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I was told the following "Theorem": Let $y^{2} =x^{3} + Ax^{2} +Bx$ be a nonsingular cubic curve with $A,B \in \mathbb{Z}$. Then the rank $r$ of this curve satisfies

$r \leq \nu (A^{2} -4B) +\nu(B) -1$

where $\nu(n)$ is the number of distinct positive prime divisors of $n$.

I can not find a name for this theorem or a reference, and I am wondering if it is a well known result, or if it is even true. Has anyone seen this result or have a suggestion on where I can find a reference. Thank you.

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@Numth: I think he means what he said... – Pete L. Clark Apr 17 '11 at 0:54
Thank you for all the help. Numth: I followed Robledo reference to Silverman's text, which matched up nicely with Pete L.Clark's answer. – pel Apr 17 '11 at 1:59
@Pete You are correct, I withdraw my objection your honor. – Graphth Apr 17 '11 at 2:32
@Numth: no biggie. – Pete L. Clark Apr 17 '11 at 5:23
up vote 7 down vote accepted

A rational elliptic curve $E_{/\mathbb{Q}}$ can be put in the form you gave if and only if it has a rational point of order $2$: in the given equation, $(0,0)$ has order $2$, and in general any point of order $2$ can be "moved" to $(0,0)$ by a change of variables.

Therefore you are in the general situation of "descent by $2$-isogeny". This is covered, for instance (but especially well) in $\S X.4$ of Silverman's seminal text Arithmetic of Elliptic Curves: see especially Example 4.8, Proposition 4.9 and Example 4.10. Although I haven't done the computation myself (at least not recently enough to remember), I believe that the upper bound on the rank that you want is exactly what comes out of this general discussion, and Example 4.10 works out a particular case.

(Of course, please let me know if this turns out not to be the case...)

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Two co-authors and I included a proof of this fact in our paper, in order to make our article self-contained (but we do not claim to be the first ones to point this out). As Pete Clark explains, it follows easily from the method of descent via 2-isogeny.

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I believe your link is broken, Álvaro. – Dylan Yott Jun 7 '14 at 2:24
@DylanYott, thanks Dylan! Fixed the link. – Álvaro Lozano-Robledo Jun 7 '14 at 3:04

It's in Alvaro Lozano-Robledo's book. In fact, you can find it online.

It's Theorem 2.6.4 on page 42.

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The reference you provided defines the curve as $y^2 = x^3 + Ax^2 + Bx$. – Alex Becker Apr 17 '11 at 1:31
@Alex I guess I'm not paying attention... and my wrongs ended up making a right. I will correct it, thanks. – Graphth Apr 17 '11 at 2:31

I think this is a reference.

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The bound is also proved in Knapp's "Elliptic Curves", p. 107, Chapter IV, Section 7.

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