counting points of varieties over finite fields: a meeting ...srg/...ghorpade_29aug2012.pdf · a...

Counting Points of Varieties over Finite Fields:A Meeting Ground for Algebra, Analysis,Geometry, Topology, and Number Theory

Sudhir R. Ghorpade

Department of MathematicsIndian Institute of Technology Bombay

Powai, Mumbai 400076http://www.math.iitb.ac.in/∼srg/

Institute Award Lecture SeriesIRCC, IIT BombayAugust 29, 2012

Sudhir R. Ghorpade Counting Points of Varieties over Finite Fields

Fields

A Working Definition of a field:a set in which we can add, subtract, multiply, and divide (bynonzero elements).Examples:

Q, the field of rational numbers

R, the field of real numbers

C, the field of complex numbers

C(X ), the field of rational functions (in one variable X ) withcoefficients in CZ/pZ, the field of (residue classes of) integers modulo aprime number p.

Note that Q ⊂ R ⊂ C ⊂ C(X ). In general, every field K contains(a subfield isomorphic to) Q or Z/pZ for a unique prime numberp; accordingly we say that the characteristic of K is 0 or p.


Finite Fields, aka Galois fields

Fact: A field with q elements exists ⇐⇒ q is a power of a prime.

Evariste Galois Finite fields were first studied byby E. Galois (1811–1832) in apaper, published in 1830, entitled“Sur la theorie des nomberes”.For any prime power q, there is,up to isomorphism, a unique fieldwith q elements; it is denoted byFq or by GF(q). If q = pe , thenFq contains Fp = Z/pZ and theonly finite fields containing Fq

are Fqn for n ≥ 1. The “Galoisgroup”’ Gal(Fqn/Fq) of Fqn overFq is cyclic of order n andgenerated by the Frobeniusautomorphism x 7−→ xq.


Algebraic Varieties

Working Definition: Set of common zeros of a bunch of polynomialequations. A little more formally, if K is a field and N a positiveinteger, then the affine N-space over K is

AN(K ) = KN = {a = (a1, . . . , aN) : ai ∈ K for i = 1, . . . ,N}

and a subset X of AN is an affine algebraic variety over K if thereare polynomials f1, . . . , fr ∈ K [T1, . . . ,TN ] such that

X = X (K ) = {a ∈ AN(K ) : fi (a) = 0 for all i = 1, . . . , r}.

Note that if K is a subfield of L, then we can also consider X (L),the set of L-rational points of X .Examples of affine algebraic varieties are: conic sections (e.g.,circle T 2

1 + T 22 = 1, parabola T2 = T 2

1 , pair of straight linesT1T2 = 0), cuspidal cubic T 2

2 − T 31 = 0, nodal cubic

T 22 − T 2

1 − T 31 = 0, sphere in 3-space T 2

1 + T 22 + T 2

3 = 1, circle in3-space given by T 2

1 + T 22 + T 2

3 = 1 and T3 = 0.Sudhir R. Ghorpade Counting Points of Varieties over Finite Fields

Projective Varieties

Even when the base field K is algebraically closed (e.g., K = C),an affine space over K is incomplete from a geometric point ofview. For example, two lines do not always meet in a point. Toremedy this, one considers• Projective Space: The projective N-space PN(K ) over a field Kis obtained by adding to AN(K ) various “points at infinity”, whichtogether form a “hyperplane at infinity”. More formally,

PN(K ) =KN+1 \ {(0, 0, . . . , 0)}

∼= AN ∪ AN−1 ∪ · · · ∪ A1 ∪ A0,

where (a0, a1, . . . , aN) ∼ (b0, b1, . . . , bN) if they are proportional,i.e., if there is c ∈ K such that bi = cai for all i = 0, 1, . . . ,N.• Projective algebraic varieties are given by common solutions inPN(K ) of a bunch of homogeneous polynomials inK [T0,T1, . . . ,TN ]. For example, T0T2 − T 2

1 defines the projectiveparabola which consists of the points (1 : a : b) with b = a2 andthe extra point (0 : 0 : 1) at infinity.


Basic Attributes of Algebraic Varieties

An algebraic variety is irreducible if it is nonempty and it isnot the union of smaller algebraic varieties. For example, theunit circle is irreducible, but a pair of straight lines isn’t.

There is a natural notion of the dimension of an algebraicvariety. Thus an algebraic variety X is a curve, surface or asolid according as dim X is 1, 2 or 3.

There is also the notion of the degree of an algebraic varietyin a given affine or projective space. For example, a conicsection as well as the standard sphere have degree 2 while thecuspidal cubic or the nodal cubic have degree 3.

