Galois Representations and Rational Points

Question

Suppose $X$ and $Y$ are two connected smooth projective varieties over $\mathbb{Q}$ (of the same dimension) that have the same $\ell$-adic Galois representations (up to semisimplification). What is the relation between $X(\mathbb{Q})$ and $Y(\mathbb{Q})$, even conjecturally?

Answer 1

In general one can say very little. There are some positive results (as indicated in the comments) in special cases, but the below example kills any hope that one can say something in general. NB "they have the same $\ell$-adic Galois representations" is vague -- I will interpret as "they have isomorphic $\ell$-adic etale cohomology in all degrees" which is the strongest reasonable interpretation I can think of.

So let $E$ be an elliptic curve over $\mathbb{Q}$ with positive rank and non-trivial Tate-Shaferevich group. Let $C$ be a torsor for $E$ corresponding to a non-trivial element of the group. Then $E(\mathbb{Q})$ is infinite and $C(\mathbb{Q})$ is empty.

However I think that the etale cohomology groups of $E$ and $C$ are all isomorphic. $H^0$ is trivial, $H^2$ is the cyclotomic character, and $H^1(C)$ is isomorphic to the dual of the Tate module of the Jacobian of $C$, and the Jacobian of $C$ is isomorphic to $E$ again, so $H^1(C)$ and $H^1(E)$ are both isomorphic to the dual of the Tate module of $E$.

In this example $E(\mathbb{Q})$ and $C(\mathbb{Q})$ are completely different, but at least $E(\mathbb{Q}_p)$ and $C(\mathbb{Q}_p)$ are the same for all $p$. So as another example, let $X$ be projective 1-space over $\mathbb{Q}$ and let $Y$ be a smooth plane conic over $\mathbb{Q}$ with no local points at some prime $p$. The problem here is that the Galois representations tell you very little about the geometry of the variety. $H^0$ is trivial, $H^2$ is cyclotomic, and $H^1$ is zero; again $X(\mathbb{Q})$ is infinite, $Y(\mathbb{Q})$ is empty, and even worse $X(\mathbb{Q}_p)$ is infinite and $Y(\mathbb{Q}_p)$ is empty. Note also that in this case $\pi_1(X)$ and $\pi_1(Y)$ are also isomorphic as Galois modules; the comments by Will Sawin and Piotr Achinger do not apply because $X$ and $Y$ do not have high enough genus to be anabelian.