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arXiv · 2103.08947

The rank of a CM elliptic curve and a recurrence formula

Keiichiro Nomoto
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Abstract

Let $p$ be a prime number and $E_{p}$ denote the elliptic curve $y^2=x^3+px$. It is known that for $p$ which is congruent to $1, 9$ modulo $16$, the rank of $E_{p}$ over $\mathbb{Q}$ is equal to $0, 2$. Under the condition that the Birch and Swinnerton-Dyer conjecture is true, we give a necessary and sufficient condition that the rank is $2$ in terms of the constant term of some polynomial that is defined by a recurrence formula.

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Keiichiro Nomoto. 2021-03-16. The rank of a CM elliptic curve and a recurrence formula. https://arxiv.org/abs/2103.08947

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