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May 29, 2026Journal of the London Mathematical Society0 citationsOpen Access

Counting 5‐isogenies of elliptic curves over Q

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SASantiago Arango‐PiñerosCHChangho HanOPOana Padurariu

Key Points

  • This research aims to determine the number of 5-isogenies of elliptic curves with specific height restrictions.
  • Utilized an explicit isomorphism between the stack of elliptic curves and generalized Fermat equations.
  • Analyzed height bounds for elliptic curves over Q.
  • Computed constants to demonstrate asymptotic behavior.
  • Established that the number of 5-isogenies is asymptotic to a computable constant.
  • Settled the asymptotic count of rational points on genus zero modular curves.

Abstract

Abstract We show that the number of 5‐isogenies of elliptic curves defined over with naive height bounded by is asymptotic to for some explicitly computable constant . This settles the asymptotic count of rational points on the genus zero modular curves as moduli stacks. We leverage an explicit ‐isomorphism between the stack and the generalized Fermat equation with ‐action of weights .

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Cite This Study

Arango‐Piñeros et al. (2026) studied this question.

synapsesocial.com/papers/6a192e95fab5b468c4417b86https://doi.org/10.1112/jlms.70572
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