David Lubicz
Contents in Brief
3.1 Definition ofQQQQQQppppppand first properties 39 3.2 Complete discrete valuation rings and fields 41
First properties•Lifting a solution of a polynomial equation
3.3 The fieldQQQQQQppppppand its extensions 43 Unramified extensions•Totally ramified extensions•Multiplicative system of representatives•Witt vectors
Thep-adic numbers play an important role in algebraic number theory. Many of the fruitful prop- erties they enjoy stem from Hensel’s lemma that allows one to lift the modulopfactorization of a polynomial. As a consequence, althoughQpis a characteristic zero field, its absolutely unramified extensions reflect the same structure as the algebraic extensions of the finite fieldFp. On the other hand, the completion of the algebraic closure ofQp can be embedded as a field, but not as a val- uation field, intoC. Consequently,p-adic numbers are used to bridge the gap between finite field algebraic geometry and complex algebraic geometry by the use of the so-called Lefschetz principle.
In this chapter, we review the definition and basic properties ofp-adic numbers. More details can be found in the excellent book by Serre [SER1979].
3.1 Definition of Q Q Q Q Q Q Q Q
ppppppppand first properties
First, we introduce the notion of inverse limit of a directed family, which is useful in the construction of thep-adic numbers.
Definition 3.1 LetIbe a set with a partial ordering relation, i.e., for alli, j, k∈Iwe have
• ii,
• ifijandjkthenik,
• ifijandjitheni=j.
ThenIis adirected setif for alli, jthere exists ak∈Isuch thatkiandkj.
Definition 3.2 Adirected family of groupsis given by 39
40 Ch. 3 Background onp-adic Numbers
• a directed setI
• for eachi∈Ia groupAiand fori, j ∈I,ija morphism of groupspij :Ai →Aj
satisfying the compatibility relation: for alli, j, k∈Iwithijk,pjk◦pij =pik. We denote by(Ai,{pij}j∈I)such a directed family.
Definition 3.3 Let(Ai,{pij}j∈I)be a directed family of groups and letAbe a group, together with a set of morphisms{pi : A →Ai}i∈I compatible with thepij, i.e.,pj =pij ◦pifori j, that satisfies the following universal property: letBbe a group and let{φi}i∈I be a set of morphisms φi:B→Aisuch that the following diagram commutes forij
B φi //
φ@@j@@@@@
@ Ai
pij
Aj
then there is a morphismφsuch that for alli, j∈I, the following diagram B φ //
φ@j@@@@@
@ A
pj
Aj
is commutative. The groupAis called theinverse limitof(Ai,{pij}j∈I)and is denoted bylim
←−Ai. The universal property implies thatlim←−Aiis unique up to isomorphism.
Proposition 3.4 Let(Ai,{pij}j∈I)be a directed family of groups withI =N∗. LetA= Ai be the product of the family. Note thatAitself is a group where the group law is defined compo- nentwise. Consider the subsetΓofAconsisting of all elements(ai)withai ∈ Aiand fori j, pij(ai) =aj. It is easily verified thatΓis a subgroup ofAwhich is isomorphic tolim
←−Aiwhere the projectionspkfork∈N∗are given bypk : (ai)→ak. In particular, the inverse limit of a directed family of groups always exists. Moreover, if theAiare rings, thenlim
←−Aiis a ring, where the ring operations are defined componentwise.
Definition 3.5 Letpbe a prime number andI=N∗. Forij ∈I, letpij :Z/piZ→Z/pjZbe the natural projections given by reduction modulopj, then(Z/piZ,{pij}j∈I)is a directed family.
Its inverse limitlim
←−Z/piZ, denoted byZp, is called thering ofp-adic integers.
The natural morphism of ringsψ : Z → Zp withψ(1Z) = 1Zp is injective, which implies that charZp= 0. The invertible elements inZpare characterized by the following proposition.
Proposition 3.6 An elementz ∈ Zp is invertible if and only ifz is not in the kernel ofp1. For every nonzero elementz∈Zpthere exists a uniquevp(z)∈Nsuch thatz=uψ(p)vp(z)withuan invertible element inZp. The integervp(z)is called thep-adic valuationofz and we extend the mapvptoZpby definingvp(0) =−∞.
