TITLE: The boundedness of the rank of twists of a fixed curve
QUESTION [14 upvotes]: It is conjectured that there are do not exist elliptic curves over $\mathbb Q$ of arbitrarily high rank. I was wondering wether someone made a similar conjecture if one restricts to a fixed $j$-invariant. If there are specific values of $j$ known such that the following question has a negative answer would also be nice to know.
Let $j \in \mathbb Q$ be fixed. Do there exist elliptic curves $E/\mathbb Q$ with $j(E)=j$ of arbitrary high rank over $\mathbb Q$?
I'm specifically interested in the value $j=0$.
Edit
I recently found an article by Bjorn Poonen called "Heuristics for the arithmetic of elliptic curves". And in that article he proves (Thm 3.7) that if elliptic curves over $\mathbb Q$ follow his heuristics then 21 is the switching point between finitely and infinitely many elliptic curves of that rank and in particular there will be only finitely many elliptic curves of rank > 21. This of course implies the boundedness of the rank of all elliptic curves over $\mathbb Q$, so the same should hold if one restricts to just one $j$-invariant.

REPLY [6 votes]: This doesn't answer your question, but I'll mention it anyway.  My thesis research shows that the average rank of $j = 0$ elliptic curves, when ordered by discriminant, is bounded by 1.5.  This follows from computing the average size of the 2-Selmer group, which is 3.  
I'll also mention that your question is much easier to answer if you replace Mordell-Weil rank with 2-Selmer rank.  In that case, Heath-Brown has two papers (The Size of the Selmer Group for the Congruent Number problem I, II) showing that for any $r$, a positive density of the curves $y^2 = x^3-D^2x$ have 2-Selmer rank $r$.  Mazur and Rubin have a paper (Ranks of Twists of Elliptic Curves and Hilbert's Tenth Problem) where they consider quadratic twists of curves with no rational 2-torsion and give some conditions for the family of twists to have arbitrary 2-Selmer rank.