Cohomological Background on Point Counting
8.1 General principle
Chapter
Cohomological Background
134 Ch. 8 Cohomological Background on Point Counting
Theorem 8.1 (Lefschetz Fixed Point Theorem) LetM be a compact complex analytic manifold andf : M → M an analytic map. Assume thatf only has isolated nondegenerate fixed points;
then {P ∈M |f(P) =P}=
i
(−1)iTr
f∗;HDRi (M) .
The HDRi (M) in the above theorem are the de Rham cohomology groups ofM and are finite dimensional vector spaces overCon whichf induces a linear mapf∗. The number of fixed points off is thus the alternating sum of the traces of the linear mapf∗on the vector spacesHDRi (M).
The dream of Weil was to mimic this situation for varieties over finite fields, i.e., construct a good cohomology theory (necessarily over a characteristic zero field) such that the number of fixed points of the Frobenius morphism is given by a Lefschetz fixed point formula.
The different approaches described in this chapter all fit in the following slightly more general framework: construct vector spaces over some characteristic zero field together with an action of the Frobenius morphismφq that provides information about the number of fixed points ofφq and thus the number ofFq-rational points onCorJC.
8.1.1 Zeta function and the Weil conjectures
LetFq be a finite field withq =pdandpprime. For any algebraic varietyX defined overFq, let Nkdenote the number ofFqk-rational points onX.
Definition 8.2 Thezeta functionZ(X/Fq;T)ofXoverFq is the generating function
Z(X/Fq;T) = exp ∞
k=1
Nk
k Tk
.
The zeta function should be interpreted as a formal power series with coefficients inQ. In 1949, Weil [WEI 1949] stated the following conjectures, all of which have now been proven.
Theorem 8.3 (Weil Conjectures) LetX be a smooth projective variety of dimension n defined over a finite field withqelements.
1. Rationality:Z(X/Fq;T)∈Q[[T]]is a rational function.
2. Functional equation:Z(T) =Z(X/Fq;T)satisfies Z
1 qnT
= +−qnE/2TEZ(T),
withEequal to the Euler–Poincaré characteristic ofX, i.e., the intersection number of the diagonal with itself in the productX×X.
3. Riemann hypothesis: there exist polynomialsPi(T)∈Z[T]fori= 0, . . . ,2n, such that Z(X/Fq;T) = P1(T)· · ·P2n−1(T)
P0(T)· · ·P2n(T) withP0(T) = 1−T,P2n(T) = 1−qnT and for1r2n−1
Pr(T) =
βr
i=1
(1−αr,iT)
where theαr,iare algebraic integers of absolute valueqr/2.
§ 8.1 General principle 135
Weil [WEI 1948] proved these conjectures for curves and abelian varieties. The rationality of the zeta function of any algebraic variety was settled in 1960 by Dwork [DWO1960] using p-adic methods. Soon after, the Grothendieck school developed -adic cohomology and gave another proof of the rationality and the functional equation. Finally, in 1973, Deligne [DEL 1974] proved the Riemann hypothesis.
Letφq be the Frobenius endomorphism ofJC, then the elements fixed byφqare exactlyJC(Fq) orker(φq−[1]) =JC(Fq). As introduced in Section 5.2.2, we can associate toφqits characteristic polynomialχ(φq)C, which is a monic polynomial of degree2gwith coefficients inZ. Furthermore, by Corollary 5.70 we have that|JC(Fq)|=χ(φq)C(1).
The relation betweenχ(φq)Cand the zeta function of the smooth projective curveCis as follows:
Proposition 8.4 LetCbe a smooth projective curve of genusgand letχ(φq)C(T)∈Z[T]be the characteristic polynomial ofφq. Define theL-polynomialofCby
L(T) =T2gχ(φq)C
1 T
,
then the zeta function ofCis given by
Z(C/Fq;T) = L(T) (1−T)(1−qT)·
LetL(T) =a0+a1T+· · ·+a2gT2g, then the functional equation shows thata2g−i=qg−iaifor i= 0, . . . , g. If we writeL(T) =2g
i=1(1−αiT), then the Riemann hypothesis implies|αi|=√ q and again by the functional equation, we can label theαisuch thatαiαi+g =qfori= 0, . . . , g.
