MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Tate showed that the functional equation for zeta functions of number fields can be proven with fourier-analytic methods on the adele ring. Can the same be done for zeta functions of varieties over finite fields?

share|cite|improve this question
Tate's methods depend heavily on the study of local fields, and so apply for example to function fields in one variable over finite fields. To generalize this, lots of people have been thinking about higher dimensional local fields, including K. Kato, Fesenko, Parshin, et al. You might find this paper useful: – Keerthi Madapusi Pera Aug 19 '12 at 22:03
Aside: Chrome's in-built reader seems to typographically disagree with the linked file. It reads perfectly under Fedora's Document Viewer. – Keerthi Madapusi Pera Aug 19 '12 at 22:07
The full volume is here: As you can see from this book, this is a very intricate and beautiful field of study. I wish I knew more about it! – Keerthi Madapusi Pera Aug 19 '12 at 22:09
@Keerthi Off topic, I know, but the bug in Chrome's native pdf viewer is long-standing. Please visit the Chromium issue log and let the dev team know if you have experienced this problem. – David Roberts Aug 20 '12 at 1:06
up vote 5 down vote accepted

This is done in Chapter 7.3 of Ramakrishnan and Valenza's Fourier Analysis on Number Fields (GTM 186) which, despite the title, describes in some details also the situation over function fields in one variable (so, for curves over finite fields).

If you want to consider curves over some global field of positive characteristic (so, not a curve $C$ over $\mathbb{F}_p$, but may be a curve over the field of rational functions of the previous $C$), Keerthi's comment applies: in particular, a recent work by Fesenko (Analysis on Arithmetic Schemes 2) treats the global functional equation.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.