# Functional equations of zeta functions over global fields

The functional equations for Dedekind zeta functions (zeta functions attached to rings of integers in algebraic number fields) come from functional equations of theta functions like $\sum_{n \in \mathbb{Z}} q^{n^2}$, where $q = e^{2 \pi i z}$, which in turn come from the Poisson summation formula for lattices in Euclidean spaces.

The functional equations for zeta functions attached to curves over finite fields come from the Riemann Roch theorem and Serre Duality (see, e.g. Chapter 2 of Sam Raskin's write-up).

Rings of integers of algebraic number fields and curves over finite fields are analogous, (see, e.g. Jordan Ellenberg's discussion of Weil's three columns).

Is there a known uniform proof of the functional equations that covers both cases?

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Tate's thesis... –  Daniel Litt Feb 20 '13 at 19:57
This reference derives the Riemann-Roch theorem from a Poisson summation formula. –  Marc Palm Feb 20 '13 at 20:09

## 1 Answer

Have a look at Bump Automorphic representations. The adelic picture is given there as does Tate's Thesis. I think Bump works in the global field context there. The main difference is that the adelic norm map has discrete image in the function field case, but that doesn't harm much.

Although Tate's proof works as well in characteristic p, he assumes the number field setting right from the start. It is a simple exercise to extent everything to global fields and Tate actually refers to Riemann Roch as an equivariant version of Poisson summation in his thesis. Certainly, also Weil's basic number theory is worth a look.

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