All the Weil

The proven Weil conjectures for varieties over finite fields outline the properties of zeta functions for such spaces. For example, consider the projective line over the finite field with $p^n$ elements. It's zeta function is

$\zeta (s) = \frac{1}{(1 - p^{-s})(1 - p^{1-s})}$

and higher dimensional projective spaces just have more factors on the bottom. Grothendieck and others eventually understood the Weil conjectures using an $l$-adic cohomology theory, where $l$ is a prime different from $p$. The zeta function was expressed in terms of determinants for $l$-adic cohomology groups, and a Poincare duality explains the famous duality symmetry of zeta functions.

In quantum gravity we find fields with $p^n$ elements in the mutually unbiased bases for dimension $p^n$. As for a projective line, there are always $p^n + 1$ elements in a set of MUBs. Considering $n$ qupits for all $n$, we are naturally led to a generalisation of $p \times p$ matrix entries to the $p$-adic numbers. But constructing the complex numbers correctly in this axiomatic setting does not mean taking the complex numbers as they are usually given to us, complete with an unmeasurable continuum and a set theoretic continuum hypothesis. As any topos theorist knows, the real and complex numbers are not even uniquely defined in a topos. Quantum gravity will use the numbers it requires, and no more. A useful complicated space is, after all, described by a finite amount of data.