Monday, February 11, 2013

The Poisson summation formula, arithmetic and geometry

François Loeser and I just uploaded a paper on arXiv about Motivic height zeta functions. That such a thing could be possible is quite funny, so I'll take this opportunity to break a long silence on this blog.

In Diophantine geometry, an established and important game consists in saying as much as possible of the solutions of diophantine equations. In algebraic terms, this means proving qualitative or quantitative properties of the set of integer solutions of polynomial equations with integral coefficients. In fact, one can only understand something by making the geometry more apparent; then, one is interested in integral points of schemes $X$ of finite type over the ring $\mathbf Z$ of integers. There are in fact two sub-games: one in which one tries to prove that such solutions are scarce, for example when $X$ is smooth and of general type (conjecture of Mordell=Faltings's theorem, conjecture of Lang) ; the other in which one tries to prove that there are many solutions —then, one can even try to count how many solutions there are of given height, a measure of their size. There is a conjecture of Manin predicting what would happen, and our work belongs to this field of thought.

Many methods exist to understand rational points or integral points of varieties. When the scheme carries an action of an algebraic group, it is tempting to try to use harmonic analysis. In fact, this has been done since the beginnings of Manin's conjecture when Franke, Manin, Tschinkel showed that when the variety is a generalized flag variety ($G/P$, where $G$ is semi-simple, $P$ a parabolic subgroup, for example projective spaces, grassmannians, quadrics,...), the solution of Manin's question was already given by Langlands's theory of Eisenstein series. Later, Batyrev and Tschinkel proved the case of toric varieties, and again with Tschinkel, I studied the analogue of toric varieties when the group is not a torus but a vector space. In these two cases, the main idea consists in introducing a generating series of our counting problem, the height zeta function, and establishing its analytic properties. In fact, this zeta function is a sum over rational points of a height function defined on the adelic space of the group, and the Poisson summation formula rewrites this sum as the integral of the Fourier transform of the height function over the group of topological characters. What makes the analysis possible is the fact that, essentially, the trivial character carries all the relevant information; it is nevertheless quite technical to establish what happens for other characters, and then to check that the behavior of the whole integral is indeed governed by the trivial character.

In mathematics, analogy often leads to interesting results. The analogy between number fields and function fields suggests that diophantine equations over the integers have a geometric analogue, which consists in studying morphisms from a curve to a given variety. If the ground field of the function field is finite, the dictionary goes quite far; for example, Manin's question has been studied a lot by Bourqui who established the case of toric varieties. But when the ground field is infinite, it is no more possible to count solutions of given height since they will generally be infinite.

However, as remarked by Peyre around 2000, all these solutions, which are morphisms from a curve to a scheme, form themselves a scheme of finite type. So the question is to understand the behavior of these schemes, when the height parameter grows to infinity. In fact, in an influential but unpublished paper, Kapranov had already established the case of flag varieties (without noticing)! The height zeta function is now a formal power series whose coefficients are algebraic varieties; one viewes them as elements of the Grothendieck ring of varieties, the universal ring generated by varieties with addition given by cutting-and-pasting, and multiplication given by the product of varieties. This ring is a standard tool of motivic integration (as invented by Kontsevich and developed by Denef and Loeser, and many people since). That's why this height zeta function is called motivic.

What we proved with François is a rationality theorem for such a motivic height function, when the variety is an equivariant compactification of a vector group. This means that all this spaces of morphisms, indexed by some integer, satisfy a linear dependence relation in the Grothendieck ring of varieties! To prove this result, we rely crucially on an analogue of the Poisson summation formula in motivic integration, due to Hrushovski and Kazhdan, which allows us to perform a similar analysis to the one I had done with Tschinkel in a paper that appeared last year in Duke Math. J.

Many things remain puzzling. The most disturbing is the following. If you read Tate's thesis, or Weil's Basic Number Theory, you'll see that the Riemann Roch formula and Serre's duality theorem for curves over finite fields are consequences of the Poisson summation formula in harmonic analysis. In motivic integration, things go the other way round: if one unwinds all the definitions (we explain this in our paper with François), the motivic Poisson summation formula boils down to the Riemann-Roch and Serre theorems. So, in principle, our proof could be understood just from these two theorems. But this is not clear at all how to do this directly: passing through the looking-glass to go computing our height zeta function in the Fourier world appears to be non-trivial and efficient...

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