Points on an algebraic variety may be singular or nonsingular.For an affine plane curve given by f (x , y) = 0, a point P issingular if f (P) = fx(P) = fy (P) = 0. An algebraic variety Xis said to be nonsingular or smooth if every point on it isnonsingular; otherwise it is said to be singular.


Genus and Betti Numbers

An interesting and important invariant of an irreducible curve is itsgenus. Here is a rough definition.

Suppose C is an irreducible curve in C2

defined by f (z ,w) = 0. Looking at the realand imaginary parts, we see that C is anintersection of two solids in R4 and hence asurface in R4. From Topology, one can seethat except for a finite number of points,the resulting surface is a sphere with acertain number of handles. The genus of Cis the number of handles.

For a general algebraic variety of dimension n one has the Bettinumbers b0, b1, . . . b2n in place of the genus. For example if X is asmooth irreducible curve of genus g , then b0 = 1 = b2 andb1 = 2g . Also if d = deg X , then 2g ≤ (d − 1)(d − 2).


Hypersurfaces and Complete Intersections

A hypersurface is a variety defined by a single polynomial.The dimension of a hypersurface (in AN or PN) is N − 1 andthe degree is the degree of the defining polynomial.

More generally, an algebraic variety X (in AN or PN) definedby the “right number” of equations, viz., r = N − n equations,where n = dim X , is said to be a complete intersection. Iff1, . . . , fr define X and deg fi = di with d1 ≥ d2 ≥ · · · ≥ dr ,then d = (d1, . . . , dr ) depend only on X and is called themultidegree of X . Moreover, deg X = d1d2 · · · dr .

If X is a nonsingular complete intersection in PN of dimensionn and multidegree d = (d1, . . . , dr ), then the Betti numbers ofX are well-understood. In fact, bi (X ) = 0 for 0 ≤ i ≤ 2n withi 6= n and bn(X ) = bn(N,d) depends only on N and d, and isgiven by an explicit (albeit complicated!) formula due toHirzebruch (1956).


Varieties over finite fields

When the base field is finite, say Fq, then our curves, surfaces,solids, etc. are finite sets and it is worthwhile to compute theircardinalities. In fact, if X is a variety defined over Fq, we can notonly consider the number |X (Fq)|, but also |X (Fqm)| for m ≥ 1.These can be nicely encoded by the Weil zeta function of X :

Z (X ,T ) = exp

( ∞∑m=1

|X (Fqm)| Tm

m

).

Example: X = Pn. Here

|Pn(Fq)| = πn := qn + qn−1 + · · ·+ q + 1.

Of course, a similar formula holds with q replaced by qm. Thus,

Z (X ,T ) = exp

∞∑m=1

Tm

m

n∑j=0

qmj

=n∏

j=0

1

1− qjT,

Note: Z (X ,T ) is a rational function in T , and it satisfiesZ (X , 1/(qnT )) = (−qn/2 T )n+1 Z (X ,T ).


The Remarkable insight of Andre Weil I

A. Weil (1906-1998) Weil’s remarkable observation around 1949was that the arithmetic or combinatorialquestion of counting the number of pointsof varieties over finite fields is intimatelyrelated to the topology of related objectssuch as the corresponding variety over thecomplex numbers. His observation wasformulated in the form of a number ofconjectures, which are now known as Weilconjectures.

Weil Conjectures: Let X be a nonsingular projective variety ofdimension n. Then:

1 Z (X ,T ) is a rational function in T .


The Remarkable insight of Andre Weil II

2 Z (X ,T ) satisfies the functional equationZ (X , 1/qnT ) = (−qn/2 T )χ Z (X ,T ) for some integer χ. Moreprecisely, there is a factorization

Z (X ,T ) =P1(X ,T )P3(X ,T ) · · ·P2n−1(X ,T )

P0(X ,T )P2(X ,T ) · · ·P2n(X ,T ),

where P0(X ,T ) = 1− T , P2n(X ,T ) = 1− qnT , and Pi (X ,T ) arepolynomials with integer coefficients such that if bi = deg Pi (X ,T ),then T bi Pi (X , 1/qnT ) = (−1)bi q(i−n)bi/2P2n−i (X ,T ).

3 (Riemann Hypothesis) If Pi (X ,T ) are as above, then there is a

factorization Pi (X ,T ) =∏bi

j=1(1− ωijT ), where the reciprocal

roots ωij are algebraic integers with the property that |ωij | = qi/2

for 1 ≤ j ≤ bi and 0 ≤ i ≤ 2n.

4 If X is obtained from ‘reduction modulo p’ of a nonsingularprojective variety Y over Z, then deg Pi (X ,T ) = bi (YC).


The case of curves and an idea for the general case

These conjectures were proved by Weil in the case of curves. Inthis case, the functional equation and the Riemann hypothesisimply that if X has dimension 1 and genus g , then∣∣∣ |X (Fq)| − π1

∣∣∣ ≤ 2g√

q.