Definition 3.7 LetRbe a ring and letv:R →Zbe a map such that for allx, y ∈ R,
• v(xy) =v(x)v(y);
• v(x+y)min
v(x), v(y)
with equality whenv(x)=v(y).
A map with the above properties is called adiscrete valuation.
§ 3.2 Complete discrete valuation rings and fields 41
Lemma 3.8 The mapvpis a discrete valuation.
The ringZpis an integral domain andQpdenotes its field of fractions. Thep-adic valuationvpcan be extended toQpby definingvp(1/x) = −vp(x)forx∈ Zp. Similarly, the natural embedding ψ: Z→Zpcan also be extended to the embeddingQ→Qp by definingψ(1/x) = 1/ψ(x), for x∈Z.
The valuation ofQpinduces a map defined by|x|p =p−vp(x)forx∈Qp. The properties ofvp
imply that| · |pis a norm onQp, which is called thep-adic norm.Thep-adic norm also defines a norm onZand onQvia the mapψ. Forx∈Z, the norm is given by|x|p =p−ν withνthe power ofpin the prime factorization ofx. Forx/y∈Q, the norm is given by|x/y|p=|x|p/|y|p. The set ψ(Q)is a dense subset ofQpfor| · |p. In fact,Qpcan also be defined as the completion ofQwith respect to| · |p. This definition is similar to the definition ofRas the completion ofQendowed with the usual archimedean norm.
3.2 Complete discrete valuation rings and fields
3.2.1 First properties
Definition 3.9 A fieldKis acomplete discrete valuation fieldif
• Kis endowed with a discrete valuationvK
• the valuation induces a norm| · |K onKby defining|x|K =λ−vK(x)withλ >1
• every sequence inKwhich is Cauchy for| · |K has a limit inK.
Remark 3.10 The topology induced by the norm|x|K =λ−vK(x)does not depend onλ.
It is easy to see that the subsetR={x∈K | |x|K 1}is a ring. This ring is an integral domain which is integrally closed, i.e., ifx∈Kis a zero of a monic polynomial with coefficients inRthen x∈ R. The ringRis called thevaluation ringofK. Clearly,M ={x∈ R | |x|K <1}is the unique maximal ideal ofR. The fieldK=R/Mis called theresidue fieldofK. In the remainder of this chapter, we will assume that the residue field is finite.
Proposition 3.11 An elementx∈ Ris invertible inRif and only ifxis not inM.
Note that ifKis a complete discrete valuation field with valuation ringRand maximal idealM, then the ringsAi =R/Mitogether with the natural projectionspij :Ai → Ajforij form a directed family of rings. It is easy to see thatRis isomorphic tolim
←−Ai.
Example 3.12 The fieldQpis a complete discrete valuation field with residue fieldFp.
Definition 3.13 An elementπ∈ Ris called auniformizing elementifvK(π) = 1. Letp1be the canonical projection fromRtoK. A mapω :K → Ris asystem of representativesofKif for all x∈ Kwe havep1
ω(x)
=x.
Definition 3.14 An elementx ∈ Ris called alift of an elementx0 ∈ Kifp1(x) = x0. Conse- quently, for allx∈ K,ω(x)is a lift ofx.
Now, letπbe a uniformizing element,ω a system of representatives ofKinRandx∈ R. Let n= vK(x), thenx/πn is an invertible element ofRand there exists a uniquexn ∈ Ksuch that vK
x−πnω(xn)
=n+ 1. Iterating this process and using the Cauchy property ofKwe obtain the existence of the unique sequence(xi)i0of elements ofKsuch that
x=
∞
i=0
ω(xi)πi.
42 Ch. 3 Background onp-adic Numbers
The following theorem classifies the complete discrete valuation fields.
Theorem 3.15 Let K be a complete discrete valuation field with valuation ring R and residue fieldK, assumed finite of characteristicp. IfcharK =pthenRis isomorphic to a power series ringK[[X1, X2, . . .]]. IfcharK= 0thenKis an algebraic extension ofQp.
