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GH from MO
GH from MO
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The trace form is a perfect pairing in the sense of harmonic analysis, i.e. it identifies $\Phi_n$ with its portyaginPontryagin dual, see Weil's "Basic Number Theory" or Tate's thesis. Therefore, your statement follows from the Pontryagin duality, for example Proposition 3.6.1 in the book "Principles of Harmonic Analysis" by Deitmar/Echterhoff.

The trace form is a perfect pairing in the sense of harmonic analysis, i.e. it identifies $\Phi_n$ with its portyagin dual, see Weil's "Basic Number Theory" or Tate's thesis. Therefore, your statement follows from the Pontryagin duality, for example Proposition 3.6.1 in the book "Principles of Harmonic Analysis" by Deitmar/Echterhoff.

The trace form is a perfect pairing in the sense of harmonic analysis, i.e. it identifies $\Phi_n$ with its Pontryagin dual, see Weil's "Basic Number Theory" or Tate's thesis. Therefore, your statement follows from the Pontryagin duality, for example Proposition 3.6.1 in the book "Principles of Harmonic Analysis" by Deitmar/Echterhoff.

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user1688
user1688

The trace form is a perfect pairing in the sense of harmonic analysis, i.e. it identifies $\Phi_n$ with its portyagin dual, see Weil's "Basic Number Theory" or Tate's thesis. Therefore, your statement follows from the Pontryagin duality, for example Proposition 3.6.1 in the book "Principles of Harmonic Analysis" by Deitmar/Echterhoff.