Weil’s Heuristics for the general case: Consider the Frobenius mapF : Fq → Fq given by x 7→ xq. For a ∈ Fq,

a ∈ Fq ⇐⇒ aq = a ⇐⇒ a is a fixed point of F

Thus |X (Fq)| (resp: |X (Fqm)|) is the number of fixed points of Xunder F (resp: Fm). Now we remember from Topology, theLefschetz Fixed Point Formula which says that if Y is a complexmanifold and f : Y → Y is a map with isolated fixed points, then

Number of fixed points of f r =2n∑i=0

(−1)iTr(f r |H i (Y ,C)

).


Status of Weil Conjectures

Alexander

Grothendieck (1928-)

and Pierre Deligne

(1944-)

The rationality of Z (X ,T ) was proved in1960 by Dwork.

Rationality and the functional equation forZ (X ,T ) proved by Grothendieck around 1963by developing (with Artin) a theory of etale`-adic cohomology spaces and proving ananalogue of the Lefschetz trace formula.Moreover, the “comparision theorem” forthese spaces yields the fourth conjecturalassertion of Weil.

Independent proofs of rationality and thefunctional equation for Z (X ,T ) were alsogiven by Lubkin around 1967.

The last remaining piece, the Riemannhypothesis for varieties over finite fields, wasproved by P. Deligne in 1973.


Lang-Weil Inequality and Deligne’s Inequality

Before the Weil Conjectures became theorems, Lang and Weil(1954) proved that if X is an irreducible projective variety in PN

defined over Fq and of dimension n and degree d , then∣∣∣ |X (Fq)| − πn∣∣∣ ≤ (d − 1)(d − 2)qn−(1/2) + Cqn−1,

where C is a constant depending only on N, n and d .Deligne (1973) proved a much sharper estimate in the case X is anonsingular complete intersection in PN , of dimension n:∣∣∣ |X (Fq)| − πn

∣∣∣ ≤ b′n qn/2.

Here b′n = bn − εn is its primitive nth Betti number of X (whereεn = 1 if n is even and εn = 0 if n is odd). In fact, if X hasmultidegree d = (d1, . . . , dr ), then b′n = b′n(N,d) equals

(−1)n+1(n + 1) +N∑

c=r

(−1)N+c

(N + 1

c + 1

) ∑ν1+···+νr=cνi≥1 ∀i

dν11 · · · dνrr


About the paper that received an IRCC Award

Gilles Lachaud Our joint paper entitled Etale cohomology,Lefschetz theorems and number of pointsof singular varieties over finite fields waspublished in Moscow Mathematical Journal,Vol. 2, No. 3 (2002), pp. 589-631. In fact,this was among the 3 special issues of thejournal dedicated to Yuri I. Manin on theoccasion of his 65th birthday. These issuesfeatured articles by some leadingmathematicians including S. Bloch, P.Deligne, B. Gross, G. Faltings, G. Margulis.A PDF version of the paper is available athttp://www.math.iitb.ac.in/ srg/Papers.html

and a revised version is put on the arXivhttp://arxiv.org/abs/0808.2169.


Estimates for singular complete intersections

Theorem (Deligne-type inequality for any complete intersection)

Let X be an irreducible complete intersection of dimension n inPNFq

, defined by r = N − n equations, with multidegree

d = (d1, . . . , dr ), and let s ∈ Z with dim Sing X ≤ s ≤ n− 1. Then∣∣∣ |X (Fq)|− πn

∣∣∣ ≤ b′n−s−1(N− s−1,d) q(n+s+1)/2 + Cs(X )q(n+s)/2,

where Cs(X ) is a constant independent of q. If X is nonsingular,then C−1(X ) = 0. If s ≥ 0, then

Cs(X ) ≤ 9× 2r × (rδ + 3)N+1 where δ = max{d1, . . . , dr}.

For normal complete intersections, this may be viewed as acommon refinement of Deligne’s inequality and the Lang-Weilinequality. Corollaries include previous results of Aubry and Perret(1996), Shparlinskiı and Skorobogatov (1990), as well as Hooleyand Katz (1991).


Estimates for irreducible varieties over finite fields

Theorem (Effective Lang-Weil inequality)

Suppose X is a projective variety in PNFq

or an affine variety in ANFq

defined over Fq. Let n = dim X and d = deg X . Then∣∣∣ |X (Fq)| − πn∣∣∣ ≤ (d − 1)(d − 2)qn−(1/2) + C+(X ) qn−1,

where C+(X ) is independent of q. Moreover if X is of type(m,N,d), with d = (d1, . . . , dm), and if δ = max{d1, . . . , dm},then we have

C+(X ) ≤

{9× 2m × (mδ + 3)N+1 if X is projective

6× 2m × (mδ + 3)N+1 if X is affine.

