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There are several ideas in arithmetic geometry that can help in proving the absence of rational points as well exhibiting rational points on algebraic curves.

I am aware there are some obstructions (e.g. étale Brauer-Manin) coming from arithmetic geometry that can prove the absence of rational points on higher-dimensional varieties.

There are also several tricks that can pass information from curves to higher-dimensional varieties:

  • take the Weil restriction of a curve over a number field
  • if the system $0=P_1(z_1, \dots, z_m)=\dots= P_n(z_1,\dots, z_m)$ defines a curve then its rational points are in bijection with the rational points on the variety $0=P_1^2(z_1, \dots, z_m)+\dots+P_n^2(z_1, \dots, z_m)$
  • take a variety dominated by the direct product of any of the above

If we exclude these tricks then can arithmetic geometry help in exhibiting rational points on higher-dimensional varieties?

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  • $\begingroup$ I know the theory of Heegner points can allow you to find points on elliptic curves. It may have analogues in higher dimensions - it is probably highly conjectural, but may be something that works in practice $\endgroup$
    – Wojowu
    Commented Feb 21, 2021 at 10:59
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    $\begingroup$ There is a conjecture called Bombieri-Lang conjecture, which is a higher-dimensional analogue of Faltings's theorem. Although this conjecture is largely open, some special cases are proved. See this question. $\endgroup$
    – Daebeom Choi
    Commented Feb 21, 2021 at 11:55
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    $\begingroup$ This question is way too broad. The vast majority of the modern study of rational points is about trying to use tools from algebraic geometry to understand them. Also your second trick (taking sums of squares) is completely useless and I have never seen a genuine application of this method. $\endgroup$
    – Daniel Loughran
    Commented Feb 21, 2021 at 12:37
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    $\begingroup$ @DanielLoughran I think many techniques used in this field do not yield clear algorithms so I personally do not feel it is too broad. The second trick is not terribly ingenious but it is there and I think if I did not point it out somebody else would. $\endgroup$
    – Grigore Milli
    Commented Feb 21, 2021 at 12:51
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    $\begingroup$ So you are looking for algorithms? This is not at all clear from your question. Can you please clarify exactly what you are after? $\endgroup$
    – Daniel Loughran
    Commented Feb 21, 2021 at 17:20

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