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Suppose we are given an arithmetic surface, $X\to \text{Spec}\mathbb{Z}[1/N]$ smooth and quasi-projective, and a finite set of closed points all in different vertical fibers.

Can we count the number horizontal sections passing through those points?

Could you direct me to the relevant literature?

Thank you in advance.

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    $\begingroup$ Are you asking about counting rational points on a projective curve defined over a number field with specified reductions modulo finitely many primes? If the genus of the curve is bigger than $1$, there should not be such rational points if you choose the primes and the reductions "general". If you are looking at $\mathbb{P}^1$, there will be infinitely many such points, but you could ask about the asymptotics as the height of the rational point grows. $\endgroup$
    – Jason Starr
    Commented Dec 8, 2015 at 19:59
  • $\begingroup$ Exactly! And by Faltings theorem we expect the number to be zero most of the time if the generic fiber has genus bigger than 1. I was wondering whether one could be more precise. $\endgroup$
    – Bear
    Commented Dec 8, 2015 at 20:26
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    $\begingroup$ I still do not understand what you are asking. $\endgroup$
    – Jason Starr
    Commented Dec 8, 2015 at 21:29
  • $\begingroup$ My hope is to be able to say something about the following question: given a bunch of elliptic curves $E_1, ... E_n$ over $\mathbb{F}_{p_1}, ....\mathbb{F}_{p_n}$ how many rational elliptic curves have those reductions? $\endgroup$
    – Bear
    Commented Dec 8, 2015 at 22:18
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    $\begingroup$ By the Chinese remainder theorem, there are always infinitely many such elliptic curves. By "how many" I suppose is meant "what proportion" (of all elliptic curves arranged by height), but then there is an obvious answer to that too: it is a product of local densities $c(p_i)$, where $c(p) = 1/n(p)$ with $n(p) := 2p+3 + \Big( \frac{-4}{p} \Big) + 2\Big( \frac{-3}{p} \Big)$, the number of isomorphy classes of elliptic curves over $\mathbb{F}_p$. $\endgroup$
    – Vesselin Dimitrov
    Commented Dec 9, 2015 at 0:33

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