Suppose $C$ is a curve of genus $2$, does $C$ admit a surjective morphism onto some elliptic curve $E$?



Can every genus $2$ curve be written as ramified cover of elliptic curve?
dhy's comment is correct, of course. You can also see that the answer is "no" by dimension counting. Fix a degree $d$. By a Riemann-Hurwitz computation, a degree $d$ cover of a genus $1$ curve by a genus $2$ curve is ramified over $2$ points (or doubly ramified over one point.) Given a positive integer $d$, a genus $1$ curve $E$ and two points $p$ and $q$, there are finitely many covers $X \to E$ of degree $d$ ramified only over $p$ and $q$. (Since each of them comes from one of finitely many maps $\pi\_1(E \setminus \{ p,q \}) \to S\_d$.) So, for fixed $d$, such covers are described by some finite covering of $M\_{1,2}$. 


Now, $\dim M\_{1,2} = 2$. So, for each $d$, we get a surface of such genus $2$ curves and, as $d$ varies, we get countably many such surfaces. Counteably many surfaces can't fill up the three-fold $M\_2$.