From now on, we restrict ourselves to complete discrete valuation ring of characteristic0with a finite residue field. By Theorem 3.15, any such ring can be viewed as the valuation ring of an algebraic extension ofQp.
3.2.2 Lifting a solution of a polynomial equation
LetKbe a complete discrete valuation field with norm| · |K and letRbe its valuation ring. Let R[X]denote the univariate polynomial ring overR. The main result of this section is Newton’s algorithm which provides an efficient way to compute a zero of a polynomialf ∈ R[X]to arbitrary precision starting from an approximate solution.
Proposition 3.16 LetK be a complete discrete valuation field with valuation ring Rand norm
| · |K. Letf ∈ R[X]and letx0∈ Rbe such that
|f(x0)|K <|f(x0)|2K
then the sequence
xn+1=xn− f(xn)
f(xn) (3.1)
converges quadratically towards a zero off inR.
The quadratic convergence implies that the precision of the approximation nearly doubles at each iteration. More precisely, letk=vK
f(x0)
and letxbe the limit of the sequence (3.1). Suppose thatxiis an approximation ofxto precisionn, i.e.,(x−xi)∈ Mn, thenxi+1=xi−f(xi)/f(xi) is an approximation ofxto precision2n−k. Very closely related to the problem of lifting the solution of a polynomial equation is Hensel’s lemma that enables one to lift the factorization of a polynomial.
Lemma 3.17 (Hensel) Letf, Ak, Bk, U, V be polynomials with coefficients inRsuch that
• f ≡AkBk (modMk),
• U(X)Ak(X) +V(X)Bk(X) = 1,withAk monic anddegU(X) <degBk(X)and degV(X)<degAk(X)
then there exist polynomialsAk+1andBk+1satisfying the same conditions as above withkreplaced byk+ 1and
Ak+1≡Ak (modMk), Bk+1≡Bk (modMk).
Iterating this lemma, we obtain an algorithm to compute a factor of a polynomial overRgiven a factor moduloM.
Corollary 3.18 With the notation of Proposition 3.16, letf ∈ R[X]be a polynomial andx0 ∈ K such thatx0is a simple zero of the polynomialf0 = p1(f). Then there exists an elementx∈ R such thatp1(x) =x0andxis a zero off.
§ 3.3 The fieldQpand its extensions 43
3.3 The field Q Q Q Q Q Q Q Q
ppppppppand its extensions
LetKbe a finite algebraic extension of Qp defined by an irreducible polynomialm ∈Qp[X]. It can be shown that there exists a unique norm| · |K onK extending thep-adic norm onQp. Let R={x∈K| |x|K 1}denote the valuation ring ofKand letM={x∈ R | |x|K <1}be the unique maximal ideal ofR. ThenK =R/Mis an algebraic extension ofFp, the degree of which is called theinertia degreeofKand is denoted byf. Theabsolute ramification indexofKis the integere=vK
ψ(p)
. The extension degree[K :Qp], the inertia degreef, and the ramification index satisfy the following fundamental relation.
Theorem 3.19 Letdbe the degree ofK/Qp, thend=ef.
3.3.1 Unramified extensions
Definition 3.20 LetK/Qpbe a finite algebraic extension, thenKis calledabsolutely unramified ife = 1. An absolutely unramified extension of degreedis denoted byQq withq = pd and its valuation ring byZq.
Proposition 3.21 Denote byP1the reduction morphismR[X]→ K[X]induced byp1and letm be the irreducible polynomial defined byP1(m). The extensionK/Qpis absolutely unramified if and only ifdegm = degm. Letd= degmandFq =Fpdthe finite field defined bym, then we havep1(R) =Fq. LetK1andK2be two unramified extensions ofQpdefined respectively bym1
andm2thenK1K2if and only ifdegm1= degm2.
As a consequence, every unramified extension ofQpis isomorphic toQp[X]/(m(X))withmbeing an arbitrary degreedlift of an irreducible polynomial overFpof degreed. Letω : Fq → Zq be a system of representatives ofFq; every element xofZq can be written as a power seriesx =
∞
i=0ω(xi)piwith(xi)i0a sequence of elements ofFq.
