Cohomological Background on Point Counting
8.3 Overview of pppppppp -adic methods
The best known application ofp-adic methods in algebraic geometry is undoubtedly Dwork’s in- genious proof of the rationality of the zeta function [DWO1960]. Although Dwork’s proof can be transformed easily in an algorithm to compute the zeta function of any algebraic variety, nobody seemed to realize this and for more than a decade only -adic algorithms were used.
At the end of 1999, Satoh [SAT 2000] introduced thep-adic approach into computational alge- braic geometry by describing ap-adic algorithm to compute the number of points on an ordinary elliptic curve over a finite field. Following this breakthrough development, many existingp-adic theories were used as the basis for new algorithms:
• Dwork’sp-adic analytic methods by Lauder and Wan [LAWA2002b]
• Serre–Tate canonical lift by Satoh [SAT 2000], Mestre [MES2000b], etc.
• Monsky–Washnitzer cohomology by Kedlaya [KED2001]
• Dwork–Reich cohomology by Lauder and Wan [LAWA2002a, LAWA2004]
• Dwork’s deformation theory by Lauder [LAU 2004].
Finally, we note that the use ofp-adic methods as the basis for an algorithm to compute the zeta function of an elliptic curve already appeared in the work of Kato and Lubkin [KALU1982].
In this section we will only review the twop-adic theories that are most important for practical applications, namely the Serre–Tate canonical lift and Monsky–Washnitzer cohomology.
8.3.1 Serre–Tate canonical lift
LetAbe an abelian variety defined overFq withq =pdandpa prime. LetQq be an unramified extension ofQp of degreedwith valuation ringZq and residue fieldZq/(pZq) Fq. Consider an arbitrary liftAofAdefined overZq, i.e.,Areduces toAmodulop, then in general there will not exist an endomorphismF ∈ End(A)that reduces to theq-th power Frobenius endomorphism φq∈End(A).
Definition 8.9 Acanonical liftof an abelian varietyAoverFq is an abelian variety AoverQq
such thatAreduces toAmodulopand the ring homomorphismEnd(A)−→End(A)induced by reduction modulopis an isomorphism.
This definition implies that ifAadmits a canonical liftAc, then there exists a liftF ∈ End(Ac) of the Frobenius endomorphismφq ∈ End(A). In fact, the reverse is also true: letA be a lift ofA and assume thatF ∈ End(A)reduces to φq ∈ End(A), then Ais a canonical lift ofA. Deuring [DEU1941] proved that for an ordinary elliptic curve, a canonical lift always exists and is unique up to isomorphism. The question of existence and uniqueness of the canonical lift for general abelian varieties was settled by Lubin, Serre and Tate [LUSE+ 1964].
Theorem 8.10 (Lubin–Serre–Tate) LetAbe anordinaryabelian variety overFq. Then there exists a canonical liftAcofAoverZqandAcis unique up to isomorphism.
§ 8.3 Overview ofp-adic methods 139
Recall that an abelian varietyAis ordinary if it has maximalp-rank, i.e.,A[p] = (Z/pZ)dim(A). The construction of ap-adic approximation ofAc givenAproceeds as follows: letA0be a lift ofAtoZq and denote withπ: A0 → Areduction modulop. Consider the subgroupA0[p]loc = A0[p]∩ker(π),i.e., thep-torsion points onA0that reduce to the neutral element ofA. As shown by Carls [CAR 2003],A1=A0/A0[p]locis again an abelian variety such that its reduction is ordinary and there exists an isogenyI0:A0−→ A1, which reduces to thep-th power Frobenius morphism σ:A −→ Aσ. Repeating this construction we can defineAi=Ai−1/Ai−1[p]locforipositive and we get a sequence of abelian varieties and isogenies
A0 I0
−→ A1 I1
−→ A2 I2
−→ A3 I3
−→. . .
Clearly we have thatAkdfork∈Nreduces toAmodulop; furthermore, the sequence{Akd}k∈N
converges to the canonical liftAcand the convergence is linear.
