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If $S_\zeta$ is the set of all separated schemes of finite type over $\mathbb{Z}$ that have the same arithmetic zeta function $\zeta$, what more can we say about $S_\zeta$ assuming it is non-empty?

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    $\begingroup$ Even restricting to smooth projective varieties over $\mathbf F_q$, this is a very difficult question. Any two smooth projective varieties with isomorphic rational Chow motives (e.g. isogenous abelian varieties) have the same zeta function, and the converse holds assuming the Tate conjecture (for arbitrary codimension). But we don't have many methods for constructing correspondences between varieties, so in general there is little one can say about this set. In my opinion, the question as currently phrased is way too broad. $\endgroup$
    – R. van Dobben de Bruyn
    Commented May 16, 2021 at 2:25
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    $\begingroup$ Perhaps the apparent fact that the question is too broad is a useful quasi-answer to the question!?! $\endgroup$
    – paul garrett
    Commented May 16, 2021 at 2:44
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    $\begingroup$ I think one good way to make the question more specific is to ask about invariants that could potentially distinguish between varieties with equivalent motives, like how, in topology, even if two manifolds are homotopy-equivalent to one another, Reidemeister torsion can still sometimes distinguish between them. Are there any known invariants along these lines for varieties with equivalent motives, rather than manifolds with the same homotopy type? $\endgroup$
    – user164898
    Commented May 16, 2021 at 4:42
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    $\begingroup$ A special case is the question of number fields with the same Dedekind zeta function. See here for some discussion. $\endgroup$
    – Wojowu
    Commented May 16, 2021 at 7:54
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    $\begingroup$ Good point. I seem to remember learning (from that paper of Perlis) that, if two number fields F, F' have the same Dedekind zeta-function, then the products of their class numbers and regulators agree, i.e., hr = h'r'. But their class numbers themselves may differ (and then also their regulators must differ). So it seems that the class number can distinguish two number fields with the same Dedekind zeta-function. Since the class number is visible in algebraic $K_0$, I guess $K$-theory is a useful invariant on the set $S_{\zeta}$ from the original question. $\endgroup$
    – user164898
    Commented May 16, 2021 at 16:37

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