[Math] Algebraic curve mapping to elliptic curve – how to check whether this is possible

Question

Question: Let $C$ be an algebraic curve over some field (like the rationals) given by a plane projective model (possibly with singularities). Is there an easy way to see if this curve has a non-trivial rational map to an elliptic curve?

Criteria involving the Jacobian (something like $J(C)$ has a subgroup of codimension $1$) wouldn't help, as the curves I am interested in have a big genus ($>10$) and high degrees in both variables, so it is very unlikely that anything about their Jacobians can be computed.

Background: This question arose from an attempt to study rational points (over $\mathbb Q$) on certain curves which possibly are coverings of lower genus curves. In particular, if one could compute a covering map to an elliptic curve, one could efficiently look for rationals points.

Answer 1

$C$ will admit a nontrivial rational map to an elliptic curve $E$ if and only if $E$ appears as an isogeny factor of the Jacobian $J(C)$. So in some sense it is impossible to avoid what you lament in your first paragraph.

However, we don't necessarily need to compute the Jacobian $J(C)$ in any sense: rather, we need to show that it does not have an elliptic curve as an isogeny factor. The conventional wisdom is that (if $C$ has genus greater than one, as you must of course intend) then it is very likely that $J(C)$ is geometrically simple and even that $\operatorname{End} J(C) = \mathbb{Z}$.

There are some sufficient conditions for that! The one I know by heart is the following beautiful theorem of Zarhin:

Let $K$ be a field of characteristic different from $2$, and let $C_{/K}$ be a hyperelliptic curve given by $y^2 = P_{2g+2}(x)$. Then if the Galois group of $P_{2g+2}(X)$ is $S_{2g+2}$ (as it usually is!!) or $A_{2g+2}$, then $\operatorname{End} J(C) = \mathbb{Z}$.

Unfortunately I have forgotten much of what I used to know about this sort of thing, but I am reasonably confident that there are further results along these lines. The name "Arsen Elkin" (whom I think was a student of Zarhin) is coming to mind.

It would be very interesting to have a criterion that one could apply to a fairly arbitrary plane curve $C$ that would be sufficient to force its Jacobian to have endomorphism ring $\mathbb{Z}$. This time I have no memories, however dim, of such a result, but it would certainly be nice...

Added: I looked up Elkin's work on MathSciNet. He has several papers in this area, but so far as I can see they all further refine the hyperelliptic case.