Dynamics of Hecke correspondences

This morning, Sebastián Herrero, Ricardo Menares and Juan Rivera-Letelier released a preprint on the arXiv entitled “p-Adic distribution of CM points and Hecke orbits. I. Convergence towards the Gauss point.” In fact, there are two other papers on the arXiv with similar titles, by two other groups of authors. The first one, “p-adic Dynamics of Hecke Operators on Modular Curves” is by
Eyal Z. Goren and Payman L Kassaei, and the second one is by Daniele Disegni, “p-adic equidistribution of CM points”.

In fact, as @_nilradical pointed out initially, the words “Hecke correspondences” are generally used in modular forms books to describe some (pairwise commuting) endomorphisms of the spaces of modular forms. So what are these authors talking about?

As he said: “literally what the hell is going on in math?”. My answer, at a time where I should have been doing something else, such as sleeping, was direct:

— This is a very nice question! Take an elliptic curve over the $p$-adics, say, and draw its images under Hecke correspondences. You get a cloud of dots on the $p$-adic modular curve. Normalize by its size. What are the limit measures ? The picture is richer than over the complex.

— what do u mean by "correspondences"? i know only about hecke operators to the extent of what is in chapter 3 of shimura's intro book and the end of serre's "course in arithmetic"

— You remember that paragraph of Serre's Book where he says modular forms of wt k are functions on pairs (E,ω) — E, ell curve ; ω, nonzero inv diff form on E — with some homogeneity and holomorphy condition ? (Idea: τ -> ell curve C/(Z+τZ) and ω = dz) ...

— yea, im ok with weight k modular forms being sections of \Omega^2k

— In this point of view, the Hecke operator T_n sends a modular form f to the function that maps (E,ω) to the sum of all f(E',ω') where E' is a quotient E/D (D, cyclic subgroup of order n) and ω' I don't tell here. Now, forget about modular forms (and differential forms ω, as well).

And I went to explain those things in a few Twitter posts, but it's time to say it at a quieter pace.

1. Elliptic curves, modular curves, and modular correspondences.

The reason for the adjective “modular” in the expression “modular forms” or in the expression “modular curves” is that they refer to the “moduli” of some extremly classical geometric objects called elliptic curves. Elliptic curves have various descriptions, all equally beautiful and important: either, complex analytically, as one-dimensional tori, quotients of the complex plane $\mathbf C$ by a lattice $\Lambda\simeq\mathbf Z^2$, or as cubic plane curves given by a (Weierstrass) equation of the form $y^2=x^3+ax+b$ in the affine plane with coordinates $(x,y)$ — forgetting a point at infinity.

In fact, the “set” of all elliptic curve can themselved be arranged in a curve provided one identifies “isomorphic” elliptic curves, so that some simplifications arise. Not any lattice must be considered, one first may assume $\Lambda=\mathbf Z+\mathbf Z\tau$, for some complex number $\tau$ with strictly positive imaginary part, but then the complex number $\tau$ and any complex number of the form $(a\tau+b)/(c\tau+d)$ gives the same elliptic curve, for any matrix $\begin{pmatrix}a & b\\c & d\end{pmatrix}$ with integer entries and determinant $1$. So, in some sense, the set of elliptic curves coincides with the quotient of the (Poincaré) upper half-space by the action of the group $\mathrm{SL}(2,\mathbf Z)$. Alternatively, in the Weierstrass model, there are two parameters $(a,b)$, but all pairs of the form $(u^4a,u^6b)$ give the same curve (via the change of variables $x'=u^2x$, $y'=u^3y$). One can guess some subtleties here, because some the matrix $-I_2$ acts trivially, and also because some specific $\tau$ have a nontrivial stabilizer (even taking $-I_2$ into acount); on the other side, $u=-1$ does not act, and some pairs $(a,b)$ are fixed by some more roots of unity. I will ignore these here; technically, they are solved in two ways, one is to add a “level structure”, the other is to consider this modular curve $M$ as an orbifold — and algebraic geometers say stack.

“Correspondence” is a long-forgotten topic in set theory, since the time where multivalued functions were considered. We forgot them because it's hard to talk consistently about them, but there are instances in which they arise naturally. In set theory, one way to define a correspondence $T$ from a set $X$ to a set $Y$ is to consider its graph $G_T$, a subset of $X\times Y$. If the graph contains a pair $(x,y)$, one says that the correspondence maps $x$ to $y$; but the graph could also contain a pair $(x,y')$, in which case it also maps $x$ to $y'$. And it could very well contain no pair with first element $x$, and then $x$ has no image under the correspondence. Correspondences can be composed: if the correspondence $S$ maps $x$ to $y$ and a correspondence $T$ maps $y$ to $z$,
then $T\circ S$ maps $x$ to $z$. They can also be inverted: just flip the graph.

