Bernard Dwork

On the zeta function of a hypersurface

Abstract : This article is concerned with the further development of the methods of p-adic analysis used in an earlier article to study the zeta function of an algebraic variety defined over a finite field. These methods are applied to the zeta function of a nonsingular hypersurface of degree d in projective n-space of characteristic p defined over the field of q elements.

Detection of a Pulse Superimposed on Fluctuation Noise

Given a known pulse superimposed on fluctuation noise having a known spectrum, we determine the frequency response of that linear device which would give the maximum value for the ratio between peak amplitude of the signal and the root-mean-square of the noise at the output. This result is applied to the case in which the fluctuation noise has a flat spectrum, and it is shown that in that case the optimal network is physically realizable if the pulse differs from zero for only a finite interval of time. The noise-suppressing efficiency of a conventional RC circuit is computed for pulse shapes of practical interest.

On natural radii of

We study the radius of p-adic convergence of power series which represent algebraic functions. We apply thep-adic theory of ordinary linear differential equations to show that the radius of convergence is the natural one, provided the degree of the function is less than p. The study of similar questions for solutions of linear differential equations is indicated. Introduction. It is well known classically that if a power series ( in one variable represents an algebraic function (resp., a solution of an ordinary linear differential equation with analytic coefficients) then the series ( converges at least up to the nearest singularity as given by the coefficients of the polynomial equation (resp., differential equation) satisfied by (. The example of the binomial expansion I x (1 y/ ( P )s s! ( ) shows that this is not the casep-adically. Scott Brown [1] has shown that this p-adic failure to converge in the natural disk cannot occur in the algebraic case if p exceeds a suitable exponential function of n, the degree of {. Following methods developed for the p-adic study of linear differential equations [9] we have given a new proof of Browns result and show furthermore that his exponential function of n may be replaced by n itself. (For a precise statement see Theorem 2.1 below.) Our result may best be explained in relation to Eisensteins theorem [4, p. 327] concerning elements of Q[[x]] which represent algebraic functions. For each prime p of Q we consider the p-adic Gauss valuation of Q(x) (the residue class field is the field Fp(x) of rational functions over the field of p elements). Let K be a finite extension of Q(x) of degree n. Trivially, this p-adic valuation of Q(x) is unramified in K for almost all p. This is the main point of Eisensteins theorem. Trivially again nontame ramification is excluded if p exceeds the degree n of K over Q(x). This is the basis of the present refinement (3.1.3.1) and (3.1.3.2) of Eisensteins theorem, since the correct formulation of Browns theorem is that p-adic failure to converge in Received by the editors October 24, 1977. AMS (MOS) subject classifications (1970). Primary 12B40, 34A25.

p

For clarity of exposition, the underlying motivation is indicated at the end of this article under concluding Remarks 3, rather than at the beginning.For clarity of exposition, the underlying motivation is indicated at the end of this article under concluding Remarks 3, rather than at the beginning.

-adic convergence

Let V be an algebraic variety, defined over GF[q]. We recall the definition of the zeta function of V

On the Rationality of Zeta Functions and L -Series

Z(V,t) = \exp (\sum\limits_{s = 1}^\infty {{N_s}{t^s}/s} )

Hypergeometric Series and Functions as Periods of Exponential Modules

where N s is the number of points of V which are rational over GF[q s ]. (For definition of L-series, see chapter II). It has been known for some time [1] that the zeta function is rational and a second exposition of the same proof has been given by Serre [2]. It is rather questionable as to whether there is any need to repeat such well known material. However because of its connection with p-adic analysis it may be in accord with the purpose of this conference to outline the old proof. This will be done in chapter I but we will use results and points of view which were not available in 1959.

On singular projective structures on Riemann surfaces

The object of this chapter is to introduce the p-adic gamma function. This will be used to explain the p-adic beta function and ultimately for our discussion of singular disks in Chapters 24–26. Our treatment is strongly influenced by Boyarsky [5]. We will be brief since similar material has been discussed by Lang [23] and by Katz [21].