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

5$\begingroup$ 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: emis.kaist.ac.kr/journals/UW/gt/ftp/main/m3/m3hlf.pdf#page=211 $\endgroup$ – Keerthi Madapusi Pera Aug 19 '12 at 22:03

$\begingroup$ Aside: Chrome's inbuilt reader seems to typographically disagree with the linked file. It reads perfectly under Fedora's Document Viewer. $\endgroup$ – Keerthi Madapusi Pera Aug 19 '12 at 22:07

1$\begingroup$ The full volume is here: maths.nott.ac.uk/personal/ibf/vlm/vlm.pdf As you can see from this book, this is a very intricate and beautiful field of study. I wish I knew more about it! $\endgroup$ – Keerthi Madapusi Pera Aug 19 '12 at 22:09

$\begingroup$ @Keerthi Off topic, I know, but the bug in Chrome's native pdf viewer is longstanding. Please visit the Chromium issue log code.google.com/p/chromium/issues/detail?id=80996 and let the dev team know if you have experienced this problem. $\endgroup$ – David Roberts Aug 20 '12 at 1:06
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.