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


From Connes p.7: "One can then apply the ideas of algebraic geometry, first developed over C, to the geometry of the curve...and obtain a geometric interpretation of the basic properties of the zeta function of k; the dictionary contains in particular the following lines:

 

spectral interpretation of the zeros Eigenvalues of Frobenius action on l-adic cohomology
functional equation (of Riemann zeta function) Riemann-Roch theorem (Poincare duality)
Explicit formulae of number theory Lefschetz formula for the Frobenius (?)
Riemann Hypothesis Castenuovo positivity" (something analogous to Weil's positivity criterion for the RH)

There's a similarity with Deninger's approach here in that a link is suggested between the Riemann-Weil Explicit Formula and Lefschetz formulae

In a personal communication, A. Juhl wrote "The ideal case would be to understand the explicit formulas (as of the RZF) as Lefschetz formulas. And as you know, this was alos the idea of Connes' recent efforts."

On p.28 (Connes?) "...in particular the expected trace formula is not a semi-classical formula but a Lefschetz formula in the spirit of [Atiyah-Bott]"

 

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