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.

• 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. – Will Sawin Mar 11 at 16:34
• Thanks @Will Sawin. On a related note, does there exist a notion of a zeta function for such spaces? – curious math guy Mar 11 at 17:32
• Again, there exists a notion of local zeta factor (which would be a simple product of local $L$-factors). – Will Sawin Mar 11 at 17:35