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  1. Given a Diophantine equation $f(x_1, \dots, x_n) \in \mathbb{Z}[x_1, \dots, x_n]$ and a family of number fields $K$ (say, the number fields of a specified degree and signature), are there techniques which can be used to determine which number fields admit a solution to the equation over the ring of integers $\mathcal{O}_K$?

  2. Relatedly, are there methods to prove analytic results about how many number fields in some family admit integer solutions to $f$? For what $f$ can we bound, for example, the number of real quadratic fields $K$ with discriminant at most $D$ such that $f$ has a solution over $\mathcal{O}_K$?

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    $\begingroup$ I think this question is quite general and you can't really say anything expect in special cases. The case of conics is fun to work out using the exact sequence for the Brauer group from class field theory. $\endgroup$
    – Daniel Loughran
    Commented Feb 7, 2021 at 16:37

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