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Given two algebraic curves defined over the rationals, is there a method for determining whether there exists a surjective map from one curve to the other? For instance, suppose X and Y are affine plane curves. How to tell whether there is a dominant rational map X->Y?

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  • $\begingroup$ What data are you putting in? Degrees of the polynomials defining is not intrinsic, since change of variables can make the degrees very different. $\endgroup$
    – Mohan
    Jul 25, 2015 at 21:40
  • $\begingroup$ I am assuming that equations for X and Y are given. $\endgroup$
    – 352506
    Jul 25, 2015 at 22:01
  • $\begingroup$ I think this is too little to say anything. Theoretically, the equations do define curves, but little can be deduced apriori about them. $\endgroup$
    – Mohan
    Jul 26, 2015 at 0:28
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    $\begingroup$ Using the equation, one could compute the zeta functions of the curves over some finite fields and check that the numerator of $\zeta(Y)$ divides the numerator of $\zeta(X)$. That's a necessary condition. $\endgroup$
    – Will Sawin
    Jul 26, 2015 at 15:08
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    $\begingroup$ If $g(Y) \geq 2$ then the degree of such a map is bounded so this can be expressed as a first-order statement in the theory of algebraically closed fields, hence can be solved at the very least using Tarski's quantifier elimination algorithm. If you want a more efficient algorithm, look into what computer algebra systems can do. If $g(Y)=0$ the answer is always yes and I'm not sure what to do in the genus $1$ case but I'm sure it's fine. $\endgroup$
    – Will Sawin
    Jul 26, 2015 at 15:10

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