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Mathematics of Computation | 1997

Generators and irreducible polynomials over finite fields

Daqing Wan

Weils character sum estimate is used to study the problem of constructing generators for the multiplicative group of a finite field. An application to the distribution of irreducible polynomials is given, which confirms an asymptotic version of a conjecture of Hansen-Mullen.


Annals of Mathematics | 1993

Newton polygons of zeta functions and L functions

Daqing Wan

In this article we give a systematic treatment of Newton polygons of exponential sums. The Newton polygon is a nice way to describe p-adic values of the zeroes or poles of zeta functions and L functions. Our main objective is to show that the Adolphson-Sperber conjecture 12], which asserts that under a simple condition the generic Newton polygon of L functions coincides with its lower bound, is false in its full form, but true in a slightly weaker form. We also show that the full form is true in various important special cases. For example, we show that for a generic projective hypersurface of degree d, the Newton polygon of the interesting part of the zeta function coincides with its lower bound (the Hodge polygon). This gives a p-adic proof of a recent theorem of Illusie, conjectured by Dwork and Mazur. For more examples, let us consider the family of affine hypersurfaces of degree d or the family of affine hypersurfaces defined by polynomials f(xi, . . ., x7n) of degree di with respect to xi (1 < i < n), where the di are fixed positive integers. Then, for all large prime numbers p, the generic Newton polygon for the zeta functions of each of the two families of hypersurfaces coincides with its lower bound. We obtain our main results, namely several decomposition theorems, using certain maximizing functions from linear programming. Our work suggests a possible connection between Newton polygons and the resolution of singularities of toric varieties. Let p be a prime, q = pa, and let Fq be the finite field of q elements and Fqm its extension of degree m. Fix a nontrivial additive character qP of Fp. For any Laurent polynomial f(xi, . . ., xn) E Fq[xl, xj1,.. . ., x, xi1] we form


Annals of Mathematics | 1999

Dwork's conjecture on unit root zeta functions

Daqing Wan

Annals of Mathematics, 150 (1999), 867–927 arXiv:math/9911270v1 [math.NT] 1 Nov 1999 Dwork’s conjecture on unit root zeta functions By Daqing Wan* 1. Introduction 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 Gouvˆea-Mazur conjecture about dimension variation of classical and p-adic modular forms. We shall introduce another *This work was partially supported by NSF. The author wishes to thank P. Deligne and N. Katz for valuable discussions at one stage of this work during 1993 and 1994. The author would also like to thank S. Sperber for his careful reading of the manuscript and for his many detailed comments which led to a significant improvement of the exposition.


Journal of the American Mathematical Society | 2000

Higher rank case of Dwork's conjecture

Daqing Wan

JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Xxxx XXXX, Pages 000–000 S 0894-0347(XX)0000-0 arXiv:math/0005309v1 [math.NT] 9 May 2000 HIGHER RANK CASE OF DWORK’S CONJECTURE DAQING WAN 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 σ-modules, see section 2 for precise definitions of the various notions used in this introduction. Our main result is the following theorem. Theorem 1.1. Let X be a smooth affine variety defined over a finite field F q of characteristic p > 0. Let (M, φ) be a finite rank overconvergent σ-module over X/F q . Then, for each rational number s, the pure slope s L-function L s (φ, T ) attached to φ is p-adic meromorphic everywhere. The proof of this theorem will be completed in two papers. In the present higher rank paper, we introduce a reduction approach which reduces Theorem 1.1 to the special case when the slope s (s = 0) part of φ has rank one and the base space X is the simplest affine space A n . This part is essentially algebraic. It depends on Monsky’s trace formula, Grothendieck’s specialization theorem, the Hodge-Newton decomposition and Katz’s isogeny theorem. In our next paper [23], we will handle the rank one case over the affine space A n . The rank one case is very much analytic in nature and forces us to work in a more difficult infinite rank setting, generalizing and improving the limiting approach introduced in [19]. Dwork’s conjecture grew out of his attempt to understand the p-adic analytic variation of the pure pieces of the zeta function of a variety when the variety moves through an algebraic family. To give an important geometric example, let us con- sider the case that f : Y → X is a smooth and proper morphism over F q with a smooth and proper lifting to characteristic zero. Berthelot’s result [1] says that the relative crystalline cohomology R i f crys,∗ Z p modulo torsion is an overconvergent F-crystal M i over X. Applying Theorem 1.1, we conclude that the pure L-functions arising from these geometric overconvergent F-crystals M i are p-adic meromorphic. In particular, this implies the existence of an exact p-adic formula for geometric p-adic character sums and a suitable p-adic equi-distribution theorem for the roots of zeta functions. For more detailed arithmetic motivations and further open prob- lems, see the expository papers [20][21]. 1991 Mathematics Subject Classification. Primary 11G40, 11S40; Secondary 11M41, 14G15. Key words and phrases. L-functions, p-adic meromorphic continuation, σ-modules. This work was partially supported by NSF. c American Mathematical Society


