I'm having some issues with abelian varieties and fields of definition. This already became clear in my previous question on Jacobians. Here's another question. If somebody can explain some nice facts on fields on definition this would help me a lot (because these aren't the only questions I have concerning fields of definition). 


Let $A$ be an abelian variety over a number field $K$. Let $L/K$ be a finite field extension. Suppose that there exists an isogeny $A\_L\to B$ defined over $L$. 


Is this isogeny defined over $K$ if $B$ can be defined over $K$?


I think the answer is negative, but
I have to admit that my set of examples is very scarce and, therefore, I can't find an easy counterexample. 



Defining isogenies over smaller fields
No: Consider the elliptic curve $E: y^2 = x^3 + x$ defined over $\mathbb{Q}$. Then the isogeny $y \mapsto iy$ and $x \mapsto -x$ is defined over $\mathbb{Q}(i)$ but obviously not over $\mathbb{Q}$.


In general if $K \subset L$ is Galois, then a morphism defined over $L$ comes from one over $K$ if and only if it is invariant under the action of $Gal(L/K)$. This follows from the so-called "theory of descent". See for example Bjorn Poonen's notes [Rational points on varieties.](http://www-math.mit.edu/~poonen/papers/Qpoints.pdf)


