Let $X$ be a (smooth projective geometrically connected) variety over a field $k$. 


Consider the set Et$(X,k)$ of finite etale covers $Y\to X$ over $k$, with $Y$ geometrically connected over $k$.


Assume Et$(X,k)$ is infinite. Consider the following question:


Does $X$ have a $k$-rational point?


The answer should be negative in general. In fact, I think one can construct a surface with infinitely many etale covers but no rational points by taking the product of two curves $C$ and $D$ over $k$, where $C$ has infinitely many etale covers and a rational point, but $D$ doesn't have any rational points. Then $C\times D$ has no rational points, but infinitely many covers.


What if $X$ is a curve? Is $X(k)$ non-empty?


Note that the converse is true if we consider curves of positive genus. That is, if $X$ is a curve of positive genus over $k$ with a $k$-rational point, then it has infinitely many etale covers. 


I'm mainly interested in the characteristic zero case, but comments on the situation in positive characteristic would also be interesting.



Varieties with infinitely many etale covers and rational points
Maybe I misunderstand something, but don't all curves have etale covers? Embed $X$ in $J^1$ (divisors of degree $1$ modulo linear equivalence). Then $J^1$ is a torsor for the Jacobian $J$ and since $J$ has etale covers, e.g. coming from multiplication by an arbitrary $n$, $J^1$ does too. Certainly, for those curves with a rational divisor of degree one, they have covers, as $J^1$ is isomorphic to $J$.


EDIT: Upon further reflection, I guess it's not true that $J^1$ always has covers, as it may not be in the divisible part of the Weil-Chatelet group of $J$. But there definitely exist curves with no points having divisors of degree one, and therefore covers of arbitrarily large degree. 


However, you question is a good one and you might be heading in the direction of Grothendieck's section conjecture: For finitely generated fields $k$, $X(k)$ is non-empty if and only if there is a section $G\_k \to \pi\_1(X)$ of the canonical projection $\pi\_1(X) \to G\_k$, where $G\_k$ is the absolute Galois group of $k$ and $\pi\_1$ is the etale fundamental group.