Proposition 3.22 An unramified extension ofQp is Galois and its Galois group is cyclic gener- ated by an elementΣthat reduces to the Frobenius morphism on the residue field. We call this automorphism theFrobenius substitution onK.
3.3.2 Totally ramified extensions
Definition 3.23 LetK/Qpbe a finite algebraic extension, thenKis called totally ramified iff = 1.
Definition 3.24 A monic degreed polynomialP(X) = d
i=0aiXi in Qp[X] is anEisenstein polynomialif it satisfies
• vp(a0) = 1,
• vp(ai)>1, fori= 1, . . . , d−1.
Such a polynomial is irreducible.
Proposition 3.25 LetKbe a totally ramified extension ofQp; then there exists an Eisenstein poly- nomialPsuch thatKis isomorphic toQp[X]/(P).
44 Ch. 3 Background onp-adic Numbers
3.3.3 Multiplicative system of representatives
LetKbe a complete discrete valuation field of characteristic zero, with valuation ringRand residue fieldK, assumed finite of characteristicp. ThenKis isomorphic to an algebraic extension ofQp. Proposition 3.26 There exists a unique system of representativesω which commutes with p-th powering, i.e., for allx∈ K,ω(xp) =ω(x)p. This systemωis multiplicative in the following way:
for allx, y∈ Kwe haveω(xy) =ω(x)ω(y).
Such a system can be obtained as follows. Letx0 ∈ K. SinceKis a perfect field, for eachr∈N there existsxr ∈ K such thatxprr =x0. SetXr = {x ∈ R | p1(x) = xr} and letYr be the set
xpr |x∈Xr
. It is easy to see that for ally ∈ Yr,p1(y) = x0. Moreover, we have for all x, y ∈Yr,vK(x−y)r. This means by the Cauchy property that there exists a unique element z ∈ Rsuch thatz ∈Yr, for allr. Then simply defineω(x0) =z. The system of representatives defined in this way is exactly the unique system that commutes withp-th powering.
Letπbe a uniformizing element ofRand letωbe the multiplicative system of representatives of KinRthat commutes withp-th powering. Writex∈ Rasx=∞
i=0ω(xi)πiwith(xi)i0the unique sequence of elements ofKas defined in Section 3.2. LetΣbe the Frobenius substitution on K; then we have
Σ(x) =
∞
i=0
ω(xi)pπi.
3.3.4 Witt vectors
Definition 3.27 Let pbe a prime number and (Xi)i∈N a sequence of indeterminates. The Witt polynomialsWn∈Z[X0, . . . , Xn]are defined as
W0 = X0, W1 = X0p+pX1, Wn =
n
i=0
piXipn−i.
Theorem 3.28 Let(Yi)i∈N be a sequence of indeterminates, then for everyΦ(X, Y) ∈ Z[X, Y] there exists a unique sequence(φi)i∈N∈Z[X0, X1, . . .;Y0, Y1, . . .]such that for alln0
Wn
φ0, . . . , φn
= Φ
Wn(X0, . . . , Xn), Wn(Y0, . . . , Yn) .
Let(Si)i∈N, resp.(Pi)i∈N, be the sequence of polynomials(φi)i∈Nassociated via Theorem 3.28 with the polynomialsΦ(X, Y) =X+Y, resp.Φ(X, Y) =X×Y. Then for any commutative ring R, we can define two composition laws onRN: leta= (ai)i∈N∈ RNandb= (bi)i∈N∈ RN, then
a+b=
Si(a, b)
i∈N and a×b=
Pi(a, b)
i∈N.
Definition 3.29 The setRNendowed with the two previous composition laws is a ring called the ring ofWitt vectorswith coefficients inRand is denoted byW(R).
The relation withp-adic numbers is the following. LetFqwithq=pdbe a finite field of characteris- ticp, thenW(Fq)is canonically isomorphic to the valuation ring of the unramified extension of de- greedofQp. Via this isomorphism, the mapF:W(Fq)→W(Fq)given byF
(ai)i∈N
= (api)i∈N
corresponds to the Frobenius substitutionΣ.