Here is the short version:
Fix an elliptic curve $E/\mathbb{Q}$. How does the torsion structure $E_d(\mathbb{Q})_{tors}$ vary, as $E_d$ runs through the quadratic twists of $E$?
Here is the longer version:
I have been playing with SAGE this morning. I inserted the elliptic curve ('11a1') $$E : y^2 + y = x^3 – x^2 – 10x – 20$$ which has rational torsion subgroup isomorphic to $\mathbb{Z}/5\mathbb{Z}$. I then computed its quadratic twist $E_d$ for all squarefree $d$ up to 2000, and observed $E_d(\mathbb{Q})_{tors}$ was always trivial.
Can it be that, in this particular family of quadratic twists, all but one of the curves have trivial torsion? Is this a general phenomenon?
(I ran this experiment for several other curves $E$ and got the same impression; that all but one of the curves in a family of twists have the same torsion structure.)
Best Answer
Theorem (originally due to Setzer?): Fix $E/\mathbb{Q}$ with $j(E)$ not 0 or 1728. Then for all but finitely many inequivalent twists $E_d$, the torsion subgroup $E_d(\mathbb{Q})_{tors}$ is isomorphic to $E[2](\mathbb{Q})$, so in particular $E_d(\mathbb{Q})_{tors}$ has order 1, 2, or 4. (Probably he also proved it for number fields.)
There's a paper of mine$^1$ with a much more general theorem using the theory of heights. I don't recall Setzer's proof except that it doesn't use heights.
Theorem: Let $K$ be a number field and let $A/K$ be an abelian variety with $\mu_n\subset {\rm Aut}(A)$. (This means we can twist $A$ by $n$'th roots of $d$.) Then every point $P\in A_d(K)$ satisfies one of the following two conditions:
Here $\hat h$ is the canonical height relative to an ample symmetric divisor, and $h^{(n)}(d)$ is a sort of "$n$'th power free height," say equal to the minimum of $h(du^n)$ for $u\in K^*$. The constants depend on $A$, but are independent of $d$.
It follows from the theorem that after discarding finitely many $d \in K^*/{K^*}^n$, a point in $A_d(K)$ is either $1-\zeta$ torsion (hence $nP=O$), or its canonical height is positive, and hence it is nontorsion.
Of course, to describe more precisely what happens for the finitely many exceptional $d$ can be a delicate matter, as some of the other answers have indicated. I think it's interesting to see how one can approach the problem via heights or via representation theory.
$^1$ J.H. Silverman, Lower bounds for height functions, Duke Math. J. 51 (1984), 395-403.
EDIT: Fixed statement of first theorem. I'd originally written that "for all but finitely many inequivalent twists $E_d$, the torsion subgroup $E_d(\mathbb{Q})_{tors}$ has at most two elements." This is clearly false, since if $E$ has the form $E:y^2=(x-a)(x-b)(x-c)$ with $a,b,c\in\mathbb{Q}$, then $E_{d}[2](\mathbb{Q})$ has order 4 for every twist.