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Say we have a set $X$ :

  • with a norm (possibly twisted by a character) and an unique factorization $$\zeta_X(s) = \sum_{x \in X} |x|^{-s} = \prod_{P \in X} \frac{1}{1-|P|^{-s}}$$
  • from some additive properties of $X$ we have a functional equation $$\zeta_X(s)= \chi(s) \zeta_X(n-s)$$

$X$ can be many different objects : the integers, a ring of integers of a number field, the effective divisors of a function field, of a curve over a finite field, the Hecke eigenvalues of a modular/automorphic form...

My question : Is something like the Poisson summation formula in the adelic setting what I need for unifying the derivation of a functional equation in all these cases ?

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  • $\begingroup$ similar to mathoverflow.net/questions/122442/… and mathoverflow.net/questions/94165/… $\endgroup$
    – reuns
    Commented Jan 10, 2017 at 2:04
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    $\begingroup$ This question is a bit vague. How about this: for an elliptic curve over a number field, we can define the $L$-function in the way you say, and we can conjecture that there's a functional equation but we can't prove it in general, so in this case it somehow doesn't make sense to ask if there is a Poisson summation formula because nobody knows. So probably we have to restrict to cases where the fnl eqn is known. But then pretty much every case is subsumed by the automorphic forms case by Langlands so what you need is nothing more than the proof of fnl eqn for autom L-fns (Godement-Jacquet). $\endgroup$
    – Kevin Buzzard
    Commented Jan 10, 2017 at 8:37
  • $\begingroup$ @KevinBuzzard I know quite well the usual L-functions, but I discovered the function field ones quite recently, and I have no idea how to link automorphic forms with function fields and Weil's conjecture. I'd like a (conjectural is not a problem) way to unify what properties of $X$ makes the zeta function suitable for a functional equation. $\endgroup$
    – reuns
    Commented Jan 10, 2017 at 9:41
  • $\begingroup$ My answer to the question of how to "unify what properties of $X$ makes the zeta function suitable for a functional equation" is that $X$ has to be motivic in nature, and then the Langlands philosophy says that it is automorphic, and the functional equation is because of Godement-Jacquet, which generalises Tate's thesis, which does indeed use Poisson summation adelically. $\endgroup$
    – Kevin Buzzard
    Commented Jan 10, 2017 at 10:18
  • $\begingroup$ @KevinBuzzard Is there a more intuitive (possibly less general) version of "being motivic" ? What is the key property we want ? $\endgroup$
    – reuns
    Commented Jan 10, 2017 at 21:30

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