On Dynamical Systems and Their Possible Significance for Arithmetic Geometry

Deninger, Christopher

doi:10.1007/978-1-4612-1314-7_3

Cited by 15 publications

(17 citation statements)

References 13 publications

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“…More recently, Deninger ([70]' [71], [72], [73], [74], [75]) developed ideas concerning the construction of geometrical frameworks which would allow us to interpret Weil's explicit formula for the Riemann zeta fUIlction as a Lefschetz fixed point formula for a certain flow. In fact, Weil's explicit formula for a global field k ( [300]) can be written in the form…”

Section: Uniform Descriptions Of the Divisors Of Zeta Functionsmentioning

confidence: 99%

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Cohomological Theory of Dynamical Zeta Functions

Juhl¹

2001

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Section: Uniform Descriptions Of the Divisors Of Zeta Functionsmentioning

confidence: 99%

“…Finally, we note that independent of all these developments Deninger suggested a wealth of ideas concerning a cohomological approach to arithmetical zeta functions and the interpretation of their explicit formulas as Lefschetz formulas ( [70], [71], [72], [73], [74], [75]). …”

Section: The Equation D-w = () In the Twisted Casementioning

confidence: 99%

Cohomological Theory of Dynamical Zeta Functions

Juhl¹

2001

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“…The latter one should then play a role similar to l-adic cohomology in finite characteristics. In particular, Deninger was able to interpret foliated Lefschetz trace-formulas [1,8,15,16,27,30] as analogues of the explicit formulas of algebraic number theory. Here finite primes of a number field should correspond to closed orbits of φ and infinite primes should correspond to fixed points lying in compact leaves.…”

Section: Introductionmentioning

confidence: 99%

A foliated analogue of one- and two-dimensional Arakelov theory

Kopei

2011

Abh. Math. Semin. Univ. Hambg.

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The analogy between number fields and Riemann surfaces was an important source of motivation for mathematicians in the last century. We improve and extend this analogy by substituting Riemann surfaces with certain foliations by Riemann surfaces. In particular we show that coverings of these foliations lead to formulas having the same structure as formulas describing number field extensions. We also study higher dimensional foliations which have properties analogous to arithmetic surfaces. This provides more evidence for a conjecture of Deninger.

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“…Interested readers can consult with [33]. There are many speculative relations of noncommutative compactifications with some structures in number theory, dynamical systems, and physics (e.g., with [3,7,9,10,24,25]). We hope to address some of them in the future.…”

Section: Introductionmentioning

confidence: 99%