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I've been learning more about different $p$-adic geometries, namely Berkovich spaces, Huber's Adic spaces and ridgid analytic spaces. In arithmetic geometry, it is often very interesting to assoicate and study the L-function of an space. My question is if this has been studied for such $p$-adic spaces. One way one could naively hope to get such L-functions is to take the L-function associated to the cohomology of such a space and study that. However, I've been unable to find a reference for this.

The issue is that google searches lead straight to $p$-adic L-functions, which as far I can tell is not what I want.

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    $\begingroup$ Because $p$-adic spaces lie over $p$-adic fields, they could only ever have $L$-functions that are local at $p$ - there would be no way to define $L$-factors at other primes. One can define these $L$-factors whenever one has a finite-dimensional cohomology theory with Frobenius acting on it (which I think is fine for at least proper rigid analytic spaces), but it's not clear what they get you. $\endgroup$
    – Will Sawin
    Commented Mar 11, 2020 at 16:34
  • $\begingroup$ Thanks @Will Sawin. On a related note, does there exist a notion of a zeta function for such spaces? $\endgroup$
    – curious math guy
    Commented Mar 11, 2020 at 17:32
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    $\begingroup$ Again, there exists a notion of local zeta factor (which would be a simple product of local $L$-factors). $\endgroup$
    – Will Sawin
    Commented Mar 11, 2020 at 17:35

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