Modular curves admit natural modular correspondences, indexed by strictly positive integers.
Specifically, the correspondence $T_n$ maps an elliptic curve $E$ to all elliptic curves $E'$ of
the form $E'=E/D$, where $D$ is a cyclic subgroup of rank $n$ of $E$.
The graph of this correspondence, when drawn on the surface $M\times M$, has a natural structure of an algebraic curve.

2. Modular dynamics

One virtue of correspondences in algebraic geometry is that they act naturally on objects that are attached “linearly” to the curve. For example, considering a function $f$ (or a differential form on $M$), rather than looking at all the images $E'$ of $E$, one could add the corresponding values $f(E')$, and consider thus sum as a function of $E$. This is exactly what is done to define the Hecke correspondence on modular forms. Geometrically this corresponds to pulling back $f$ from $M$ to $M\times M$ (using the first projection), then restricting on the graph of the correspondence, then “pushing-out” (a kind of trace) to the second factor — this is where the sum happens.

Another way to linearize the correspondence is to formally add all images $E'$, for example considering that the correspondence maps a Dirac measure $\delta_E$ at a point $E$
to the sum of the Dirac measures $\delta_{E'}$ at its images; maybe dividing by the number of images so that one keeps a probability measure. In this framework, the correspondence $T_n$ is a map from the space of probability measures on the modular curve to itself.

Which is cool because you can now iterate the process and wonder about the possible limit measures.

This is analogous to what is done in complex dynamics, for example when you study the dynamics
of the map $z\mapsto z^2+c$ (here, $c$ is a parameter): a construction of the Julia set consists in taking the 2 preimages of a point $z$ (any point, with a few exceptions), their 4 preimages, etc. As long as the construction goes on, the cloud of points that one gets becomes closer and closer to the Julia set.

In the case of the modular curve and the dynamics of Hecke correspondences (on can compose a given Hecke correspondence, or consider $T_n$ for large $n$, it does not really matter), what happens is described by theorems of William Duke / Laurent Clozel and Emmanuel Ullmo / Rodolphe Richard.
Whatever point one starts with, whatever probability measure one starts with, the probability measures on $M$ that are constructed by the dynamics converge to the Poincaré measure on the modular curve — that is, to the measure that is given by the hyperbolic measure $dx\, dy/ y^2$ on the Poincaré upper half-plane, restricted to a fundamental domain of the action.

In fact, Duke and Clozel/Ullmo have different goals. What they consider is a sequence of probability measures formed by considering elliptic curves with complex multiplication and all their conjugates. The limit theorem that they prove is then quite subtle and relies on deep properties of Maass forms of half integral weight.

3. Modular dynamics: $p$-adic fields

Over the $p$-adics, the dynamics is even more complicated, although its behaviour does not seem to be governed by analytic properties of analytic objects such as Maass forms, but by the arithmetic of the elliptic curve one starts with.

A first difficulty comes from the framework required to define $p$-adic dynamics. One wants to start from the field $\mathbf Q_p$, but it is not algebraically closed, and so the construction of the images of a curve by the correspondence require to take its algebraic closure $\overline{\mathbf Q_p}$. But then $p$-adic analysis is not so cool, because although that field has a natural $p$-adic absolute value, it is not complete anymore — so let's take its completion, $\mathbf C_p$. And this field is algebraically closed.

However, and this is a reflection that the Galois theory of $\mathbf Q_p$ is complicated, the field $\mathbf C_p$ is not locally compact, so that measure theory on such a field is not very well behaved. For example, a basic tool in measure theory over compact (metrizable, say) spaces is that the set of probability measures is itself compact for the vague topology on measures, so that any sequence of probability measures has a converging subsequence, etc.

In our case, this won't hold anymore and one needs to consider a suitable “compactification” of the $p$-adic modular curve — in fact, its analytification $M_p$ in the sense of Berkovich. One then gets a locally compact topological space on which the Hecke correspondences still act naturally, and
questions of dynamics now can be formulated properly.

The only thing I'll write about the dynamics is that it is subtle: there are domains which are stable by the correspondences. For example, if the reduction mod $p$ of an elliptic curve $E$ is supersingular, then all of its images $E' = E/D$ also have supersingular reduction. In other words, the supersingular locus in the Berkovich modular curve $M_p$ is totally invariant by the Hecke correspondence — this locus is a finite union of open disks. This shows that there are many possible limit measure and the papers that I have quoted above study the various limit phenomaena.

