 | MR | Zbl

[18] Martin N. Huxley Exponential sums and lattice points III, Proc. Lond. Math. Soc., Volume 87 (2003) no. 3, pp. 591-609 | DOI | MR | Zbl

[19] Nicholas M. Katz Galois properties of torsion points on abelian varieties, Invent. Math., Volume 62 (1981), pp. 481-502 | DOI | MR | Zbl

[20] Nicholas M. Katz; Barry Mazur Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, 108, Princeton University Press, 1985 | DOI

[21] Irwin Kra On lifting Kleinian groups to $SL (2, ℂ)$ , Differential geometry and complex analysis, Springer, 1985, pp. 181-193 | Zbl

[22] The LMFDB Collaboration The L-functions and Modular Forms Database, 2020 (http://www.lmfdb.org, accessed 28 June 2020)

[23] Barry Mazur Modular curves and the Eisenstein ideal, Publ. Math., Inst. Hautes Étud. Sci., Volume 47 (1977), pp. 33-186 | DOI | Numdam | Zbl

[24] Maggie Pizzo; Carl Pomerance; John Voight Counting elliptic curves with an isogeny of degree three, Proc. Am. Math. Soc., Volume 7 (2020), pp. 28-42 | DOI | MR | Zbl

[25] Carl Pomerance; Edward F. Schaefer Elliptic curves with Galois-stable cyclic subgroups of order 4, Res. Number Theory, Volume 7 (2021) no. 2, 35, 19 pages | MR | Zbl

[26] Abdellah Sebbar Classification of torsion-free genus zero congruence groups, Proc. Am. Math. Soc., Volume 129 (2001) no. 9, pp. 2517-2527 | DOI | MR | Zbl

[27] Jean-Pierre Serre Lectures on the Mordell-Weil theorem, Aspects of Mathematics, E15, Vieweg & Sohn, 1997 | DOI

[28] Jean-Pierre Serre Abelian $ℓ$ -adic representations and elliptic curves, Research Notes in Mathematics, 7, A K Peters, 1998

[29] Goro Shimura Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, 11, Princeton University Press, 1994

[30] Andrew V. therland; David Zywina Modular curves of prime-power level with infinitely many rational points, Algebra Number Theory, Volume 11 (2017) no. 5, pp. 1199-1229 | MR | Zbl

[31] Jacques Vélu Isogénies entre courbes elliptiques, C. R. Acad. Sci. Paris, Volume 273 (1971) | Zbl

[32] John Voight; David Zureick-Brown The canonical ring of a stacky curve (to appear in Mem. Amer. Math. Soc.)

[33] David Zywina Possible indices for the Galois image of elliptic curves over $ℚ$ (2015) (https://arxiv.org/abs/1508.07663)

Cited by Sources: