Sunday, February 5, 2017

Counting points and counting curves on varieties — Tribute to Daniel Perrin

$\require{enclose}\def\VarC{\mathrm{Var}_{\mathbf C}}\def\KVarC{K_0\VarC}$
Daniel Perrin is a French algebraic geometer who turned 70 last year. He his also well known in France for his wonderful teaching habilities. He was one of the cornerstones of the former École normale supérieure de jeunes filles, before it merged in 1985 with the rue d'Ulm school. From this time remains a Cours d'algèbre which is a must for all the students (and their teachers) who prepare the agrégation, the highest recruitment process for French high schools. He actually taught me Galois theory (at École normale supérieure in 1990/1991) and Algebraic Geometry (the year after, at Orsay). His teaching restlessly stresses  the importance of examples. He has also been deeply involved in training future primary school teachers, as well as in devising the mathematical curriculum of high school students: he was responsible of the report on geometry. It has been a great honor for me to be invited to lecture during the celebration of his achievements that took place at Orsay on November, 23, 2016.

Diophantine equations are a source of numerous arithmetic problems. One of them has been put forward by Manin in the 80s and consists in studying the behavior of the number of solutions of such equations of given size, when the bound grows to infinity. A geometric analogue of this question considers the space of all curves with given degree which are drawn on a fixed complex projective, and is interested in their behavior when the degree tends to infinity. This was the topic of my lecture and is the subject of this post.

Let us first begin with an old problem, apparently studied by Dirichlet around 1840, and given a rigorous solution by Chebyshev and Cesáro around 1880: the probability that two integers be coprime is equal to 6/π26/\pi^2. Of course, there is no probability on the integers that has the properties one would expect, such as being invariant by translation, and the classical formalization of this problem states that the numbers of pairs (a,b)(a,b) of integers such that 1a,bn1\leq a,b\leq n and gcd(a,b)=1\gcd(a,b)=1 grows as n26/π2n^2 \cdot 6/\pi^2 when n+n\to+\infty,

This can be proved relatively easily, for example as follows. Without the coprimality condition, there are n2n^2 such integers. Now one needs to remove those pairs both of which entries are multiples of 22, and there are n/22\lfloor n/2\rfloor^2 of those, those where a,ba,b are both multiples of 33 (n/32\lfloor n/3\rfloor^2), and then comes 55, because we have already removed those even pairs, etc. for all prime numbers. But in this process, we have removed twice the pairs of integers both of which entries are multiples of 23=62\cdot 3=6, so we have to add them back, and then remove the pairs of integers both of which are multiples of 2352\cdot 3\cdot 5, etc. This leads to the following formula for
the cardinality C(n)C(n) we are interested in:

$\displaystyle
 C(n) = n^2 - \lfloor\frac n2\rfloor^2 - \lfloor \frac n3\rfloor^2-\lfloor \frac n5\rfloor^2 - \dots
+ \lfloor \frac n{2\cdot 3}\rfloor^2+\lfloor\frac n{2\cdot 5}\rfloor^2+\dots
- \lfloor \frac n{2\cdot 3\cdot 5} \rfloor^2 - \dots $.

Approximating n/a\lfloor n/a\rfloor by n/an/a, this becomes

$\displaystyle
C(n) \approx  n^2 - \left(\frac n2\right)-^2 - \left (\frac n3\right)^2-\left( \frac n5\rfloor\right)^2 - \dots
+ \left (\frac n{2\cdot 3}\right)^2+\left(\frac n{2\cdot 5}\right)^2+\dots
- \left (\frac n{2\cdot 3\cdot 5} \right)^2 - \dots $

which we recognize as

$\displaystyle
C(n)\approx n^2 \left(1-\frac1{2^2}\right) \left(1-\frac1{3^2}\right)\left(1-\frac1{5^2}\right) \dots
=n^2/\zeta(2)$,

where ζ(2)\zeta(2) is the value at s=2s=2 of Riemann's zeta function ζ(s)\zeta(s). Now, Euler had revealed the truly arithmetic nature of π\pi by proving in 1734 that ζ(2)=π2/6\zeta(2)=\pi^2/6. The approximations we made in this calculation can be justified, and this furnishes a proof of the above claim.

We can put this question about integers in a broader perspective if we recall that the ring Z\mathbf Z is a principal ideal domain (PID) and study the analogue of our problem in other PIDs, in particular for F[T]\mathbf F[T], where F\mathbf F is a finite field; set q=Card(F)q=\operatorname{Card}(\mathbf F). The above proof can be adapted easily (with simplifications, in fact) and shows that number of pairs (A,B)(A,B) of monic polynomials of degrees n\leq n such that gcd(A,B)=1\gcd(A,B)=1 grows as qn(11/q)q^n(1-1/q) when n+n\to+\infty. The analogy becomes stronger if one observes that 1/(11/q)1/(1-1/q) is the value at s=2s=2 of 1/(1q1s)1/(1-q^{1-s}), the Hasse-Weil zeta function of the affine line over F\mathbf F.

What can we say about our initial question if we replace the ring Z\mathbf Z with the PID C[T]\mathbf C[T]? Of course, there's no point in counting the set of pairs (A,B)(A,B) of coprime monic polynomials of degree n\leq n in C[T]\mathbf C[T], because this set is infinite. Can we, however, describe this set? For simplicity, we will consider here the set VnV_n of pairs of coprime monic polynomials of degree precisely nn. If we identify a monic polynomial of degree nn with the sequence of its coefficients, we then view VnV_n as a subset of Cn×Cn\mathbf C^{n}\times\mathbf C^n. We first observe that VnV_n is an Zariski open subset of C2n\mathbf C^{2n}: its complement WnW_n is defined by the vanishing of a polynomial in 2n2n variables — the resultant of AA and BB.

When n=0n=0, we have V0=C0={pt}V_0=\mathbf C^0=\{\mathrm{pt}\}.

Let's look at n=1n=1: the polynomials A=T+aA=T+a and B=T+bB=T+b are coprime if and only if aba\neq b;
consequently, V1V_1 is the complement of the diagonal in C2\mathbf C^2.

For n=2n=2, this becomes more complicated: the resultant of the polynomials T2+aT+bT^2+aT+b and T2+cT+dT^2+cT+d is equal to a2dabcadc+b22bd+bc2+d2a^2d-abc-adc+b^2-2bd+bc^2+d^2; however, it looks hard to guess some relevant properties of VnV_n (or of its complement) just by staring at this equation. In any case, we can say that V2V_2 is the complement in C4\mathbf C^4 of the union of two sets, corresponding of the degree of the gcd of (A,B)(A,B). When gcd(A,B)=2\gcd(A,B)=2, one has A=BA=B; this gives the diagonal, a subset of C4\mathbf C^4 isomorphic to C2\mathbf C^2; the set of pairs of polynomials (A,B)(A,B) whose gcd has degree 11 is essentially C×V1\mathbf C\times V_1: multiply a pair (A1,B1)(A_1,B_1) of coprime polynomials of degree 11 by an arbitrary polynomial of the form (Td)(T-d).
Consequently,
\begin{align}V_2&=\mathbf C^4 - \left( \mathbf C^2 \cup \mathbf C\times V_1\right)\\
&= \mathbf C^4 - \left( \enclose{updiagonalstrike}{\mathbf C^2}\cup \left(\mathbf C\times (\mathbf C^2-\enclose{updiagonalstrike}{\mathbf C})\right)\right)\\
&=\mathbf C^4-\mathbf C^3
\end{align}
if we cancel the two C2\mathbf C^2 that appear. Except that this makes no sense!

However, there is a way to make this computation both meaningful and rigorous, and it consists in working in the Grothendieck ring $\KVarC$ of complex algebraic varieties. Its additive group is generated by isomorphism classes of algebraic varieties, with relations of the form [X]=[U]+[Z][X]=[U]+[Z] for every Zariski closed subset ZZ of an algebraic variety XX, with complement U=XZU=X-Z. This group has a natural ring structure for which [X][Y]=[X×Y][X][Y]=[X\times Y]. Its unit element is the class of the point, [A0][\mathbf A^0] if one wishes. An important element of this ring $\KVarC$ is the class L=[A1]\mathbf L=[\mathbf A^1] of the affine line. The natural map $e\colon \VarC\to \KVarC$ given by e(X)=[X]e(X)=[X] is the universal Euler characteristic: it is the universal map from $\VarC$ to a ring such that e(X)=e(XZ)+e(Z)e(X)=e(X-Z)+e(Z) and e(X×Y)=e(X)e(Y)e(X\times Y)=e(X)e(Y), where X,YX,Y are complex varieties and ZZ is a Zariski closed subset of XX.

In particular, it generalizes the classical Euler characteristic, the alternate sum of the dimensions of the cohomology groups (with compact support, if one wishes) of a variety. A subtler invariant of $\KVarC$ is given by mixed Hodge theory: there exists a unique ring morphism $\chi_{\mathrm H}\KVarC\to\mathbf Z[u,v]$ such that for every complex variety XX, χH([X])\chi_{\mathrm H}([X]) is the Hodge-Deligne polynomial of XX. In particular, if XX is projective and smooth, χH([X])=supp,qdimhq(X,ΩXp)upvq\chi_{\mathrm H}([X])=\sup_{p,q} \dim h^q(X,\Omega^p_X) u^pv^q. If one replaces the field of complex numbers with a finite field F\mathbf F, one may actually count the numbers of F\mathbf F-points of XX, and this furnishes yet another generalized Euler characteristic.

The preceding calculation shows that e(V0)=1e(V_0)=1, e(V1)=L2Le(V_1)=\mathbf L^2-\mathbf L and e(V2)=L4L3e(V_2)=\mathbf L^4-\mathbf L^3; more generally, one proves by induction that e(Vn)=L2nL2n1e(V_n)=\mathbf L^{2n}-\mathbf L^{2n-1} for every integer n0n\geq 0.

Equivalently, one has e(Wn)=L2n1e(W_n)=\mathbf L^{2n-1} for all nn. I have to admit that I see no obvious reason for the class of WnW_n to be equal to that of an affine space. However, as Ofer Gabber and Jean-Louis Colliot-Thélène pointed out to me during the talk, this resultant is the difference of two homogeneous polynomials pqp-q of degrees d=2d=2 and d+1=3d+1=3; consequently, the locus it defines is a rational variety — given a,b,ca,b,c, there is generically a unique tt such that pqp-q vanishes at (at,bt,ct,t)(at,bt,ct,t).

These three results have a common interpretation if one brings in the projective line P1\mathbf P_1. Indeed, pairs (a,b)(a,b) of coprime integers (up to ±1\pm1) correspond to rational points on P1\mathbf P_1, and if F\mathbf F is a field, then pairs (A,B)(A,B) of coprime polynomials in F[T]\mathbf F[T] correspond (up to F×\mathbf F^\times) to elements of P1(F(T))\mathbf P_1(\mathbf F(T)).
In both examples, the numerical datum max(a,b)\max(|a|,|b|) or max(deg(A),deg(B))\max(\deg(A),\deg(B)) is called the height of the corresponding point.

In the case of the ring Z\mathbf Z, or in the case of the ring F[T]\mathbf F[T] where F\mathbf F is a finite field, one has an obvious but fundamental finiteness theorem: there are only finitely many points of P1\mathbf P_1 with bounded height. In the latter case, C[T]\mathbf C[T], this naïve finiteness does not hold. Nevertheless, if one sees P1(C(T))\mathbf P_1(\mathbf C(T)) as an infinite dimensional variety — one needs infinitely many complex numbers to describe a rational function, then the points of bounded height constitute what is called a bounded family, a “finite dimensional” constructible set.

The last two examples have a common geometric interpretation. Namely, F(T)\mathbf F(T) is the field of functions of a projective smooth algebraic curve CC over F\mathbf F; in fact, CC is the projective line again, but we may better ignore this coincidence. Then a point xP1(F(T))x\in\mathbf P_1(\mathbf F(T))
corresponds to a morphism εx ⁣:CP1\varepsilon_x\colon C\to\mathbf P_1, and the formula H(x)=deg(ϵxO(1))H(x)=\deg(\epsilon_x^*\mathscr O(1)) relates the height H(x)H(x) of xx to the degree of the morphism εx\varepsilon_x.

Since the notion of height generalizes from P1\mathbf P_1 to projective spaces Pn\mathbf P_n of higher dimension (and from Q\mathbf Q to general number fields), this suggests a general question. Let VPnV\subset\mathbf P_n be a projective variety over a base field kk hat can one say about the set of points xV(k)x\in V(k) such that H(x)BH(x)\leq B, when the bound BB grows to \infty?
The base field kk can be either a number field, or the field of functions F(C)\mathbf F(C) of a curve CC over a finite field F\mathbf F, or the field of functions C(C)\mathbf C(C) of a curve over the complex numbers. In the last two cases, the variety can even be taken to be constant, deduced from a variety V0V_0 over F\mathbf F or C\mathbf C.

  1. When kk is a number field, this set is a finite set; how does its cardinality grows? This is a question that Batyrev and Manin have put forward at the end of the 80s, and which has attracted a lot of attention since.
  2. When k=F(C)k=\mathbf F(C) is a function field over a finite field, this set is again a finite set; how does its cardinality grows? This question has been proposed by Emmanuel Peyre by analogy with the question of Batyrev and Manin.
  3. When k=C(C)k=\mathbf C(C) is a function field over C\mathbf C, this set identifies with a closed subscheme of the Grothendieck-Hilbert scheme of VV; what can one say about its geometry, in particular about its class in $\KVarC$? Again, this question has been proposed by Emmanuel Peyre around 2000.

In a forthcoming post, I shall recall some results on these questions, especially the first one, and in particular explain an approach based on the Fourier summation formula. I will then explain a theorem proved with François Loeser where we make use of Hrushovski–Kazhdan's motivic Fourier summation formula in motivic integration to prove an instance of the third question.