Pure L-Functions from Algebraic Geometry over Finite Fields

This survey gives a concrete and intuitively self-contained introduction to the theory of pure L-functions arising from a family of algebraic varieties defined over a finite field of characteristic p. The standard fundamental questions in any theory of L-functions include the meromorphic continuation, functional equation, Riemann hypothesis (RH for short), order of zeros at special points and their special values. Our emphasis here will be on the meromorphic continuation and the RH. These two questions can be described in a general setup without introducing highly technical terms. The construction of the pure L-function depends on the choice of an absolute value of the rational number field Q. In the case that the absolute value is the complex or ℓ-adic absolute value (ℓ ≠ p), the rationality of the pure L-function requires the full strength of the ℓ-adic cohomology including Deligne’s main theorem on the Weil conjectures. In the case that the absolute value is the p-adic absolute value, the pure L-function is no longer rational but conjectured by Dwork to be p-adic meromorphic. This conjecture goes beyond all existing p-adic cohomology theories. Its truth opens up several new directions including a possible p-adic RH for such pure L-functions. The guiding principle of our exposition in this paper is to describe all theorems and problems as simple as possible, directly in terms of zeta functions and L-functions without using cohomological terms. In the case that this is not easy to do so, we simply give an intuitive discussion and try to convey a little feeling. Along the way, a number of natural open questions and conjectures are raised, some of them may be accessible to certain extent but others may be somewhat wild due to the lack of sufficient evidences.