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It's a popularising article and the research seems to have been done mostly with physicists and probably a biography of Hilbert. Don't expect too much of this kind of journalism.
Everything you are bringing up concerns, it seems, a vector space of two complex dimensions spanned by the P function for the lattice of Gaussian integers, and the constant function. A function in that space can be identified by the leading term of its Laurent expansion at 0, and a single value. Since P vanishes at (1 + i)/2 for this lattice, the value there tells you the constant part. Your question concerns the action of a form of a kind of "complex conjugation" of order 2 (linear over the real numbers) on what is a real vector space of dimension four. Linear algebra.
It is a more interesting question, and one that can be answered given Weil's collected papers, whether Weil published anything about it. I recall that he did some road-testing of the sheaf idea around maybe 1947, and this was probably unpublished material/Bourbaki early drafting.
The framing of the issue matters. If TVS is taken to be a theory supporting rigorous quantum field theory, then pronouncing it dead is obviously premature. I was implicitly suggesting that the axiomatic approach, and consideration of the circle of ideas introduced by Laurent Schwartz, might reasonably explain the content of the statement. It doesn't mean, for example, that Fréchet algebras will never have a good theory.
I don't know why you need this, but you could try thinking of the ideles as a hyperbola in the square of the adeles. In a local sense it is not very hard to see the relationshop of the measures.
