Showing that $E(\mathbb Q)_{\text{tors}} \cong \left\{\mathcal O\right\}$? Suppose you have an elliptic curve $E/\mathbb{Q}: y^2= x^3+ax+b$. Suppose that $E$ has good reduction at primes $p$ and $q$, and suppose that you want to show $E(\mathbb Q)_{\text{tors}} \cong \left\{\mathcal O\right\}.$ Further more, suppose that $|E(\mathbb F_p)| = m$ and $|E(\mathbb F_q)| = n$. Then $|E(\mathbb Q)_{\text{tors}}| $ divides $\gcd(m, n)$. If $\gcd(m, n) = 1$ then $E(\mathbb Q)_{\text{tors}} \cong \left\{\mathcal O\right\}$. But say $\gcd(m, n)=2$. How do you rule out the curve having torsion subgroup of size $2$? And the same question for $\gcd(m, n)=3$. What's the process in general? Normally I would think of checking the order of non-trivial points over $E(\mathbb Q)$, but in this case there are no such points and this is what we're trying to prove.
My other question is if you have a set of points $\left\{a_0 \cdots, a_m\right\}$ torsion points over $E(\mathbb{F}_p)$ and $\left\{a_0 \cdots, a_n\right\}$ torsion points over $E(\mathbb{F}_q)$, can this give me explicit solutions over $\mathbb{Q}$? I know that you can have solutions over $\mathbb F_p$ but none over $\mathbb{Q}$, but when solutions exist over $\mathbb Q$ is there a way to determine these from the solutions over $\mathbb F_p$?
I'm new to elliptic curves, so I apologise if these are not good questions.
 A: A point $P = (x_0,y_0)$ is two torsion if and only if $y_0= 0$, you can see this from the definition of doubling on an elliptic curve, the tangent line goes to infinity only for a vertical tangent line, which happens only for points with $y$-coordinate zero (take the partial derivatives of the defining equation).
So to find the two torsion points we simply have to factor $x^3 + ax + b$ over the rationals.
More generally however this sort of analysis of tangent lines to find torsion points gets harder, but there is a theorem by Lutz-Nagell (https://en.wikipedia.org/wiki/Nagell%E2%80%93Lutz_theorem) that classifies torsion points in a very explicit way for elliptic curves over the rationals.
This theorem always gives us a finite list of possible points that we can check if they are torsion (if they even define points on the curve) by repeatedly adding one point to itself. Note that as soon as we get non-integer coordinates for a multiple of our point we know it cannot have been torsion so we can stop.
