Galois embeddings of abelian varieties and a question raised by Ekedahl-Serre

 For a smooth projective $n$-dimensional variety $X\subseteq\mathbb{P}^N$, let $W$ be a linear subspace of $\mathbb{P}^N$ of dimension $N-n-1$ that is disjoint from $X$ and let $\pi_W:X\to\mathbb{P}^n$ be the linear projection associated to $W$. A natural question to ask is: When does this projection induce a Galois extension of function fields? We will address this question in the case that $X$ is an abelian variety. Moreover, we will relate this discussion to a question asked by Ekedahl and Serre on Jacobian varieties that are isogenous to the product of elliptic curves.