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.
On
It's in Alvaro Lozano-Robledo's book. In fact, you can find it online.
http://www.math.uic.edu/~wgarci4/pcmi/PCMI_Lectures.pdf
It's Theorem 2.6.4 on page 42.
On
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.
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...)