Let $S$ be a finite set of places of a global function field $K$. Are there finitely many Abelian varieties over $K$ with good reduction outside $S$? What if we exclude isotrivial families?
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5$\begingroup$ Presumably you want to fix the dimension $g$ of $A$, otherwise once you have a single such $A$ you have infinitely many, namely its powers $A^n$. Even then, I think this conjecture can fail already for $|S|=1$ due to Jacobians of Artin-Schreier curves $y^p - y = P(x)$. For $p=2$ and $p=3$ this already happens for $d=1$ ($y^2-y = x^3 + c t x$, $x^3-x = y^2 + c t y$). Those examples must be isotrivial (constant $j$ invariant) but I think that in characteristic $2$ the genus-3 curves $y^2-y = x^7 + ct$ give a nonisotrivial family with good reduction away from $t = \infty$. $\endgroup$– Noam D. ElkiesCommented Feb 27 at 15:36
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1$\begingroup$ For every $g\geq 2$, there is a non-isotrivial morphism $\mathbb{P}^1\to \mathcal{A}_g$ (i.e., a non-isotrivial family $X\to\mathbb{P}^1$ of $g$-dimensional principally polarized abelian schemes) whose fibres are all supersingular. Pre-composing with arbitrary endomorphisms of $\mathbb{P}^1$ (i.e., pulling-back $X\to \mathbb{P}^1$ along arbitrary endomorphisms of $\mathbb{P}^1$) gives you infinitely many pairwise non-isomorphic families. $\endgroup$– Ariyan JavanpeykarCommented Apr 23 at 6:39
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