2023-11-27
https://www.youtube.com/watch?v=AGqjsDBUzUw
Zeta functions and structure types. There is a kind of structure R_L we can put on finite sets, i.e. a species in Joyal's sense, such that an R_L-structure on a finite set is a way of making it into a finite field. We can extract from this a Dirichlet series:
https://ncatlab.org/johnbaez/show/Dirichlet+species+and+the+Hasse-Weil+zeta+function
The exponential of this is the Riemann zeta function. The slice category of species over R_L is a Grothedieck topos: an object here is a way of making a finite set into a field and putting some further structure on it. Similarly, there's a species R_{L,p} such that an R_{L,p}-structure on a finite set is a way of making it into a finite field of characteristic p. Again the slice category of species over R_{L,p} is a Grothendieck topos. Putting the double negation topology on this and forming the category of sheaves, we get a Boolean topos, which is the geometric theory of algebraic closures of 𝔽ₚ. This topos is also the category of G-sets where G is the absolute Galois group of 𝔽ₚ, namely the profinite completion of ℤ:
https://en.wikipedia.org/wiki/Profinite_integer
If you take a commutative ring A and hom it into the the algebraic closure of 𝔽ₚ, you get an object in this topos. From this you can perhaps get one Euler factor of the L-series of this commutative ring (or its corresponding affine scheme).
John Carlos Baez /
npub17u…8pd6m
2025-01-10 18:27:22
in reply to nevent1q…ka68
Author Public Key
npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6mPublished at
2025-01-10 18:27:22Event JSON
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