Warning! This set of interlinked pages is very much under construction. If you are already familiar with the subject matter, I would appreciate any help you might provide. If you are seeking information on the subject matter, I would be interested to hear what you would ultimately like this page to contain. Thank you.

 Gutzwiller Trace Formula ___  [link]   ___ Selberg Trace Formula | | [link] | | | | [link] | | Riemann's zeta function ___  [link]   ___ Riemann-Weil Explicit Formula | | [link] | | | | [link] | | Connes'(hypothetical) trace formula ___  [link]   ___ (Deninger's) Lefschetz Trace Formulae

On p.1 of C. Deninger's "Lefschetz trace formulas and explicit formulae in analytic number theory" we read

"...In particular, we were lead to a Lefschetz trace formula...If we specialized to the motive M(chi) of an algebraic Hecke character chi which for simplicity we suppose to be unitary, [the LTF] implies the following relation

delta(Phi(0)+Phi)(1)) - Sum rho Phi(rho) = Sumscript p(Phi;Chi)

This looks very similar to the version of the Riemann-Weil Explicit Formula which Deninger gives in the first paragraph ("It takes the following form if chi is unitary and normalized.")

delta(Phi(0)+Phi(1)) - SumrhoPhi(rho) = alpha F(0) + Sumscript p Wscript p (F;chi)

Here F is a rapidly decreasing function on the reals, Phi is its Fourier transform, rho are nontrivial zeros of L(chi, s), delta is 0 or 1 (depending on the triviality of chi as a character), script p are places of the number field K.

Wscript p(F;chi) are local terms. The field K completes at place script p locally as Kscript p. alpha is a constant depending on K and chi, but not on F.

So to summarise: Deninger's investigation of global arithmetic cohomology theory lead to a Lefschetz trace formula. By then specialising to the motive of an algebraic Hecke character chi, he was lead to a definition of local traces which produces a relation analogous to the Riemann-Weil explicit formula. This suggests that the local terms Wscript p in the explicit formula should be interpreted as traces. In particular, Deninger argues that they are traces of an operator on certain infinite dimensional spaces script Fscript p(M(chi)).

To summarise the summary: A particular Lefschetz trace formula, appropriately specialized, produces a relation analogous to the Riemann-Weil Explicit Formula.

I have yet to grasp how Deninger then precedes from this observation.

Andreas Juhl wrote to me "The ideal case would be to understand the explicit formulas (as of the Riemann zeta function) as Lefschetz formulas. And as you know, this was also the idea of Connes' recent efforts."

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