Curves Over Finite Fields and Deligne's Conjectures

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deJong9970049We propose to study certain groups that are associated with algebraiccurves over finite fields. The groups in question can be viewed as theautomorphism groups of unramified coverings of a fixed algebraic curve over a finite field which is also fixed throughout the discussion. As we let vary the coverings we obtain a system of groups, which iscalled the algebraic fundamental group of the curve; this concept was introduced by Grothendieck. The algebraic fundamental group mixes ina fascinating way the arithmetic of the finite field and the geometryof the curve in question. In particular, one can associate to apoint on the curve a certain conjugacy class in this group, consistingof the so-called Frobenius elements. The question which we would liketo answer is: What is the (relative) position of these Frobenius elementsin the group? The general area of research of the project is Arithmetic AlgebraicGeometry. Roughly speaking, the Algebra refers to the fact that wework mainly with polynomials as far as functions are concerned.We define geometric objects byequating to zero a few of these polynomials. Such an object is called an algebraic set or a variety. It turns out that there is a surprisingly rich geometry of these objects, especially if we consider equations in higher dimensions and of higher degree. The arithmetical aspect comes into play when we consider only those polynomials which have integers(or rational numbers) as coefficients. These objects have wide-rangingapplications in number theory, geometry, and the theory of data security.