Tate's Thesis

Tate’s Thesis used an adelic method to deal with functional equations of different types of L-functions, where the character could be made into Riemann-Zeta L-functions, Dirichlet L-functions and Hecke L-functions. He mainly considered about global and local eigendistributions, getting the basis of Schwartz-Bruhat functions and the corresponding distribution space, where global and local L-factors serves as coefficients of different basis, i.e.

\[z_0(s,\omega) L_v(s,\omega) = z(s,\omega).\]

for local eigendistributions.

Using the Fourier transformation, we get two basis in the one-dim eigendistribution, which induces an functional equation:

\[\widehat{z_0(1-s,\omega^{-1})} = \epsilon_v(s,\omega,\psi) z_0(s,\omega).\]

where \(\psi\) is for the Fourier transformation. \(v\) is a (local) place. Actually we can calculate all \(\epsilon_v\) factors at \(v\) whenever \(\omega,\psi\) have different ramification properties or being archimedean.

However, globally, using Poisson summation and Riemann-Roch theorem, we can also get an global functional equation for \(z(s,\omega)\).

\[z(s,\omega) = \widehat{z(1-s,\omega^{-1})}.\]

Defining the global L-function to be \(\Lambda(s,\omega) = \prod_{v} L_v(s,\omega)\), and combine all coefficients together, we get the final functional equation:

\[\Lambda(s,\omega) = \epsilon(s,\omega)\cdot \Lambda(1-s,\omega^{-1}).\]

Where all equations are on where those functions are defined, and those functions have meromorphic continuation to the whole plane, where reader can refer to:

Chapter of Stephen S. Kudla \(\href{https://platoeinsyu.github.io/assets/pdf/Articles/Tate/TatesThesis.pdf}{\text{Tate's Thesis}}\), Notes by BAPTISTE DEJEAN \(\href{https://platoeinsyu.github.io/assets/pdf/Articles/Tate/DeJean.pdf}{\text{Tate's Thesis}}\).