A Quick Introduction To Dwork’s Conjecture

This paper is an expanded version of my lecture delivered at the 97 Seattle summer research conference on finite fields. It gives a quick exposition of Dwork’s conjecture about p-adic meromorphic continuation of his unit root L-function arising from a family of algebraic varieties defined over a finite field of characteristic p. As a simple illustration, we discuss the classical example of the universal family of elliptic curves where the conjecture is already known to be true and where the conjecture is closely related to arithmetic of modular forms such as the Gouvêa-Mazur conjecture. Special attention is given to questions related to the p-adic absolute values of the unit root L-function. In particular, it is observed that an average version of a suitable p-adic Riemann hypothesis is true for the elliptic family. Following a suggestion of Mike Fried, I also include a section describing some of my personal interactions with Dwork. This extra section serves as a dedication to the memory of Dwork who actively attended the conference and died nine months later. 1991 Mathematics Subject Classification: 11G40, 11G20, 14G15. 1. One version of Dwork’s conjecture This section gives a quick reformulation of Dwork’s conjecture in the general case. There are several different but essentially equivalent languages, such as p-adic Galois representations, p-adic ètale sheaves and unit root F-crystals, that could be used to describe Dwork’s conjecture. To be compatible with the general theme of the conference, I will use the language of p-adic representations and Galois groups. This provides a short although not the simplest reformulation of Dwork’s conjecture. In one sentence, the conjecture simply says that if ρ is a continuous p-adic Galois representation coming from algebraic geometry over a finite field of characteristic p, then the L-function L(ρ, T ) is p-adic meromorphic. We now make this a little more precise. Let q be a power of a fixed prime number p and let Fq be the finite field of q elements. For a geometrically connected algebraic variety X defined over Fq, let πarith 1 (X) denote the arithmetic fundamental group of X . This is a profinite group. More precisely, πarith 1 (X) is the profinite completion of the finite Galois groups of pointed finite unramified Galois coverings of X . If X is integral and normal with function field Fq(X), then π arith 1 (X) is simply the profinite Galois group Gal(Fq(X) /Fq(X)) modulo the normal subgroup generated by the inertia c ©1998 American Mathematical Society