  I give a new derivation of the Explicit Formula for an arbitrary number field
and abelian Dirichlet-Hecke character, which treats all primes in exactly the
same way, whether they are discrete or archimedean, and also ramified or not.
This is followed with a local study of a Hilbert space operator, the
``conductor operator'', which is expressed as H = log(|x|) + log(|y|) (where x
and y are Fourier dual variables on a nu-adic completion of the number field).
I also study the commutator operator K = i[log(|y|),log(|x|)] (which shares
with H the property of complete dilation invariance, and turns out to be
bounded), as well as the higher commutator operators. The generalized
eigenvalues of these operators are given by the derivatives on the critical
line of the Tate-Gel'fand-Graev Gamma function, which itself is in fact closely
related to the additive Fourier Transform viewed in multiplicative terms. This
spectral analysis is thus a natural continuation to Tate's Thesis in its local
aspects. (combines my earlier papers math/9809119, math/9811040, math/9812012,
one result added, new references)
