Towards a classification of isolated 𝑗-invariants

Abbey Bourdon; Sachi Hashimoto; Timo Keller; Zev Klagsbrun; David Lowry-Duda; Travis Morrison; Filip Najman; Himanshu Shukla

doi:10.1090/mcom/3956

journal article Apr 25, 2024

Towards a classification of isolated 𝑗-invariants

Abbey Bourdon

Sachi Hashimoto Timo Keller Zev Klagsbrun David Lowry-Duda Travis Morrison Filip Najman Himanshu Shukla

Mathematics of Computation Vol. 94 No. 351 pp. 447-473 · American Mathematical Society (AMS)

View at Publisher Save 10.1090/mcom/3956

Abstract

We develop an algorithm to test whether a non-complex multiplication elliptic curve

E

/

Q

E/\mathbf {Q}

gives rise to an isolated point of any degree on any modular curve of the form

X
1

(
N
)

X_1(N)

. This builds on prior work of Zywina which gives a method for computing the image of the adelic Galois representation associated to

E
E

. Running this algorithm on all elliptic curves presently in the

L
L

-functions and Modular Forms Database and the Stein–Watkins Database gives strong evidence for the conjecture that

E
E

gives rise to an isolated point on

X
1

(
N
)

X_1(N)

if and only if

j
(
E
)
=
−
140625

/

8
,
−
9317

j(E)=-140625/8, -9317

,

351

/

4

351/4

, or

−
162677523113838677

-162677523113838677

.

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References

37

[1]

[Avi] J. B. Avila, Galois representations of elliptic curves and abelian entanglements, Doctoral Thesis, Leiden University, 2015. ↑

[2]

The Magma Algebra System I: The User Language

WIEB BOSMA, JOHN CANNON, CATHERINE PLAYOUST

Journal of Symbolic Computation 1997 10.1006/jsco.1996.0125

[3]

Bourdon, Abbey "On the level of modular curves that give rise to isolated 𝑗-invariants" Adv. Math. (2019) 10.1016/j.aim.2019.106824

[4]

Odd degree isolated points on $$X_1(N)$$ with rational j-invariant

Abbey Bourdon, David R. Gill, Jeremy Rouse et al.

Research in Number Theory 2024 10.1007/s40993-023-00488-0

[5]

[BM] J. S. Balakrishnan and B. Mazur, Ogg’s torsion conjecture: fifty years later, arXiv:2307.04752. ↑

[6]

[BN] A. Bourdon and F. Najman, Sporadic points of odd degree on 𝑋₁(𝑁) coming from ℚ-curves, Preprint, arXiv:2107.10909. ↑

[7]

Bilu, Yuri "Serre’s uniformity problem in the split Cartan case" Ann. of Math. (2) (2011) 10.4007/annals.2011.173.1.13

[8]

Clark, Pete L. "The least degree of a CM point on a modular curve" J. Lond. Math. Soc. (2) (2022) 10.1112/jlms.12518

[9]

Derickx, Maarten "Sporadic cubic torsion" Algebra Number Theory (2021) 10.2140/ant.2021.15.1837

[10]

Derickx, Maarten "Torsion points on elliptic curves over number fields of small degree" Algebra Number Theory (2023) 10.2140/ant.2023.17.267

[11]

Derickx, Maarten "Rational families of 17-torsion points of elliptic curves over number fields" 10.1090/conm/701/14142

[12]

Deligne, P. "Les schémas de modules de courbes elliptiques" (1973)

[13]

Duke, William "Elliptic curves with no exceptional primes" C. R. Acad. Sci. Paris S\'{e}r. I Math. (1997) 10.1016/s0764-4442(97)80118-8

[14]

Derickx, Maarten "Gonality of the modular curve 𝑋₁(𝑁)" J. Algebra (2014) 10.1016/j.jalgebra.2014.06.026

[15]

Isolated points on $$X_1(\ell ^n)$$ with rational j-invariant

Özlem Ejder

Research in Number Theory 2022 10.1007/s40993-022-00313-0

[16]

Faltings, G. "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" Invent. Math. (1983) 10.1007/bf01388432

[17]

Faltings, Gerd "The general case of S. Lang’s conjecture" (1994)

[18]

Frey, Gerhard "Curves with infinitely many points of fixed degree" Israel J. Math. (1994) 10.1007/bf02758637

[19]

Fulton, William (1998) 10.1007/978-1-4612-1700-8

[20]

Greicius, Aaron "Elliptic curves with surjective adelic Galois representations" Experiment. Math. (2010) 10.1080/10586458.2010.10390639

[21]

Liu, Qing (2002)

[22]

[{LMF}23] The LMFDB Collaboration, The L-functions and modular forms database, \url{https://www.lmfdb.org}, 2023 [Online; accessed August 2023]. ↑

[23]

Lozano-Robledo, Álvaro "On the field of definition of 𝑝-torsion points on elliptic curves over the rationals" Math. Ann. (2013) 10.1007/s00208-013-0906-5

[24]

Mazur, B. "Rational isogenies of prime degree (with an appendix by D. Goldfeld)" Invent. Math. (1978) 10.1007/bf01390348

[25]

Milne, J. S. "Abelian varieties" (1986)

[26]

Milne, J. S. "Jacobian varieties" (1986)

[27]

Najman, Filip "Torsion of rational elliptic curves over cubic fields and sporadic points on 𝑋₁(𝑛)" Math. Res. Lett. (2016) 10.4310/mrl.2016.v23.n1.a12

[28]

Rouse, Jeremy "ℓ-adic images of Galois for elliptic curves over ℚ (and an appendix with John Voight)" Forum Math. Sigma (2022) 10.1017/fms.2022.38

[29]

Propri�t�s galoisiennes des points d'ordre fini des courbes elliptiques

Jean-Pierre Serre

Inventiones mathematicae 1972 10.1007/bf01405086

[30]

Serre, Jean-Pierre (1997) 10.1007/978-3-663-10632-6

[31]

Shimura, Goro (1971)

[32]

Sutherland, Andrew V. "Computing images of Galois representations attached to elliptic curves" Forum Math. Sigma (2016) 10.1017/fms.2015.33

[33]

Stein, William A. "A database of elliptic curves—first report" (2002) 10.1007/3-540-45455-1_22

[34]

[vH] M. van Hoeij, Low degree places on the modular curve 𝑋₁(𝑛), Preprint, arXiv:1202.4355. ↑

[35]

[Zywa] D. Zywina, Explicit open images for elliptic curves over ℚ, Preprint, arXiv:2206.14959. ↑

[36]

[Zywb] D. Zywina, Explicit open images for elliptic curves over ℚ, Github repository, \url{https://github.com/davidzywina/OpenImage}. ↑

[37]

[Zywc] D. Zywina, On the possible image of the mod ℓ representations associated to elliptic curves over ℚ, arXiv:1508.07660. ↑

Metrics

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Citations

37

References

Details

Published: Apr 25, 2024
Vol/Issue: 94(351)
Pages: 447-473
License: View

Authors

Funding

National Science Foundation Award: DMS-2145270

Deutsche Forschungsgemeinschaft Award: DMS-2145270

Simons Foundation Award: DMS-2145270

Hrvatska zaklada za znanost Award: DMS-2145270

Cite This Article

Abbey Bourdon, Sachi Hashimoto, Timo Keller, et al. (2024). Towards a classification of isolated 𝑗-invariants. Mathematics of Computation, 94(351), 447-473. https://doi.org/10.1090/mcom/3956

Towards a classification of isolated 𝑗-invariants

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