Arithmetic differential operators over the p-adic integers

148 Pages·2012·0.856 MB·English

by  Claire C Ralph, S R Simanca

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES
ManagingEditor:ProfessorM.Reid,MathematicsInstitute,
UniversityofWarwick,CoventryCV47AL,UnitedKingdom
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LONDON MATHEMATICAL SOCIETY LECTURE
NOTE SERIES: 396
Arithmetic Differential Operators
over the p-adic Integers
CLAIRE C. RALPH
CornellUniversity,USA
SANTIAGO R. SIMANCA
UniversitédeNantes,France
cambridge university press
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Singapore,SãoPaulo,Delhi,Tokyo,MexicoCity
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Informationonthistitle:www.cambridge.org/9781107674141
©C.RalphandS.Simanca,2012
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Contents
1 Introduction page 1
2 The p-adic numbers Qp 8
2.1 A pragmatic realization of Qp 11
2.2 The p-adic integers Zp and their field of fractions 13
2.3 The topology of Qp 15
2.4 Analytic and algebraic properties of Qp 17
2.5 (p−1)-roots of unity in Qp 20
3 Some classical analysis on Qp 23
3.1 The Artin–Hasse exponential function 30
3.2 The completion of the algebraic closure of Qp 33
3.3 Zeta functions 38
4 Analytic functions on Zp 48
4.1 Strassmann’s theorem 53
5 Arithmetic differential operators on Zp 55
5.1 Multiple primes I 61
6 A general view of arithmetic differential operators 65
6.1 Basic algebraic concepts 67
6.2 General δp-functions: arithmetic jet spaces 73
6.3 The analogue of a δp-linear operators for group
schemes 82
6.4 Multiple primes II 86
7 Analyticity of arithmetic differential operators 103
8 Characteristic functions of discs in Zp: p-adic
coordinates 107
8.1 Characteristic functions of discs of radii 1/p 107
8.2 Characteristic functions of discs of radii 1/pn 116
v
vi Contents
9 Characteristic functions of discs in Zp: harmonic co-
ordinates 119
9.1 A matrix associated to pm 120
9.2 Analytic functions and arithmetic differential
operators 122
10 Some differences between δ-operators over Zp and
Z(cid:2)ur 127
p
References 135
Index 138
1
Introduction
Our purpose in this monograph is to provide a concise and complete
introduction to the study of arithmetic differential operators over the
p-adic integers Zp. These are the analogues of the usual differential op-
erators over say, the ring C[x], but where the role of the variable x is
replaced by a prime p, and the roles of a function f(x) and its deriva-
tive df/dx are now played by an integer a ∈ Z and its Fermat quotient
δpa=(a−ap)/p.
Inmakingourpresentationofthesetypeofoperators,wefindnobet-
ter way than discussing the p-adic numbers in detail also, and some of
the classical differential analysis on the field of p-adic numbers, empha-
sizing the aspects that give rise to the philosophy behind the arithmetic
differentialoperators.Thereaderisurgedtocontrasttheseideasatwill,
while keeping in mind that our study is neither exhaustive nor intended
to be so, and most of the time we shall content ourselves by explain-
ing the differential aspect of an arithmetic operator by way of analogy,
rather than appealing to the language of jet spaces. But even then, the
importance of these operators will be justified by their significant ap-
pearanceinnumbertheoreticconsiderations.Oneofourgoalswillbeto
illustrate how different these operators are when the ground field where
they are defined is rather coarse, as are the p-adic integers Zp that we
use.
In order to put our work in proper perspective, it is convenient to
introduce some basic facts first, and recall a bit of history. Given a
prime p, we may define the p-adic norm (cid:3) (cid:3)p over the field of rational
numbersQ.Thecompletionoftherationalsinthemetricthatthisnorm
induces is the field Qp of p-adic numbers, and this field carries a non-
Archimedeanp-adicnormextendingtheoriginalp-adicnormonQ.This
isthedescriptionofQpasgivenbyK.Henselcirca1897(see,forinstance,
1
2 Introduction
[28]). Two decades later, A. Ostrovski [39] proved that any nontrivial
normonQisequivalenttoeithertheEuclideannormortoap-adicnorm
forsomeprimep.Inthisway,therearosethephilosophicalprinciplethat
treats the real numbers and all of the p-adic numbers on equal footing.
In the twentieth century, the p-adic numbers had a rich history. We
briefly mention some major results.
The idea that studying a question about the field Q can be answered
by putting together the answers to the same question over the fields R
and Qp for all ps was born with the Hasse–Minkowski’s theorem. This
states that a quadratic form over Q has a nontrivial zero in Qn if, and
only if, it has a nontrivial zero in Rn and a nontrivial zero in Qn for
p
each prime p. This theorem was proven by Hasse in his thesis around
1921 [27], the problem having been proposed to him by Hensel who had
proventhen=2caseafewyearsearlier.Suchaprinciplefailsforcubics.
Thedevelopmentabovecameafterseveralinterestingresultsthatpre-
cededtheintroductionofthep-adicnumbers.Thelocal-to-globalprinci-
ple embodied in the Hasse–Minkowski theorem had a precedent in Rie-
mannian geometric, since as recently as 1855, Bonnet had proved that
if the curvature of a compact surface was bounded below by a positive
constant,thenitsdiameterwasboundedabovebyaquantitydepending
only on the said constant. Strictly on the arithmetic side of things, in
the seventeenth century J. Bernoulli defined the(cid:3)Bernoulli numbers Bk,
the coefficients in the expansion et/(et −1) = k(cid:3)Bktk/k!, used them
to compute closed-form expressions for the sums m jn, and devel-
j=0
oped several identities that these numbers satisfy. A century later, the
Bernoulli numbers were used by Euler to show heuristically that if ζ is
(cid:3)
theRiemmanzetafunction,thenζ(1−k)= ∞n=11/n1−k =−Bk/k for
any integer k ≥2. In the mid nineteenth century, Riemman proved that
(cid:3)
ζ(s) = ∞ 1/ns is a meromorphic function on the complex plane C,
n=1
giving Euler’s argument complete sense. Further, he used the Gamma
(cid:4) (cid:5)
functiontodefineΛ(s)=π−s2Γ 2s ζ(s)andprovedthefunctionalequa-
tion Λ(s) = Λ(1−s). The intimate relationship between the Bernoulli
numbers and the values of ζ(s) at negative integers led to the idea that
these numbers have profound arithmetical properties, a fact discovered
by Kummer in his work on Fermat’s last theorem circa 1847. The ideal
class group of Q(ζN), ζN a primitive N-th root of unity, is the quotient
of the fractional ideals of Q(ζN) by the set of principal ideals, and it
turnsouttobeagroupoffiniteorderhN withrespecttoidealmultipli-
cation. A prime p is said to be regular if p(cid:2)hp, and irregular otherwise.
Kummer proved that p is regular if, and only if, p does not divide the