---
tag: QYW79XKJTP
kind: concept
name: Tate's thesis
msc: [11R42, 11R56, 11S40]
sources:
  - url: https://en.wikipedia.org/wiki/Tate%27s_thesis
same_as:
  wikidata: Q7687933
  wikipedia: Tate's thesis
---

## Summary

Tate's thesis is the 1950 doctoral thesis of John Tate, written under Emil Artin.
It proves the meromorphic continuation and the functional equation of the L-functions of [[Hecke characters|R65XKHN4D1]].
The tool is harmonic analysis on the [[adele ring|SEH68BA1QQ]] and on the idele group.
The L-function appears as a zeta integral of a test function against a character over the ideles.
The functional equation then follows from the Poisson summation formula for the discrete subgroup $K$ of $\mathbb{A}_K$.
The method made adeles a standard tool and is a model for the theory of automorphic L-functions.

## Explanations

### L1 human

The [[Riemann zeta function|1XWBY4N0VT]] has a symmetry that relates its values at $s$ and $1 - s$.
Tate found one uniform reason for this symmetry and for its analogues over every number field.
He split the zeta function into local pieces, one for each completion, and studied each piece with Fourier analysis on that completion.
The pieces are glued with the Poisson summation formula, and the symmetry drops out.
For the rational numbers, this gives a new proof of the functional equation of the Riemann zeta function.