LetCbe a smooth projective curve defined overFqof genusg, with Jacobian varietyJC. Assum- ing thatJC is ordinary, we can consider its canonical liftAc. Note thatAc itself does not have to be the Jacobian variety of a curve [OOTS1986]. SinceEnd(Ac)is isomorphic toEnd(JC), there exists a liftFof the Frobenius endomorphismφq.
To recover the characteristic polynomial ofφq, we proceed as follows: letD0(Ac,Qq)denote the space of holomorphic differential forms of degree1 on Ac defined overQq, then we have dim(D0
Ac,Qq)
=g, sincedim(JC) =g. Given a basisBofD0(Ac,Qq), every endomorphism λ∈EndQq(Ac)can be represented by ag×gmatrixM defined overQq by considering the action ofλ∗onB, i.e.,λ∗(B) =M B. The link with the characteristic polynomial of Frobeniusχ(φq)C is then given by the following proposition.
Proposition 8.11 LetF ∈EndQq(Ac)be the lift of the Frobenius endomorphismφq ∈EndFq(JC) and letMF be the matrix through whichφ∗q acts onD0(Ac,Qq). IfP(T)∈Zq(T)is the charac- teristic polynomial ofMF+qMF−1, then the characteristic polynomialχ(φq)Cis given by
χ(φq)C(T) =TgP
T+ q T
· (8.1)
Note that we can also writeχ(φq)C(T) =P1(X)P2(X)withP1the characteristic polynomial of MFandP2the characteristic polynomial ofqMF−1.
The point-counting algorithms based on the canonical lift thus proceed in two stages: in the first stage, a sufficiently precise approximation of the canonical lift of JC (or its invariants) is com- puted and in the second stage, the action of the lifted Frobenius endomorphismFis computed on D0(Ac,Qq).
8.3.2 Monsky–Washnitzer cohomology
In this section we will specialize the formalism of Monsky–Washnitzer cohomology as described in the seminal papers by Monsky and Washnitzer [MOWA1968, MON1968, MON1971], to smooth affine plane curves. Further details can be found in the lectures by Monsky [MON1970] and in the survey by van der Put [PUT1986].
LetC be a smooth affine plane curve over a finite fieldFq withq = pd elements, and letQq
be a degreedunramified extension ofQp with valuation ringZq, such thatZq/pZq = Fq. The aim of Monsky–Washnitzer cohomology is to express the zeta function of the curveCin terms of a Frobenius operatorF acting onp-adic cohomology groupsHi(C,Qq)defined overQq associated toC. Note that it is necessary to work over a field of characteristic0; otherwise, it would only be possible to obtain the zeta function modulop. For smooth curves, most of these groups are zero as illustrated in the next proposition.
140 Ch. 8 Cohomological Background on Point Counting
Proposition 8.12 LetCbe a nonsingular affine curve over a finite fieldFq, then the only nonzero Monsky–Washnitzer cohomology groups areH0(C,Qq)andH1(C,Qq).
In the remainder of this section, we introduce the cohomology groupsH0(C,Qq)andH1(C,Qq) and review their main properties.
SinceCis plane,Ccan be given by a bivariate polynomial equationg(x, y) = 0withg∈Fq[x, y].
LetA=Fq[x, y]/
g(x, y)
be the coordinate ring ofC. Take an arbitrary liftg(x, y)∈Zq[x, y]of g(x, y)and letCbe the curve defined byg(x, y) = 0with coordinate ringA=Zq[x, y]/
g(x, y) . To compute the zeta function ofCin terms of a Frobenius operator, we need to lift the Frobenius endomorphismφqonAto theZq-algebraA, but as illustrated in the previous section, this is almost never possible. Furthermore, theZq-algebraAdepends essentially on the choices made in the lifting process as the following example illustrates.
Example 8.13 ConsiderC:xy−1 = 0overFpwith coordinate ringA=Fp[x,1/x], and consider the two lifts
g1(x, y) =xy−1 g2(x, y) =x(1 +px)y−1
then we have thatA1=Zp[x,1/x]andA2=Zp[x,1/(x(1 +px))], which are not isomorphic.
A first attempt to remedy both difficulties is to work with thep-adic completionA∞ofA, which is unique up to isomorphism and does admit a lift ofφq toA∞. But then a new problem arises since the de Rham cohomology ofA∞, which provides the vector spaces we are looking for, is too big.
