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Weil conjectures with a difference


The classical Hassle-Weil zeta function of a variety defined over a finite field contains all the number-theoretic information about the variety because it encodes the number of rational points of the variety over all finite fields extending the field of definition. Weil conjectures, proved in full by Deigned, state that the zeta function, which is a priori just a formal power series, is in fact a rational function and that its form is governed by the geometry of the variety. Trying to understand how ' geometric' this zeta function really is, we might ask whiter it is ' motivic' does there exist a rational function with coefficients in the ring of isomorphism classes of varieties (Grothendieck ring) or some other ring of motivic/geometric nature wich specialise to the usual Weil-Weil zeta function? At the heart of the co homological proof of the rationality of the classical Weil-Weil zeta function lays the Escheats fixed point formula applied to the Freebies auto orphism. I propose a Model Theoretic context of difference fields (fields with a distinguished auto orphism) and difference varieties, where the Escheats trace formalism can be mimicked with the distinguished auto orphism playing the role of the Freebies. In this framework I introduce a difference analogue of the zeta function. My main objective is proving the rationality of the difference zeta function. This would entail a variant of the motives nature of the Weil-Weil zeta function, because the difference zeta would specialise to the usual zeta function over finite fields GF (q) for almost all prime powers q. This development would be of great importance in Algebraic Geometry and Number Theory. Upon successful completion of the rationality part, difference analogues of other statements of the Weil conjectures will be investigated. To achive this goal I propose to develop intersection theory and étale cohomology theory for difference varieties and schemes.

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