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Poisson summation formula is widely used in many parts of the litterature, its classical formulation for sums over integers as well as its adelic version. What is its corresponding form for more general number fields? I was expecting to find it used in proofs of the functional equation for Dedekind L-functions attached to a number field, but I only found Poisson formulas for elements summed over an ideal rather than sums over ideals.

Is there anything of the form, for $K$ a number field, $E$ its ring of integer, and $f$ a suitable real valued function, $$\sum_{a \subset E} f(a) = C \sum_{a' \in E'} \hat{f}(a')$$

where the sum runs over integer ideals $a$ of $E$? What are the notions of dual and Fourier transforms here? Should we rather have something with norms of ideals instead of ideals? That is to say, is $N$ is the norm of the number field $K$,

$$\sum_{a \subset E} f(N(a)) = C \sum_{a' \in E'} \hat{f}(N(a'))$$

Thanks in advance, I do not find anything suitable in the litterature.

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    $\begingroup$ Maybe this?: londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/… E. Friedman, N-P Skoruppa, "A Poisson Summation Formula for Extensions of Number Fields" $\endgroup$ – Nemo Nov 7 '18 at 14:34
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    $\begingroup$ Is there a reason why you object to the adelic version, as in Iwasawa-Tate theory? $\endgroup$ – paul garrett Nov 7 '18 at 16:08
  • $\begingroup$ @paulgarrett I would like a summation formula mostly of the second form, as a sum over ideals more than a sum over elements in the ideal. All the references I find, like the link Nemo provided, or the adelic versions, give a sum over elements. $\endgroup$ – Automorphic Nov 8 '18 at 3:43
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    $\begingroup$ You can in some cases cut your sum over classes modulo a given ideal and sum over the elements of this ideal, in order to use Friedman-Skoruppa formula. However this still does not give you a formula for sums over elements, I confess I am widely interested in such a result, could it come from more general results of abelian harmonic analysis of groups, along the same lines then the usual Poisson formulas? $\endgroup$ – Desiderius Severus Nov 22 '18 at 1:56
  • $\begingroup$ You can also apply Friedman-Skoruppa in the case of principal ideals, since it mainly reduces to elements modulo the units, therefore why not trying to cut by the class group? $\endgroup$ – Desiderius Severus Nov 22 '18 at 2:02
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Perhaps Section 3 of the paper "A quadratic large sieve over number fields" by Goldmakher and Louvel (https://arxiv.org/abs/1112.1642) might be of some assistance.

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