
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 . 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 of integers such that and grows as when ,
This can be proved relatively easily, for example as follows. Without the coprimality condition, there are such integers. Now one needs to remove those pairs both of which entries are multiples of , and there are of those, those where are both multiples of (), and then comes , 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 , so we have to add them back, and then remove the pairs of integers both of which are multiples of , etc. This leads to the following formula for
the cardinality 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 by , 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 is the value at of Riemann's zeta function . Now, Euler had revealed the truly arithmetic nature of by proving in 1734 that . 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 is a principal ideal domain (PID) and study the analogue of our problem in other PIDs, in particular for , where is a finite field; set . The above proof can be adapted easily (with simplifications, in fact) and shows that number of pairs of monic polynomials of degrees such that grows as when . The analogy becomes stronger if one observes that is the value at of , the Hasse-Weil zeta function of the affine line over .
What can we say about our initial question if we replace the ring with the PID ? Of course, there's no point in counting the set of pairs of coprime monic polynomials of degree in , because this set is infinite. Can we, however, describe this set? For simplicity, we will consider here the set of pairs of coprime monic polynomials of degree precisely . If we identify a monic polynomial of degree with the sequence of its coefficients, we then view as a subset of . We first observe that is an Zariski open subset of : its complement is defined by the vanishing of a polynomial in variables — the resultant of and .
When , we have .
Let's look at : the polynomials and are coprime if and only if ;
consequently, is the complement of the diagonal in .
For , this becomes more complicated: the resultant of the polynomials and is equal to ; however, it looks hard to guess some relevant properties of (or of its complement) just by staring at this equation. In any case, we can say that is the complement in of the union of two sets, corresponding of the degree of the gcd of . When , one has ; this gives the diagonal, a subset of isomorphic to ; the set of pairs of polynomials whose gcd has degree is essentially : multiply a pair of coprime polynomials of degree by an arbitrary polynomial of the form .
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 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 for every Zariski closed subset of an algebraic variety , with complement . This group has a natural ring structure for which . Its unit element is the class of the point, if one wishes. An important element of this ring $\KVarC$ is the class of the affine line. The natural map $e\colon \VarC\to \KVarC$ given by is the universal Euler characteristic: it is the universal map from $\VarC$ to a ring such that and , where are complex varieties and is a Zariski closed subset of .
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 , is the Hodge-Deligne polynomial of . In particular, if is projective and smooth, . If one replaces the field of complex numbers with a finite field , one may actually count the numbers of -points of , and this furnishes yet another generalized Euler characteristic.
The preceding calculation shows that , and ; more generally, one proves by induction that for every integer .
Equivalently, one has for all . I have to admit that I see no obvious reason for the class of 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 of degrees and ; consequently, the locus it defines is a rational variety — given , there is generically a unique such that vanishes at .
These three results have a common interpretation if one brings in the projective line . Indeed, pairs of coprime integers (up to ) correspond to rational points on , and if is a field, then pairs of coprime polynomials in correspond (up to ) to elements of .
In both examples, the numerical datum or is called the height of the corresponding point.
In the case of the ring , or in the case of the ring where is a finite field, one has an obvious but fundamental finiteness theorem: there are only finitely many points of with bounded height. In the latter case, , this naïve finiteness does not hold. Nevertheless, if one sees 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, is the field of functions of a projective smooth algebraic curve over ; in fact, is the projective line again, but we may better ignore this coincidence. Then a point
corresponds to a morphism , and the formula relates the height of to the degree of the morphism .
Since the notion of height generalizes from to projective spaces of higher dimension (and from to general number fields), this suggests a general question. Let be a projective variety over a base field hat can one say about the set of points such that , when the bound grows to ?
The base field can be either a number field, or the field of functions of a curve over a finite field , or the field of functions 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 over or .
- When 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.
- When 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.
- When is a function field over , this set identifies with a closed subscheme of the Grothendieck-Hilbert scheme of ; 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.