# A local-global principle for rational isogenies of prime degree

• [1] Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139
• Volume: 24, Issue: 2, page 475-485
• ISSN: 1246-7405

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## Abstract

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Let $K$ be a number field. We consider a local-global principle for elliptic curves $E/K$ that admit (or do not admit) a rational isogeny of prime degree $\ell$. For suitable $K$ (including $K=ℚ$), we prove that this principle holds for all $\ell \equiv 1\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4$, and for $\ell <7$, but find a counterexample when $\ell =7$ for an elliptic curve with $j$-invariant $2268945/128$. For $K=ℚ$ we show that, up to isomorphism, this is the only counterexample.

## How to cite

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Sutherland, Andrew V.. "A local-global principle for rational isogenies of prime degree." Journal de Théorie des Nombres de Bordeaux 24.2 (2012): 475-485. <http://eudml.org/doc/251084>.

@article{Sutherland2012,
abstract = {Let $K$ be a number field. We consider a local-global principle for elliptic curves $E/K$ that admit (or do not admit) a rational isogeny of prime degree $\ell$. For suitable $K$ (including $K=\mathbb\{Q\}$), we prove that this principle holds for all $\ell \equiv 1\;\@mod \;4$, and for $\ell &lt; 7$, but find a counterexample when $\ell =7$ for an elliptic curve with $j$-invariant $2268945/128$. For $K=\mathbb\{Q\}$ we show that, up to isomorphism, this is the only counterexample.},
affiliation = {Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139},
author = {Sutherland, Andrew V.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {elliptic curve; isogeny; local-global principle; elliptic curves},
language = {eng},
month = {6},
number = {2},
pages = {475-485},
publisher = {Société Arithmétique de Bordeaux},
title = {A local-global principle for rational isogenies of prime degree},
url = {http://eudml.org/doc/251084},
volume = {24},
year = {2012},
}

TY - JOUR
AU - Sutherland, Andrew V.
TI - A local-global principle for rational isogenies of prime degree
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2012/6//
PB - Société Arithmétique de Bordeaux
VL - 24
IS - 2
SP - 475
EP - 485
AB - Let $K$ be a number field. We consider a local-global principle for elliptic curves $E/K$ that admit (or do not admit) a rational isogeny of prime degree $\ell$. For suitable $K$ (including $K=\mathbb{Q}$), we prove that this principle holds for all $\ell \equiv 1\;\@mod \;4$, and for $\ell &lt; 7$, but find a counterexample when $\ell =7$ for an elliptic curve with $j$-invariant $2268945/128$. For $K=\mathbb{Q}$ we show that, up to isomorphism, this is the only counterexample.
LA - eng
KW - elliptic curve; isogeny; local-global principle; elliptic curves
UR - http://eudml.org/doc/251084
ER -

## References

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8. Nicholas M. Katz, Galois properties of torsion points on abelian varieties. Inventiones Mathematicae 62 (1981), no. 3, 481–502. Zbl0471.14023MR604840
9. Serge Lang, Introduction to modular forms, Springer, 1976. Zbl0344.10011MR429740
10. —, Elliptic functions, second ed., Springer-Verlag, 1987. MR890960
11. Barry Mazur, An introduction to the deformation theory of Galois representations. Modular forms and Fermat’s last theorem, Springer, 1997. Zbl0901.11015MR1638481
12. Pierre J. R. Parent, Towards the triviality of ${X}_{0}^{+}\left({p}^{r}\right)\left(ℚ\right)$ for $r>1$. Compositio Mathematica 141 (2005), 561–572. Zbl1167.11310MR2135276
13. René Schoof, Counting points on elliptic curves over finite fields. Journal de Théorie des Nombres de Bordeaux 7 (1995), 219–254. Zbl0852.11073MR1413578
14. Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Inventiones Mathematicae 15 (1972), no. 2, 259–331. Zbl0235.14012MR387283

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