They also consider the analogue of Duke's result in that setting. Disegni also considers the analogous problem for Shimura curves (instead of elliptic curves, they parameterize abelian surfaces with real multiplication).

This will be all for this blog spot, now go and read these papers!

Radon measures form a sheaf for a natural Grothendieck topology on topological spaces

First post of the year, so let me wish all of you a happy new year!

Almost two years ago, Antoine Ducros and I released a preprint about differential forms and currents on Berkovich spaces. We then embarked in revising it thoroughly; unfortunately, we had to correct a lot of inaccuracies, some of them a bit daunting. We made a lot of progress and we now have a much clearer picture in mind. Fortunately, all of the main ideas remain the same.

A funny thing emerged, which I want to explain in this blog.

One of our mottos was to define sheaves of differential forms, or of currents. Those differential forms were defined in two steps : by definition, they are locally given by tropical geometry, so we defined a presheaf of tropical forms, and passed at once to the associated sheaf. What we observed recently is that it is worth spending some time to study the presheaf of tropical forms.

Also, Grothendieck topologies play such an important rôle in analytic geometry over non-archimedean fields; this is obvious for classical rigid spaces, but they are also important in Berkovich geometry, in particular if you want to care about possibly non-good spaces for which points may not have a neighborhood isomorphic to an affinoid space. So it was natural to sheafify the presheaf of tropical forms for the G-topology, giving rise to a G-sheaf of G-forms.

Now, every differential form of maximal degree $\omega$ on a Berkovich space $X$ gives rise to a measure on the topological space underlying $X$. Our proof of this is a bit complicated, and was made more complicated by the fact that we first tried to define the integral $\int_X \omega$, and then defined $\int_X f\omega$ for every smooth function $f$, and then got $\int_X f\omega$ for every continuous function with compact support $f$ by approximation, using a version of the Stone-Weierstrass theorem in our context.

In the new approach, we directly concentrate on the measure that we want to construct. For G-forms, this requires to glue measures defined locally for the G-topology. As it comes out (we finished to write down the required lemmas today), this is quite nice.

Since Berkovich spaces are locally compact, we may restrict ourselves to classical measure theory on locally compact spaces. However, we may not make any metrizability assumption, nor any countability assumption, since the most basic Berkovich spaces lack those properties. Assume that the ground non-archimedean field $k$ is the field $\mathbf C((t))$ of Laurent series over the field $\mathbf C$ of complex numbers. Then the projective line $\mathrm P^1$ over $k$ is not metrizable, and the complement of its ``Gauss point'' $\gamma$ has uncountably many connected components (in bijection with the projective line over $\mathbf C$). Similarly, the complement of the Gauss point in the projective plane $\mathrm P^2$ over $k$ is connected, but is not countable at infinity, hence not paracompact.

As always, there are two points of view on measure theory: Borel measures (countably additive set functions on the $\sigma$-algebra of Borel sets) and Radon measures (linear forms on the vector space of continuous compactly supported functions). By the theorem of Riesz, they are basically equivalent: locally finite, compact inner regular Borel measures are in canonical bijection with Radon measures. Unfortunately, basic litterature is not very nice on that topic; for example, Rudin's book constructs an outer regular Borel measure which may not be inner regular, while for us, the behavior on compact sets is really the relevant one.

Secondly, we need to glue Radon measures defined on the members of a G-cover of our Berkovich space $X$. This is possible because Radon measures on a locally compact topological space naturally form a sheaf for a natural Grothendieck topology!

Let $X$ be a locally compact topological space and let us consider the category of locally compact subspaces, with injections as morphisms.  Radon measures can be restricted to a locally compact subspace, hence form a presheaf on that category.

Let us decree that a family $(A_i)_{i\in I}$ of locally compact subspaces of a locally compact subspace $U$ is a B-cover (B is for Borel) if for every point $x\in U$, there exists a finite subset $J$ of $I$ such that $x\in A_i$ for every $i\in J$ and such that $\bigcup_{i\in J}A_i$ is a neighborhood of $x$. B-covers form a G-topology on the category of locally compact subsets, for which Radon measures form a sheaf! In other words, given Radon measures $\mu_i$ on members $A_i$ of a B-cover of $X$ such that the restrictions to $A_i\cap A_j$ of $\mu_i$ and $\mu_j$ coincide, for all $i,j$, then there exists a unique Radon measure on $X$ whose restriction to $A_i$ equals $\mu_i$, for every $i$.

This said, the proof (once written down carefully) is not a big surprise, nor specially difficult,  but I found it nice to get a natural instance of sheaf for a Grothendieck topology within classical analysis.