Q-isogeny and Q-torsion subgroup

Question

What is meant by a Q-isogeny and the Q-torsion subgroup? (And by Q, I mean rational 'Q')`

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Answer 1

I assume you have an elliptic curve $E$ defined over $\mathbb{Q}$ -- in simplest terms, this means a Weierstrass equation $y^2 = x^3 + Ax + B$ with $A,B \in \mathbb{Q}$.

Then a $\mathbb{Q}$-isogeny is a finite morphism $\varphi: (E,O) \rightarrow (E',O')$ which is defined over $\mathbb{Q}$: in other words, given locally by rational functions with $\mathbb{Q}$-coefficients. An equivalent perspective is that an isogeny is essentially determined -- i.e., up to an automorphism on the target -- by its kernel $E[\phi]$, a finite subgroup of $E(\overline{\mathbb{Q}})$. Then the definedness over $\mathbb{Q}$ is equivalent to invariance under the group $\operatorname{Aut}(\overline{\mathbb{Q}}/\mathbb{Q})$: for every Galois automorphism $\sigma$, we want $\sigma E[\phi] = E[\phi]$.

Similarly, the $\mathbb{Q}$-torsion subgroup is the subgroup of the full torsion subgroup which is defined over $\mathbb{Q}$. This means $E[\operatorname{tors}] \cap E(\mathbb{Q})$: it is just the subgroup of $E(\mathbb{Q})$ consisting of points of finite order. It can also (equivalently) be defined as the Galois invariants of $E[\operatorname{tors}](\overline{\mathbb{Q}})$ (= $E[\operatorname{tors}](\mathbb{C})$).