## Counting higher dimensional abelian varieties of a given conductor

This question is a follow up to an earlier question of mine on enumerating elliptic curves of a given conductor.

I've heard people say that studying higher dimensional varieties via explicit defining equations often leads to hopelessly unmanageable complexity. As apparent evidence of this, on page 65 of Cassels' and Flynn's book titled Prolegomena to a Middlebrow Arithmetic of Curves of Genus $2$, the authors state that the defining equations that they find for Jacobian varieties of genus $2$ curves consist of $72$ quadratic equations in $\mathbb P^{15}$. People say that rather than studying higher dimensional algebraic varieties as solution sets to explicit equations, one typically studies such varieties in a more abstract and geometric way.

This makes sense. Yet I wonder how one can get look at concrete examples without defining equations. I know that there are some varieties such as moduli spaces which provide examples. But suppose, say, you want to prove that there are finitely abelian varieties of an arbitrary fixed dimension $d$ over a fixed arbitrary number field $K$ with a fixed conductor $N$ without looking at the automorphic side of things.

Is there a (conjectural) method of proving finiteness without writing down explicit defining equations?

[Edit: As Barinder Banwait points out, this follows immediately from the Shafarevich conjecture, which was proved by Faltings.]

Pushing the envelop further,

Is there a (conjectural) algorithm for enumerating these objects?

Pushing the envelop still further,

Suppose beyond enumerating such varieties, you want to determine, e.g., how many of them have surjective (mod $5$) Galois representation (say, attached to $H^1$) - is there a conjectural algorithm for doing so?

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The answer to your first question is YES; it is known as "Shafarevich conjecture for abelian varieties" (though it's not a conjecture anymore). See this blog post of Martin Orr for an overview: martinorr.name/blog/2011/09/19/… – Barinder Banwait Sep 18 at 18:46
Warning: Even for Jacobians $J$ of curves $C$ of genus $g \geq 2$, it can happen that $C$ has good reduction at a prime $p$ but $C$ does not. For $g=2$ the list of curves over ${\bf Q}$ with good reduction outside $\lbrace 2 \rbrace$ is known, but not so the list of Jacobians with good reduction outside $\lbrace 2 \rbrace$. Yes, it follows from Faltings that this list is finite, but the proof is not effective (though I think it does yield an upper bound on the size of the list, at least in principle). – Noam D. Elkies Sep 18 at 19:02
@ Barinder Banwait - yes, of course - I'm sheepish for having missed this, as it was already mentioned in the answers to my last question! – Jonah Sinick Sep 18 at 19:33
@ Noam D. Elkies - I was asking about abelian varieties rather than curves and so don't understand the relevance of your comment - am I missing something? – Jonah Sinick Sep 18 at 19:36
In the case $d=2$, every abelian surface is isogenous to a Jacobian or to a product of elliptic curves, so we are reduced to the case of Jacobians. But as Noam D. Elkies shows, this problem is strictly more difficult than that of curves. – François Brunault Sep 18 at 20:36
Here is a variant of this problem. Consider the number of $g$-dimensional p.p. abelian varieties with everywhere good reduction over a varying number field $K$. In genus $g = 1$, it follows from the uniform finiteness theorem for the unit equation that this number is bounded solely in terms of $[K:\mathbb{Q}]$. If you consider more generally elliptic curves on $K$ with good reduction outside $S$, the bound becomes simply exponential in $[K:\mathbb{Q}] + |S|$.
Are there any results available, for higher genus $g > 1$, on whether or not the number of $g$-dimensional p.p. abelian schemes on $R := \mathcal{O}_{K,S}$ could be bounded solely in terms of $g$ and the rank of $R$?