Restriction of scalars with simple endomorphism algebra

Abstract

Suppose $L/K$ be a finite abelian extension of number fields of odd degree and suppose an abelian variety $A$ defined over $L$ is a $K$-variety. If the endomorphism algebra of $A/L$ is a field $F$, the followings are equivalent : 
(1) The enodomorphiam algebra of the restriction of scalars from $L$ to $K$ is simple. 
(2) There is no proper subfield of $L$ containing $L^{G_F}$ on which $A$ has a $K$-variety descent.