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Can a smooth projective variety over $\mathbb{Q}_p$ have two smooth projective models with non-isomorphic $\mathrm{mod}\:p$ fibers? Can the $\mathrm{mod}\:p$ fibers have different number of rational points?

I guess you can do blow-ups but let's impose some kind of minimality hypothesis.

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The answer to question 1 is yes. Examples almost like what you are after can be found here:

Does isomorphic generic fibre imply isomorphic special fibre for smooth morphisms?

Though admittedly the examples there are non-minimal as the generic fibre is isomorphic to $\mathbb{P}^2$ blown up in two points. But it is possible to come up with minimal examples as $\mathbb{P}^1 \times \mathbb{P}^1$ can be specialised to both $\mathbb{P}^1 \times \mathbb{P}^1$ and a Hirzebruch surface; I can try to find the details if you would like.

The answer to question 2 is however no. The smooth proper base change theorem implies that the $\ell$-adic cohomology of the special fibres is isomorphic. Therefore the Weil conjectures implies that they have the same number of points modulo $p$.

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  • $\begingroup$ Minor point: I think it's $\mathbf{P}^1\times \mathbf{P}^1$ that specializes to a Hirzebruch surface, not $\mathbf{P}^2$ (wrong Picard number). $\endgroup$
    – Piotr Achinger
    Commented Sep 16, 2021 at 9:25
  • $\begingroup$ Yes quite, thanks! $\endgroup$
    – Daniel Loughran
    Commented Sep 16, 2021 at 9:40

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