I am expanding naf's comments to make a self-contained community wiki answer. By an elliptic fibration we mean a smooth projective relatively minimal surface $f: X \to C$ with general fiber given by an elliptic curve.

**Main Proposition.** If $C$ is a smooth projective curve and $S \subset C$ a finite subset, then the number of nonisotrivial elliptic surfaces with a section $f: X \to C$ and having singular fibers only over points of $S$ is finite.

The proof consists in making a base change of finite degree, ramified only at $S$ such that the new elliptic surface is rigidified by $n$-level structure and using that moduli spaces of elliptic curves with $n$-level structure is representable, together with some standard finiteness results. We go through the arguments in detail. To deal with elliptic fibrations without a section, one needs to use the Jacobian fibration trick.

**Warm up.** Let us first give a simple proof of a related but easier statement: if $C$ is a smooth projective curve and $f: X \to C$ is an elliptic surface with no singular fibers, then $f$ is isotrivial (all fibers isomorphic).

Proof: the $j$ invariant of each fiber gives an algebraic map $C \to \mathbb{A}^1$ (the $j$-line). Since $C$ is projective, such a map is constant, and so all fibers have the same $j$-invariant, hence are isomorphic.

**Representing moduli spaces of elliptic curves**. The $j$-line $\mathbb{A}^1$ is the coarse moduli space of the stack $\mathcal{M}_{1,1}$ of elliptic curves. The standard way to get a representing variety, not a stack, is to get rid of automorphisms by parametrizing pairs $(E, \phi)$ where $E$ is an elliptic curve and $\phi: E[n] \simeq (\mathbb{Z}/n)^2$; in other words we choose a basis of $n$-torsion points on $E$. The corresponding stack is the modular curve $X(n)$. It is a usual smooth curve for $n \ge 4$ and has positive genus for $n$ large enough. As we have defined it $X(n)$ is not projective; one can compactify and then points at infinity parametrize certain singular curves with level structure. There are other variants of modular curves such as $X_1(n)$ and $X_0(n)$; any of them will work.

**Level structure and monodromy.** Of course, our elliptic surface $f: X \to C$ admits level structure at each smooth fiber, but not necessarily in a family. This is controlled by the monodromy representation: if $U = C \setminus S$ is the locus of smooth fibers, and $0 \in U$ is any point, $E = f^{-1}(0)$, then $\pi_1(U, 0)$ acts on $E[n] \simeq (\mathbb{Z}/n)^2$; we get a homomorphism
$$
\gamma: \pi_1(U) \to \mathrm{GL}_2(\mathbb{Z}/n).
$$
By definition, a family over $U$ admits a level $n$ structure if $\gamma$ is trivial, that is you can canonically identify $n$-torsion points at all fibers.

**Lemma 1.** Fix $n > 0$. For any elliptic surface $f: X \to C$ with singular fibers only at $S \subset C$ we can make a base change $C' \to C$ of bounded degree and ramified only at $S$ such that the pullback of $f$ to $C'$ admits $n$-level structure on the smooth part.

*Proof.* Let $G = \gamma^{-1}(e)$. This is a subgroup in $\pi_1(U)$ of index bounded by $|\mathrm{GL}_2(\mathbb{Z}/n)|$ and by the covering theory there exists $U' \to U$ of degree equal to $[\pi_1(U):G]$ such that $\pi_1(U) = G$ so that the new mondoromy representation is $\gamma|_G$ which is trivial. Hence the pullback of $f$ to $U$ admits the $n$-level structure. We may extend our unramified cover to a finite map $C' \to C$ between smooth projective curves ramified only at $S$.

In the proof of the Proposition we will use finiteness of such covers:

**Lemma 2.** Given $S \subset C$ there are only finitely many covers $C' \to C$ of bounded degree ramified only at $S$.

This Lemma is a starting point for various beautiful theories such as Hurwitz numbers counting covers of $\mathbb{P}^1$ and Grothendieck's dessin's d'enfant dealing with covers of $\mathbb{P}^1$ branched at $3$ points. The proof is standard and also uses representations of the fundamental group of $C \setminus S$.

**Proof of the Proposition**

Take $n$ large enough so that $X(n)$ is a curve of genus $g \ge 2$. By Lemma 1 and Lemma 2 we may make a base change and assume that $f$ admits an $n$-level structure over $U$. In this case due to representability of the modular curves above to give such an elliptic fibration is the same as to give a map $U \to X(n)$. Such maps are in bijection with maps $C \to \overline{X}(n)$ (the smooth projective model). Finally result follows from:

**Lemma 3.** Given smooth projective curves $C_1$, $C_2$ with $g(C_1) \ge 2$, $g(C_2) \ge 2$ there are at most finitely many nonconstant maps $C_1 \to C_2$. (This is of course not true if genus is $0$ or $1$; it suffices to ask for $g(C_2) \ge 2$.) This is the De Franchis Theorem.

**Exit quiz.** Use the same proof to show that any elliptic fibration $f: X \to \mathbb{P}^1$ with only two singular fibers is isotrivial!

**Historical notes.** Results of this kind in much larger generality go back to Shafarevich 1962 (hyperelliptic curves), Parshin 1968, Arakelov 1971, Faltings 1983 (abelian varieties with some restrictions) and Deligne 1987 (Hodge structures). The form discussed here was presumably known to Shafarevich in 1962.

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