The aim of the project is to define and study a new Grothendieck topology (the "Weil-etale topology") in connection with special values of Hasse-Weil L-functions of varieties over number fields (the "Tamagawa Number conjecture"). The idea of such a topology, as well as a first definition, is due to Lichtenbaum who also shows the expected relationship to the L-function in the simplest case of a zero-dimensional variety. However, truncation of the complex computing Weil-etale cohomology is necessary because in even degrees greater than two the cohomology group is nonvanishing and of infinite rank. One goal of the project would be to redefine the Weil etale topos of the ring of integers of a number field so that it has bounded cohomology as well as a number of other properties such as a map to the etale topos and to the classifying topos of the real numbers. A second goal is to find a definition of the Weil etale topos for higher dimensional arithmetic schemes of characteristic zero (the case of finite characteristic is already well understood due to work of Lichtenbaum and Geisser). A third, more speculative goal would be to reprove the analytic class number formula using the Weil-etale topology with the aim of generalizing it to Artin L-functions. One interesting aspect of this project is the interaction of topos theory and logic with more classical and well established number theory, such as the analytic class number formula.
Diophantine equations and their solutions have occupied the imagination of mathematically interested people for centuries but they also have found real world applications in coding theory. Modern mathematics provides a bewildering array of techniques, ranging from the disarmingly simple to the highly abstract, to study diophantine equations. The geometric perspective, viewing the solution set as a "space", has been particularly useful. The project aims to contribute to this line of thought by defining and studying a new cohomology theory for diophantine equations.
