There are many fascinating conjectures relating arithmetic and special values of zeta functions. The first project (with Milne) is the construction of a triangulated category of integral motives over a finite field with applications to the Lichtenbaum-Milne conjecture on special values. The second project is to prove formulas for special values at half-integers of zeta functions; this is a completely unexplored area. Recent work of the PI has led to a formula for the special value at half using supersingular elliptic curves. The third project involves specific questions about the Weil-group of number fields and class field theory.
The study of special values of zeta functions was pioneered by Euler; it is a subject of very high contemporary interest and research. It involves geometry, algebra and arithmetic in a very sophisticated way. It leads to beautiful identities between discrete (integers) and continuous (transcendental numbers like pi). These arise even in the study of Feynman diagrams in quantum field theory. There are many applications in coding theory, physics, and cryptography. There are many challenging problems for undergraduate and graduate students. The interdisciplinary nature will help broad dissemination and enhance scientific and technological understanding.