Saturday, February 16, 2013

The Poisson summation formula, Minkowski's first theorem, and the density of sphere packings

I opened by accident a paper by Henry Cohn and Noam Elkies, New upper bounds on sphere packings I, Annals of Mathematics, 157 (2003), 689–714, and I had the good surprise to see a beautiful application of the Poisson summation formula to upper bounds for the density of sphere packings.

In fact, their argument is very close to a proof of the first theorem of Minkowski about lattice points in convex bodies which I had discovered in 2009. However, a final remark in their paper, appendix C, shows that this proof is not really new. Anyway, the whole story is nice enough to prompt me to discuss it in this blog.

1. The Poisson formula

I first recall the Poisson formula: let $f\colon\mathbf R^n\to \mathbf R$ be a continuous function whose Fourier transform $\hat f$ is integrable, let $L$ be a lattice in $\mathbf R^n$, and let $L'$ be the dual lattice of $L$. Then, 
\[ \sum_{x\in L} f(x) = \frac1{\mu(\mathbf R^n/L)}\sum_{y\in L'} \hat f(y). \]

In fact, one needs a bit more about $f$ that the above hypotheses (but we'll ignore this in the sequel). For example, it is sufficient that $f(x)$ and $\hat f(x)$ be bounded from above by a multiple of $1/(1+\| x\|)^{n+\epsilon}$, for some strictly positive real number $\epsilon$.

2. Minkowski's first theorem

Let $B$ a closed symmetric convex neighborhood of $0$. Minkowski's first theorem bounds from below the cardinality of $B\cap L$. A natural idea would be to apply the Poisson summation formula to the indicator function $f_B$ of $B$. However, $f_B$ is not continuous, so we need to replace $f_B$ by some function $f$ which satisfies the following properties:
  • $f\leq f_B$, so that $\#(B\cap L)= \sum_{x\in L} f_B(x)\geq \sum_{x\in L} f(x) $;
  • $\hat f\geq 0$, so that $\sum_{y\in L'} \hat f(y) \geq \hat f(0)$.
The second condition suggest to take for $f$ a function of the form $g*\check g$, so that $\hat f=|\hat g| ^2$. Then, the first condition will hold outside of $B$ if $g$ is supported in $\frac12 B$. Let's try $g=cf_{B/2}$ for some positive real number $c$. For all $x\in\mathbf R^n$,  
\[ f(x) \leq f(0) =c^2 \int_{\mathbf R^n} f_{B/2}(x) f_{B/2}(-x) = c^2 \mu(B/2) = c^2 \mu(B)/2^n. \]
It suffices to choose $c=(2^n/\mu(B))^{1/2}$.
On the other hand, 
\[ \hat g(0) = \int_{\mathbf R^n} f_{B/2}(x) = \mu(B/2)=c\mu(B)/2^n=1/c. \]
Finally, the Poisson formula implies
$\displaystyle \#(B\cap L) \geq \sum_{x\in L} f(x)  = \frac1{\mu(\mathbf R^n/L)} \sum_{y\in L'}\hat f(y)  \geq \frac1{\mu(\mathbf R^n/L)} \hat f(0) $
$\displaystyle = \frac1{\mu(\mathbf R^n/L)} | \hat g(0)|^2  = \frac{\mu(B)}{2^n}\mu(\mathbf R^n/L).$

This is exactly Minkowski's first theorem!

Of course, one may consider apply this argument to other functions $f$. In the case where $B$ is an Euclidean ball, a natural choice consists in taking a Gaussian $f(x)=a\exp(-b\|x\|^2)$, since then $\hat f$ has the same form. This is what has essentially been done by Schoof and van der Geer in their paper Effectivity of Arakelov divisors and the analogue of the theta divisor of a number field, Selecta Math. New Ser. 6 (2000), 377–398, and Damian Rössler independently. Indeed, up to normalization factors, the left hand side of the Poisson summation formula is then interpreted as the exponential of the $h^0$ of a line bundle over an arithmetic curve, and the Poisson summation formula itself is the analogue of Serre's duality theorem in Arakelov geometry. As Jean-Benoît Bost explained to me, Gaussian functions provide an inequality for $\#(B\cap L)$ which is sharper than Minkowski's first theorem. The details of the computation can be found in my notes about Arakelov geometry (beware, these are mostly a work in slow progress). I have not tried to look for optimal functions beyond that case.

3. The theorem of Cohn-Elkies

We consider a sphere packing, that is a set of points in $\mathbf R^n$ with mutual distances at least 1,
and we want to bound from above its density, that is the ratio of volume occupied by balls of radius $1/2$ centered at these spheres. One may think of a lattice packing, the particular case where the centers of these spheres are exactly the points of a lattice $L$ or, more generally, of periodic packing when the centers of the spheres are finitely many translates $v_1+L,\dots,v_N+L$ of  a lattice $L$. In fact, Cohn and Elkies argue that it suffices to study such periodic packings (repeating periodically an arbitrarily large part of the sphere packing), so we shall do like them and assume that our sphere packing is periodic. 

Now, let $f$ be a real valued function on $\mathbf R^n$ satisfying the following properties:
  • $f$ is continuous, integrable on $\mathbf R^n$ as well as its Fourier transform.
  • $f(0)>0$ and $f(x)\leq 0$ for $\| x\|\geq 1$;
  • $\hat f(x)\geq 0$ for all $x$.
Then, the density $\Delta$ of the sphere packing satisfies 
\[ \Delta \leq 2^{-n} \mu(B) \frac{f(0)}{\hat f(0)}, \]
where $\mu(B)$ is the volume of the unit ball.

We assume that no difference $v_i-v_j$ is a point of the lattice $L$ (otherwise, we can exclude one translate from the list). With the above notation, the fundamental parallelepiped of the lattice contains exactly $N$ balls of radius $1/2$, hence 
\[ \Delta= 2^{-n}\mu(B) \frac N{\mu(\mathbf R^n/L)}. \]
For any $v\in\mathbf R^n$, the Poisson summation formula for $x\mapsto f(x+v)$ writes
\[ \sum_{x\in L} f(x+v) = \frac1{\mu(L)} \sum_{y\in L'} e^{2\pi i \langle v,y\rangle} \hat f(y), \]
$\displaystyle \sum_{1\leq j,k\leq N} \sum_{x\in L} f(x+v_j-v_k)  = \frac1{\mu(L)} \sum_{y\in L'}\hat f(y)  \sum_{j,k=1}^N e^{2\pi i \langle v_j-v_k,y\rangle}$ 
$\displaystyle = \frac1{\mu(L)} \sum_{y\in L'}\hat f(y) \left| \sum_{j=1}^N e^{2\pi i \langle v_j,y\rangle} \right|^2.$
All terms of the right hand side are positive or null, so that we can bound it from below by
the term for $y=0$, hence
\[ \sum_{1\leq j,k\leq N} \sum_{x\in L} f(x+v_j-v_k) \geq N^2 \frac{\hat f(0)}{\mu(\mathbf R^n/L)}. \]
Now, there is a sphere of our packing centered at $x+v_j$, and another at $v_k$,
so that $\| x+v_j-v_k\|\geq 1$ unless $x+v_j=v_k$, that is $v_k-v_j=x$ hence $x=0$ and $v_j=v_k$. In the first case, the value of $f$ at $x+v_j-v_k$ is negative or null; in the latter, it equal $f(0)$. Consequently, the left hand side of the previous inequality is at most $Nf(0)$.  Finally, $\displaystyle N f(0) \geq N^2 \hat f(0)/\mu(\mathbf R^n/L)$, hence the desired inequality.

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...