Journal of the American Mathematical Society | 2000

Rank one case of Dwork's conjecture

Daqing Wan

JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Xxxx XXXX, Pages 000–000 S 0894-0347(XX)0000-0 arXiv:math/0005308v1 [math.NT] 9 May 2000 RANK ONE CASE OF DWORK’S CONJECTURE DAQING WAN 1. Introduction In the higher rank paper [17], we reduced Dwork’s conjecture from higher rank case over any smooth affine variety X to the rank one case over the simplest affine space A n . In the present paper, we finish our proof by proving the rank one case of Dwork’s conjecture over the affine space A n , which is called the key lemma in [17]. The key lemma had already been proved in [16] in the special case when the Frobenius lifting σ is the simplest q-th power map σ(x) = x q . Thus, the aim of the present paper is to treat the general Frobenius lifting case. Our method here is an improvement of the limiting method in [16]. It allows us to move one step further and obtain some explicit information about the zeros and poles of the unit root L-function. As in [16], to handle the rank one case, we are forced to work in the more difficult infinite rank setting, see section 2 for precise definitions of the various basic infinite rank notions. Let F q denote the finite field of characteristic p > 0. Our main result of this paper is the following theorem. Theorem 1.1. Let φ be a nuclear overconvergent σ-module over the affine n-space A n /F q , ordinary at the slope zero side. Let φ unit be the unit root (slope zero) part of φ. Assume that φ unit has rank one. Let ψ be another nuclear overconvergent σ-module over A n /F q . Then for each integer k, the L-function L(ψ ⊗ φ ⊗k unit , T ) is p-adic meromorphic. Furthermore, the family L(ψ ⊗ φ ⊗k T of L-functions unit parametrized by integers k in each residue class modulo (q − 1) is a strong family of meromorphic functions with respect to the p-adic topology of k. A finite rank σ-module is automatically nuclear. Thus, Theorem 1.1 includes the key lemma of [17] over A n as a special case. The basic ideas of the present paper are the same as the limiting approach in [16]. The details are, however, quite different. In the simplest q-th power Frobenius lifting case, one has the fundamental Dwork trace formula available, which is completely explicit for uniform estimates. This makes it easy to extend the Dwork trace formula to infinite rank setting. It also makes it possible to see the various analytic subtleties involved in a concrete case. As a result, we were able to prove analytically optimal results in [16]. For a general Frobenius lifting (even over the simplest affine n-space A n as we shall work in this paper), one has to use the much more difficult Monsky trace formula which is a generalization of Dwork’s trace formula. Thus, the first task of this paper is to extend the Monsky trace formula to infinite rank setting and to make it sufficiently 1991 Mathematics Subject Classification. Primary 11G40, 11S40; Secondary 11M41, 14G15. Key words and phrases. L-functions, Fredholm determinants, p-adic meromorphic continua- tion, nuclear σ-modules and Banach modules. This work was partially supported by NSF. c American Mathematical Society


Journal of the American Mathematical Society | 1996

-functions of -sheaves and Drinfeld modules

Y. Taguchi; Daqing Wan

In this paper, we apply Dworks p-adic methods to study the meromorphic continuation and rationality of various L-functions arising from 7r-adic Galois representations, Drinfeld modules and (p-sheaves. As a consequence, we prove some conjectures of Goss about the rationality of the local L-function and the meromorphic continuation of the global L-function attached to a Drinfeld module. Let Fq be a finite field of q elements with characteristic p. Let 7r be a prime of the polynomial ring A = Fq [t]. Let A, be the completion of the ring A at 7r. This is an analogue of the classical ring 7p of p-adic integers. Let X be an irreducible algebraic variety defined over Fq and let 7r, (X) be the arithmetic fundamental group of X/IFq with respect to some base point. The group 7r, (X) may be regarded as the Galois groiip of a separable closure of the function field of X/Fq modulo the inertia groups at the closed points of X/lq. Suppose now that we are given a continuous 7r-adic representation


