Papers › Counting 5-isogenies of elliptic curves over ℚ
Counting 5-isogenies of elliptic curves over ℚ
Santiago Arango-Piñeros, Changho Han, Oana Padurariu, Sun Woo Park
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We show that the number of 5-isogenies of elliptic curves defined over ℚ with naive height bounded by H > 0 is asymptotic to C₅·H^(1/6) (logH)² for some explicitly computable constant C₅ > 0. This settles the asymptotic count of rational points on the genus zero modular curves X₀(m). We leverage an explicit ℚ-isomorphism between the stack 𝒳₀(5) and the generalized Fermat equation x² + y² = z⁴ with 𝔾ₘ-action of weights (4, 4, 2).
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