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Let $K=\mathbb{F}_q(C)$ be the function field of a smooth projective curve $C$ over a finite field $\mathbb{F}_q$ with $\text{cha}(K)>3$ and let $E$ be an elliptic curve over $E$. To $E$ we may associate its conductor $\mathfrak{n}_E$, which is a divisor on $C$.

In his paper "Elliptic curves with large ranks over function fields" (https://arxiv.org/abs/math/0109163) Ulmer writes on page 313 "Here we ignore the finitely many elliptic curves over $K$ with trivial conductor." From the context, this should mean that $\mathfrak{n}_E$ has degree 0.

Does anyone have a reference for the claim that (up to $K$-isomorphism) there are only finitely many elliptic curves over $K$ with $\text{deg}(\mathfrak{n}_E)=0$? I already tried searching standard references, but couldn't find anything.

Any help would be much appreciated.

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    $\begingroup$ As the conductor is an effective divisor, $\operatorname{deg}(n_E)=0\iff n_E=0$. $\endgroup$
    – Chris Wuthrich
    Commented Jan 5 at 19:12
  • $\begingroup$ Okay I guess that imples that $E$ is isotrivial since then the $j$-invariant lies in $\mathbb{F} _q$ and so $E$ must be a twist of a constant curve. But how does this imply that there are only finitely many $E$? $\endgroup$
    – MightyGuy
    Commented Jan 5 at 19:58

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