Example 8.14 Consider the affine line overFq, thenA=Zq[x]andA∞is the ring of power series ∞
i=0
rixi with ri∈Zq and lim
i→∞ri= 0.
We would like to defineH1(A,Qq)as A∞dx/d(A∞)⊗Zq Qq, but this turns out to be infinite dimensional. For example, it is clear that each term in the differential form∞
i=0pixpi−1dxis exact but its sum is not, since∞
i=0xpi is not inA∞. The fundamental problem is that∞
i=0pixpi−1 does not converge fast enough for its integral to converge as well.
Monsky and Washnitzer therefore work with a subalgebraA†ofA∞, whose elements satisfy growth conditions.
Definition 8.15 LetA=Zq[x, y]/
g(x, y)
, then thedagger ringorweak completionA†is defined asA† =Zqx, y †/(g(x, y)), whereZqx, y †is the ring of overconvergent power series
ri,jxiyj∈Zq[[x, y]]| ∃δ, ε∈R, ε >0,∀(i, j) : vp(ri,j)ε(i+j) +δ
.
The ringA† satisfiesA†/(pA†) = A and depends up toZq-isomorphism only on A. Further- more, Monsky and Washnitzer show that ifϕ is anFq-endomorphism ofA, then there exists a Zq-endomorphismϕofA†liftingϕ. In particular, we can lift the Frobenius endomorphismφq on Ato aZq-endomorphismFonA†.
To each elements ∈ A† we can associate the differentialdssuch that the usual Leibniz rule applies: fors, t ∈A† : d(st) = sdt+tds, which implies thatd(a) = 0fora ∈ Zq. The set of all these differentials clearly is a module overA†and is denoted byD1(A†). The following lemma gives a precise description of this module.
Lemma 8.16 The universal moduleD1(A†)of differentials satisfies D1(A†) =
A†dx+A†dy A†
∂g
∂xdx+∂g
∂ydy
.
§ 8.3 Overview ofp-adic methods 141
Taking the total differential of the equationg(x, y) = 0gives∂x∂gdx+∂g∂ydy = 0,which explains the moduleA†(∂g∂xdx+∂y∂gdy)in the above lemma. The mapd:A† −→D1(A†)is a well defined derivation, so it makes sense to consider its kernel and cokernel.
Definition 8.17 The cohomology groupsH0(A,Qq)andH1(A,Qq)are defined by
H0(A,Qq) = ker(d)⊗ZqQq and H1(A,Qq) = coker(d)⊗ZqQq. (8.2) By definition we haveH1(A,Qq) =
D1(A†)/d(A†)
⊗Zq Qq; the elements ofd(A†)are called exact differentials. One can prove thatH0(A,Qq)andH1(A,Qq)are well defined, only depend on A, and are finite dimensional vector spaces overQq.
Proposition 8.18 LetCbe a nonsingular affine curve of genusg, thendim
H0(A,Qq)
= 1and dim
H1(A,Qq)
= 2g+m−1, where m is the number of points needed to complete C to a smooth projective curve.
LetFbe a lift of the Frobenius endomorphismφqtoA†, thenFinduces an endomorphismF∗on the cohomology groups. The main theorem of Monsky–Washnitzer cohomology is the following Lefschetz fixed point formula.
Theorem 8.19 (Lefschetz fixed point formula) LetC/Fq be a nonsingular affine curve overFq, then the number ofFqk-rational points onCis equal to
C(Fqk)= Tr
qkF−k∗;H0(C,Qq)
−Tr
qkF−k∗;H1(C,Qq) .
SinceH0(C,Qq)is a one-dimensional vector space on whichF∗acts as the identity, we conclude thatTr
qkF−k∗;H0(C,Qq)
=qk. To count the number ofFqk-rational points onC, it thus suf- fices to compute the action ofF∗onH1(C,Qq).
The algorithms based on Monsky–Washnitzer cohomology thus also proceed in two stages: in the first stage, a sufficiently precise approximation of the liftFis computed and in the second stage, a basis ofH1(C,Qq)is constructed together with reduction formulas to express any differential form on this basis. More algorithmic details can be found in Section 17.3.