Dwork ’ s conjecture on unit root zeta functions
نویسندگان
چکیده
In this article, we introduce a systematic new method to investigate the conjectural p-adic meromorphic continuation of Professor Bernard Dwork’s unit root zeta function attached to an ordinary family of algebraic varieties defined over a finite field of characteristic p. After his pioneer p-adic investigation of the Weil conjectures on the zeta function of an algebraic variety over a finite field, Dwork went on to study the p-adic analytic variation of a family of such zeta functions when the variety moves through an algebraic family. In the course of doing so, he was led to a new zeta function called the unit root zeta function, which goes beyond the reach of the existing theory. He conjectured [8] that such a unit root zeta function is p-adic meromorphic everywhere. These unit root zeta functions contain important arithmetic information about a family of algebraic varieties. They are truly p-adic in nature and are transcendental functions, sometimes seeming quite mysterious. In fact, no single “nontrivial” example has been proved to be true about this conjecture, other than the “trivial” overconvergent (or ∞ log-convergent) case for which Dwork’s classical p-adic theory already applies; see [6]–[11], [26] for various attempts. In this article, we introduce a systematic new method to study such unit root zeta functions. Our method can be used to prove the conjecture in the case when the involved unit root F-crystal has rank one. In particular, this settles the first “nontrivial” case, the rank one unit root F-crystal coming from the family of higher dimensional Kloosterman sums. Our method further allows us to understand reasonably well about analytic variation of an arithmetic family of such rank one unit root zeta functions, motivated by the Gouvêa-Mazur conjecture about dimension variation of classical and p-adic modular forms. We shall introduce another
منابع مشابه
RANK ONE CASE OF DWORK ' S CONJECTURE 3 Theorem 1
1. Introduction In the higher rank paper 17], we reduced Dwork's conjecture from higher rank case over any smooth aane variety X to the rank one case over the simplest aane space A n. In the present paper, we nish our proof by proving the rank one case of Dwork's conjecture over the aane space A n , which is called the key lemma in 17]. The key lemma had already been proved in 16] in the specia...
متن کاملHigher Rank Case of Dwork
Dedicated to the memory of Bernard Dwork 1. Introduction In this series of two papers, we prove the p-adic meromorphic continuation of the pure slope L-functions arising from the slope decomposition of an overconvergent F-crystal, as conjectured by Dwork 6]. More precisely, we prove a suitable extension of Dwork's conjecture in our more general setting of-modules, see section 2 for precise deen...
متن کاملArtin’s Conjecture and Elliptic Analogues
A well-known conjecture of Emil Artin predicts that every natural number a unequal to unity or a perfect square is a primitive root (mod p) for infinitely many primes p. In 1967, C. Hooley [Ho1] proved this conjecture assuming the generalized Riemann hypothesis (GRH) for the Dedekind zeta functions of certain Kummer extensions. In fact, he establishes an asymptotic formula for the number of suc...
متن کاملT-ADIC L-FUNCTIONS OF p-ADIC EXPONENTIAL SUMS
The T -adic deformation of p-adic exponential sums is defined. It interpolates all classical p-power order exponential sums over a finite field. Its generating L-function is a T -adic entire function. We give a lower bound for its T -adic Newton polygon and show that the lower bound is often sharp. We also study the variation of this L-function in an algebraic family, in particular, the T -adic...
متن کامل