Finite Fields and Their Applications | 2008

On the subset sum problem over finite fields

Jiyou Li; Daqing Wan

The subset sum problem over finite fields is a well-known NP-complete problem. It arises naturally from decoding generalized Reed-Solomon codes. In this paper, we study the number of solutions of the subset sum problem from a mathematical point of view. In several interesting cases, we obtain explicit or asymptotic formulas for the solution number. As a consequence, we obtain some results on the decoding problem of Reed-Solomon codes.


SIAM Journal on Computing | 2007

On the List and Bounded Distance Decodability of Reed-Solomon Codes

Qi Cheng; Daqing Wan

For an error-correcting code and a distance bound, the list decoding problem is to compute all the codewords within a given distance to a received message. The bounded distance decoding problem is to find one codeword if there is at least one codeword within the given distance, or to output the empty set if there is not. Obviously the bounded distance decoding problem is not as hard as the list decoding problem. For a Reed-Solomon code [n, k]/sup q/, a simple counting argument shows that for any integer 0 < g < n, there exists at least one Hamming ball of radius n - g, which contains at least (/sup n//sub g/)/q/sup g-k/ many codewords. Let g(n, k, q) be the smallest positive integer g such that (/sup n//sub g/)/q/sup g-k/ < 1. One knows that k /spl les/ g(n, k, q) /spl les/ /spl radic/nk /spl les/ n. For the distance bound up to n- /spl radic/nk;, it is well known that both the list and bounded distance decoding can be solved efficiently. For the distance bound between n - /spl radic/nk and n - g(n, k, q), we do not know whether the Reed-Solomon code is list, or bounded distance decodable, nor do we know whether there are polynomially many codewords in all balls of the radius. It is generally believed that the answers to both questions are no. There are public key cryptosystems proposed recently, whose security is based on the assumptions. In this paper, we prove: (1) List decoding can not be done for radius n - g(n, k: q) or larger, otherwise the discrete logarithm over F/sub qg(m, k, q)-k/ is easy. (2) Let h and g be positive integers satisfying q /spl ges/ max(g/sup 2/, (h-l)/sup 2+/spl epsiv//) and g /spl ges/ (4//spl epsiv/ + 2)(h + 1) for a constant /spl epsiv/ > 0. We show that the discrete logarithm problem over F/sub qh/ can be efficiently reduced by a randomized algorithm to the bounded distance decoding problem of the Reed-Solomon code [q, g - h]/sub q/ with radius q - g. These results show that the decoding problems for the Reed-Solomon code are at least as hard as the discrete logarithm problem over finite fields. The main tools to obtain these results are an interesting connection between the problem of list-decoding of Reed-Solomon code and the problem of discrete logarithm over finite fields, and a generalization of Katzs theorem on representations of elements in an extension finite field by products of distinct linear factors.


Lms Journal of Computation and Mathematics | 2002

Computing Zeta Functions of Artin–schreier Curves over Finite Fields

Alan G. B. Lauder; Daqing Wan

The authors present a practical polynomial-time algorithm for computing the zeta function of certain Artin–Schreier curves over finite fields. This yields a method for computing the order of the Jacobian of an elliptic curve in characteristic 2, and more generally, any hyperelliptic curve in characteristic 2 whose affine equation is of a particular form. The algorithm is based upon an efficient reduction method for the Dwork cohomology of one-variable exponential sums.


Crelle's Journal | 2005

L-functions for symmetric products of Kloosterman sums

Lei Fu; Daqing Wan

Abstract The classical Kloosterman sums give rise to a Galois representation of the function field unramified outside 0 and ∞. We study the local monodromy of this representation at ∞ using l -adic method based on the work of Deligne and Katz. As an application, we determine the degrees and the bad factors of the L-functions of the symmetric products of the above representation. Our results generalize some results of Robba obtained through p -adic method.

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Qi Cheng

University of Oklahoma

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Jiyou Li

Shanghai Jiao Tong University

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Jun Zhang

Capital Normal University

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Liang Xiao

University of Connecticut

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