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I'd like to know if there is a (conjectural) criterion for a meromorphic function on $\mathbb{C}$ to be the zeta function of an arithmetic scheme, i.e., a statement of the form

"If a meromorphic function on $\mathbb{C}$ has the following properties, then it is the zeta function of a regular, irreducible scheme of finite type over $\mathbb{Z}$ of dimension n."

One kind of criterion would say that if the function has a Dirichlet series with appropriate Euler product and condition on growth of its terms, an appropriate functional equation, zeros and poles at expected locations etc., then it comes from an arithmetic scheme. This would be an analog of the converse theorems of Hecke, Weil et al. for automorphic forms which, of course, are relevant to the question due to their conjectural association with arithmetic schemes via Galois representations.

An extension of the question would ask for a characterization of the set of all arithmetic schemes whose zeta functions are equal to a given meromorphic function.

I should perhaps pose the question for varieties over finite fields first where an answer is more likely to exist, or is even obvious, in view of Weil conjectures.

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    $\begingroup$ Concerning your last paragraph, just for the case of curves, see mathoverflow.net/questions/70605/from-zeta-functions-to-curves. $\endgroup$
    – KConrad
    Mar 23, 2018 at 15:38
  • $\begingroup$ @KConrad, thank you for pointing it out. It brings out the complexity of the problem even in the finite field case, and gives very useful references. $\endgroup$
    – user122285
    Mar 23, 2018 at 21:48

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