This shows that Theorem 5.76 immediately follows from the Weil conjectures.
Taking the logarithm of both expressions for the zeta function leads to
lnZ(C/Fq;T) = ∞ k=1
Nk
k Tk= 2g
i=1
ln(1−αiT)−ln(1−T)−ln(1−qT).
Sinceln(1−sT) =−∞
i=1
(sT)k
k , we conclude that for all positivek
Nk =qk+ 1− 2g
i=1
αki.
The zeta functionZ(C/Fq;T)of a curveCcontains important geometric information aboutCand its JacobianJC. For example, Stichtenoth [STI1979] proved the following theorem.
Theorem 8.5 LetL(T) =a0+· · ·+a2gT2g, then thep-rank ofJCis equal to max{i|ai ≡0 (modp)}
Furthermore, Stichtenoth and Xing [STXI1995] showed thatJC is supersingular, i.e., isogenous overFqto a product of supersingular elliptic curves, if and only ifpdk/2 |akfor all1kg.
8.1.2 Cohomology and Lefschetz fixed point formula
In this section we indicate how the Weil conjectures, except for the Riemann hypothesis, almost im- mediately follow from a good cohomology theory. LetXbe a projective, smooth algebraic variety
136 Ch. 8 Cohomological Background on Point Counting
of dimensionnover a finite fieldFqof characteristicpand letX⊗FqFqdenote the corresponding variety over the algebraic closureFqofFq.
Let denote a prime different frompand letQbe the field of -adic numbers. Grothendieck introduced the -adic cohomology groupsHi(X,Q)(see [SGA 4]), which he used to prove the rationality and functional equation of the zeta function. The description of these cohomology groups is far beyond the scope of this book and we will simply state their main properties. However, forX, a smooth projective curve, we have the following theorem.
Theorem 8.6 LetCbe a smooth projective curve over a finite fieldFqof characteristicpand let be a prime different fromp; then there exists an isomorphism
H1(C,Z)T(JC).
To prove the rationality of the zeta function and the factorization of its numerator and denomina- tor, we only need the following two properties:
• The -adic cohomology groupsHi(X,Q)are finite dimensional vector spaces overQ
andHi(X,Q) = 0fori <0andi >2n.
• Letf : X →X be a morphism with isolated fixed points and suppose moreover that each fixed point has multiplicity1. Then the numberN(f, X)of fixed points off is given by a Lefschetz fixed point formula:
N(f, X) = 2n
i=0
(−1)iTr
f∗;Hi(X,Q) .
Recall that the numberNkofFqk-rational points onXequals the number of fixed points ofφkq with φqthe Frobenius morphism. By the Lefschetz fixed point formula, we have
Nk = 2n
i=0
(−1)iTr
φkq∗;Hi(X,Q) .
Substituting this in the definition of the zeta function proves the following theorem.
Theorem 8.7 LetXbe a projective, smooth algebraic variety overFq of dimensionn, then
Z(X/Fq, T) =P1(T). . . P2n−1(T) P0(T). . . P2n(T) with
Pi(T) = det(1−φ∗qT;Hi X,Q)
.
The above theorem constitutes the first cohomological approach to computing the zeta function of a projective, smooth algebraic variety: construct a basis for the -adic cohomology groupsHi(X,Q) and compute the characteristic polynomial of the representation ofφqonHi(X,Q). Unfortunately, the definition of theHi(X,Q)is very abstract and thus useless from an algorithmic point of view.
For curves not all is lost, since by Theorem 8.6 we have the isomorphismH1(C,Z)T(JC).
The second cohomological approach constructsp-adic cohomology groups defined over the un- ramified extensionQqofQp. Several different theories that satisfy a Lefschetz fixed point formula exist, e.g., Monsky–Washnitzer cohomology [MOWA1968, MON1968, MON 1971], Lubkin’sp- adic cohomology [LUB 1968], crystalline cohomology by Grothendieck [GRO1968] and Berth- elot [BER 1974], and finally, rigid cohomology by Berthelot [BER1986]. The main algorithmic advantage over the -adic cohomology theory is the existence of comparison theorems that provide