Constant Tamagawa numbers of special elliptic curves

2021/06/01 by Li, Luying, Lv, Chang
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2106.00340

Abstract

For the elliptic curves E_σ2D : y² = x³ + σ2Dx , which has 2-isogeny curve E'_σ2D : y² = x³ -σ8Dx, σ= ± 1, D = p₁^e₁p₂^e₂⋯ p_n^e_n, where p_i are different odd prime numbers and e_i = 1 or 3, we demonstrate that Tamagawa numbers of these elliptic curves are always one or zero by the use of matrix in finite field \mathbb F₂. The specific number depends on the value of σ. By our proofs of these results, we find a method to quickly sieve a part of the elliptic curves with Mordell-Weil rank zero or rank one in this form as an application.

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