Does a homogenous space for an elliptic curve E/$\Bbb Q$ always have a $\Bbb Q _p$ rational point for every prime $p$? And why?
I know that a homegenous space has $\Bbb Q$ rational points only if it is not a trivial class.
But I have no information on local fields. Any reference（webpage, book, etc...）is also appreciated. Thank you in advance.
No, this is not true. Consider the curve $$C : 2w^2 = 4 - 20z^4$$ which has Jacobian $$E : y^2 = x^3 + 5x$$ with an isomorphism $E \to C$ over $\mathbb{Q}(\sqrt{5})$ given by $$ (x, y) \to \left(\frac{\sqrt{2}x}{y}, \sqrt{2}\left(x - \frac{5}{x}\left(\frac{x}{y}\right)^2 \right) \right)$$ for details see Silverman X3.7.
I claim that $C$ has no $5$-adic points. To see this note that it suffices to show that there are no $u, v, w \in \mathbb{Z}_5$ such that $$w^2 = 2u^4 - 10v^4$$
Suppose otherwise, then we may assume that $(u, v, w) = (1)$. Then $u,w \not\equiv 0 \pmod{5}$, hence $$w^2 \equiv 2u^4 \pmod{5}$$ but in that case $2 \equiv (w/u^2)^2 \pmod{5}$, a contradiction since $2$ is not a square in $\mathbb{F}_5$.
Thus $C(\mathbb{Q}_p) = \emptyset$.
The standard reference for such things is Silverman Chapter X.