As a corollary, one obtains an analogue of a result of Schmidt(1974) that gives a lower bound for the number of points ofirreducible hypersurfaces over Fq.


A Conjecture of Lang and Weil

When Lang and Weil proved the inequality , namely,∣∣∣ |X (Fq)| − πn∣∣∣ ≤ (d − 1)(d − 2)qn−(1/2) + Cqn−1, (1)

they showed in the same paper that if K is an algebraic functionfield of dimension n over k = Fq, then there is a constant γ forwhich (1) holds with (d − 1)(d − 2) replaced by γ, for any model Xof K/k , and moreover, the smallest such constant γ is a birationalinvariant. Subsequently, Lang and Weil went on to conjecture thatthis constant γ can be described algebraically as being twice thedimension of the associated Picard variety P, at least when X isnonsingular. They made further conjectural statements relating theWeil zeta function of X and the “characteristic polynomial” of Pwhen X is projective and nonsingular.In effect, we show that these conjectures hold in the affirmativeprovided one uses the “correct” Picard variety.


Albanese and Picard

For a variety X defined over Fq of dimension n ≥ 2, denote byb+2n−1,`(X ) = dim H2n−1

+ (X ,Q`) be the (2n − 1)th virtual Betti

number and P+2n−1,`(X ,T ) = det

(1− TF |H2n−1

+ (X ,Q`))

be the“pure part” of P2n−1,`(X ,T ). Also let Albw X denote theAlbanese-Weil variety of X .

Theorem

Let X be any variety defined over Fq of dimension n ≥ 2, and letg = dim Albw X . Then P+

2n−1,`(X ,T ) = q−g fc(Albw X , qnT ). Inparticular, if φ is the map induced by the Frobenius on the Tatemodule of Albw X (tensored with Q`), then

b+2n−1,`(X ) = 2g and Tr(F | H2n−1

+ (X ,Q`)) = qn−1Tr(ϕ).

If X is normal, then Picw X and Albw X are the duals of eachother, and fc(Albw X ,T ) = fc(Picw X ,T ). Thus, the conjecturalstatements of Lang and Weil follow as a particular case.


Tools and Techniques

Proofs of the above theorems use a variety of techniques fromalgebraic geometry and topology, and to a lesser extent complexanalysis and algebra. These include

a variant of Bertini’s theorem to successively construct goodhyperplane sections.

a suitable generalization of the Weak Lefschetz Theorem forsingular varieties, which is proved in the paper.

Grothendieck-Lefschetz trace formula and Deligne’s MainTheorem for general varieties over finite fields.

Katz’s estimates for sums of Betti numbers.

Analysis of zeros and poles of the Weil zeta function andrelated objects.

Combinatorial methods to find suitable bounds using theformulae of Hirzebruch and Jouanolou for nonsingularcomplete intersections.


Feedback and Impact

The work has had a gratifying feedback from some peers includingJ.-P. Serre who called it vraiment interessant. There have beenseveral applications, some quite surprising. These include:

Work of T. Bandman, G.-M. Gruel, F. Grunewald, B. Kunyavskiı, G.Pfister and E. Plotkin (2003 and 2006) on a long standing problemin group theory

topics in diophantine equations (Waring’s problem in functionfields), by Y.-R. Liu and T. Wooley (2007)

the study of Boolean functions by F. Rodier (2008)

arithmetic progressions over finite fields, by B. Cook and A. Magyar(2010)

the study of primitive semifields by R. Gow and J. Sheekey (2011)

to coding theory, by Nakashima (2009), F. Edoukou, S. Ling, and C.Xing (2009), and also J. B. Little (2011).

There have also been several extensions and generalizations ofsome of the results, mainly due to A. Cafure and G. Matera(2007-2012).


References

For those interested in more details, can refer to the original papermentioned earlier. See also a supplement (Corrigenda andAddenda) published in Moscow Mathematical Journal, Vol. 9, No.2 (2009), pp. 431–438. These are available on my home page at:

http://www.math.iitb.ac.in/ srg/Papers.html

An expository account is available in the following paper, also jointwith G. Lachaud, and also available on my home page:

Number of solutions of equations over finite fields, and aconjecture of Lang and Weil, in: Number Theory andDiscrete Mathematics (Chandigarh, 2000), Trends inMathematics, Birkhauser, Basel (2002), pp. 269-291.

And of course one can look up the references within thesereferences to gain much knowledge!


Thank you for your attention!


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