# Shafarevich's theorem for elliptic curves defined over function field of algebraic curve over algebraically closed field

Let $K$ be a number field and $S$ a finite set of places of $K$. Then Shafarevich's theorem states that there are only finitely many isomorphism classes of elliptic curves $E$ over $K$ with good reduction outside $S$.

My question is whether the above holds when $K$ is the function field of an algebraic curve over an algebraically closed field of characteristic zero, and $E / K$ has non-constant $j$-invariant?

-
In positive characteristic there are the Moret-Bailly pencils. –  Jason Starr May 7 '13 at 17:53
Oops, Moret-Bailly's examples are pencils of Abelian surfaces, not elliptic curves. –  Jason Starr May 7 '13 at 17:55
Yes. Let $U$ be the open affine curve away from $S$ over the constant field $k$. For any elliptic curve $E \rightarrow U$ there's a finite etale cover $U' \rightarrow U$ of universally bounded degree over which $E$ acquires a point of order 1728. Since $\pi_1(U)$ has only finitely many quotients of any given size, there are only finitely many possibilities for $U'$ up to $U$-isomorphism. Each connected component $U'_i$ of $U'$ has a specified map to $Y_1(1728)$ and it is dominant if $j(E) \not\in k$. There are only finitely many such maps since $X_1(1728)$ has genus $> 1$. QED –  user28172 May 7 '13 at 20:09
@nosr: nice answer! Why post it as comment? There are however two points I didn't understand: 1) why is the degree of $U'\to U$ universally bounded? 2) your proof shows that given $p:E\to U$ we can associate to it a pair $(f:U' \to U, g:U'\to X_1(1728))$ and that there are finitely many such pairs, but I fail to see why there can't be infinitely many $p:E\to U$ to which the same pair $(f, g)$ is assigned - i.e. there could be many ways of descending $g*(\mathcal{E})$ along $f$, where $\mathcal{E}\to X_1(1728)$ is the universal curve. –  Piotr Achinger May 8 '13 at 1:35
I think the following problem is lurking in the background here: the action of the automorphism group of $K$ on $j$. In order to get finiteness statements, this group should be seen to be finite (eg when $K$ is the function field of a curve over a finite field). –  Damian Rössler May 9 '13 at 7:40
I don't know if this would be of any help but there is a nice paper of Bandini, Longhi and Vigni (https://www.mat.unical.it/~bandini/igusa.pdf) that contains a proof for admissible elliptic curves over global function fields of characteristic $p>3$.