Optimal curves differing by a 5-isogeny

• Autorzy: Dongho Byeon , Taekyung Kim
• Języki publikacji: EN

Abstrakty

EN

For i = 0,1, let $E_i$ be the $X_i(N)$-optimal curve of an isogeny class 𝓒 of elliptic curves defined over ℚ of conductor N. Stein and Watkins conjectured that E₀ and E₁ differ by a 5-isogeny if and only if E₀ = X₀(11) and E₁ = X₁(11). In this paper, we show that this conjecture is true if N is square-free and is not divisible by 5. On the other hand, Hadano conjectured that for an elliptic curve E defined over ℚ with a rational point P of order 5, the 5-isogenous curve E' := E/⟨P⟩ has a rational point of order 5 again if and only if E' = X₀(11) and E = X₁(11). In the process of the proof of Stein and Watkins's conjecture, we show that Hadano's conjecture is not true.

Kategorie tematyczne

• 14K02: Isogeny
• 11G05: Elliptic curves over global fields

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 165
• Numer: 4
• Strony: 351-359

Daty

wydano

2014

Twórcy

autor

Dongho Byeon

• Department of Mathematics, Seoul National University, Seoul, Korea

autor

Taekyung Kim

• Department of Mathematics, Seoul National University, Seoul, Korea

Identyfikatory

DOI

10.4064/aa165-4-5