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In algebraic geometry there are examples when a variety $X$ is somehow related (I call it double-mirror) to another variety $Y$, together with a 2-torsion Brauer class. By "related" I mean statements like eqiuvalence of derived categories or other more crude invariants. I am wondering whether there are similar examples in number theory.

Specifically, is there a concept of a zeta function of a pair $(Y,A)$ where $X$ is a smooth variety $Y$ over $\mathbb Z$ and $A$ a Brauer class on it? If so, are there examples where the zeta function of $X$ is equal to that of $(Y,A)$?

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  • $\begingroup$ I'm confused by your assumptions: do you assume that $X$ and $Y$ are both smooth over $\mathbb{Z}$? Such schemes are quite rare. $\endgroup$
    – Daniel Loughran
    Commented Dec 4, 2014 at 21:45
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    $\begingroup$ I think it's already an interesting question without the twist. For example, for smooth projective varieties over finite fields, it is unknown if derived equivalence implies that the zeta functions are the same. See for example this poster of Katrina Honigs: mimuw.edu.pl/~gael/Poster/poster_Honigs.pdf. $\endgroup$
    – Benjamin Antieau
    Commented Dec 5, 2014 at 0:04
  • $\begingroup$ I know little of arithmetic geometry, so smooth may not be the right term. I am fine with having bad reduction at some primes. $\endgroup$
    – Lev Borisov
    Commented Dec 5, 2014 at 1:06

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