Naser Sardari is finishing a postdoc at Wisconsin this year and just gave a beautiful talk about his new paper. Now Naser thinks of this as a paper about automorphic forms — and it is — but I want to argue that it is *also* a paper which develops an unexpected new form of the Chabauty method! As I will now explain. Tell me if you buy it.

First of all, what does Naser prove? As the title might suggest, it’s a statement about the multiplicity of Hecke eigenvalues a_p; in this post, we’re just going to talk about the eigenvalue zero. The Hecke operator T_p acts on the space of weight-k modular forms on Gamma_0(N); how many zero eigenvectors can it have, as k goes to infinity with N,p fixed? If you believe conjectures of Maeda type, you might expect that the Hecke algebra acts irreducibly on the space S_k(Gamma_0(N)); of course this doesn’t rule out that one particular Hecke operator might have some zeroes, but it should make it seem pretty unlikely.

And indeed, Naser proves that the number of zero eigenvectors is *bounded independently of k*, and even gives an explicit upper bound. (When the desired value of a_p is nonzero, T_p has finite slope and we can reduce to a problem about modular forms in a single p-adic family; in that context, a uniform bound is easy, and one can even show that the number of such forms of weight <k grows very very very very slowly with k, where each "very" is a log; this is worked out on Frank Calegari’s blog.. On the other hand, as Naser points out below in comments, if you ask about the “Hecke angle” a_p/p^{(k-1)/2}, we don’t know how to get any really good bound in the nonzero case. I think the conjecture is that you always expect finite multiplicity in either setting even if you range over all k.)

What I find most striking is the method of proof and its similarity to the Chabauty method! Let me explain. The basic idea of Naser’s paper is to set this up in the language of deformation theory, with the goal of bounding the number of weight-k p-adic Galois representations rho which *could* be the representations attached to weight-k forms with T_p = 0.

We can pin down the possible reductions mod p of such a form to a finite number of possibilities, and this number is independent of k, so let’s fix a residual representation rhobar once and for all.

The argument takes place in R_loc, the ring of deformations of rhobar|G_{Q_p}. And when I say “the ring of deformations” I mean “the ring of deformations subject to whatever conditions are important,” I’m just drawing a cartoon here. Anyway, R_loc is some big p-adic power series ring; or we can think of the p-adic affine space Spec R_loc, whose Z_p-points we can think of as the space of deformations of rhobar to p-adic local representations. This turns out to be 5-dimensional in Naser’s case.

Inside Spec R_loc, we have the space of local representations which extend to global ones; let’s call this locus Spec R_glob. This is still a p-adic manifold but it’s cut out by *global* arithmetic conditions and its dimension will be given by some computation in Galois cohomology over Q; it turns out to be 3.

But also inside Spec R_loc, we have a submanifold Z cut out by the condition that a_p is not just 0 mod p, it is 0 on the nose, and that the determinant is the kth power of cyclotomic for the particular k-th power you have in mind. This manifold, which is 2-dimensional, is something you could define without ever knowing there was such a thing as Q; it’s just some closed locus in the deformation space of rhobar|Gal(Q_p).

But the restriction of rho to Gal(Q_p) is a point psi of R_loc which has to lie in *both* these two spaces, the local one which expresses the condition “psi looks like the representation of Gal(Q_P) attached to a weight-k modular form with a_p = 0” and the global one which expresses the condition “psi is the restriction to Gal(Q_p) of representation of Gal(Q) unramified away from some specified set of primes.” So psi lies in the intersection of the 3-dimensional locus and the 2-dimensional locus in 5-space, and the miracle is that you can prove this intersection is transverse, which means it consists of a finite set of points, and what’s more, it is a set of points whose cardinality you can explicitly bound!

If this sounds familiar, it’s because it’s *just like Chabauty.* There, you have a curve C and its Jacobian J. The analogue of R_loc is J(Q_p), or rather let’s say a neighborhood of the identity in J(Q_p) which looks like affine space Q_p^g.

The analogue of R_glob is (the p-adic closure of) J(Q), which is a proper subspace of dimension r, where r is the rank of J(Q), something you can compute or at least bound by Galois cohomology over Q. (Of course it can’t be a proper subspace of dimension r if r >= g, which is why Chabauty doesn’t work in that case!)

The analogue of Z is C(Q_p); this is something defined purely p-adically, a locus you could talk about even if you had no idea your C/Q_p were secretly the local manifestation of a curve over Q.

And any rational point of C(Q), considered as a point in J(Q_p), has to lie in both C(Q_p) and J(Q), whose dimensions 1 and at most g-1, and once again the key technical tool is that this intersection can be shown to be *transverse*, whence finite, so C(Q) is finite and you have Mordell’s conjecture in the case r < g. And, as Coleman observed decades after Chabauty, this method even allows you to get an explicit bound on the number of points of C(Q), though not an effective way to compute them.

I think this is a very cool confluence indeed! In the last ten years we've seen a huge amount of work refining Chabauty; Matt Baker discusses some of it on his blog, and then there’s the whole nonabelian Chabauty direction launched by Minhyong Kim and pushed forward by Jen Balakrishnan and Netan Dogra and many others. Are there other situations in which we can get meaningful results from “deformation-theoretic Chabauty,” and are the new technical advances in Chabauty methods relevant in this context?

Thanks for your post! I have a comment regarding the following:

If I understand him correctly, such a uniform bound on the multiplicity of an eigenvalue lambda isn’t known for any lambda other than 0. (Though a bound of the form o(dim(S_k)) is given for nonzero lambda by Frank Calegari on his blog.)

The bound o(dim(S_k)) is due to Serre which is proved in the following paper and he discusses some arithmetic applications:

https://www.college-de-france.fr/media/jean-pierre-serre/UPL942235006964043982_Serre_Re__partition_asymptotique_….pdf

There are two natural normalizations of the Hecke operators T_p which are important from the arithmetic and analytic point of view. I denote them by the arithmetic and analytic normalization.

1) Arithmetic normalization: Let f be a Hecke weight k modular form of fixed level N where gcd(N,p)=1. We normalize T_p so that

-2p^{(k-1)/2}<a_f(p)<2p^{(k-1)/2} (this is the Ramanujan bound)

where a_f(p) is the p-th Hecke eigenvalue of f. It follows from the argument of Frank Calegari on his blog, that the number of weight k modular forms f with a_f(p)=\lambda is bounded by

O(v_p (\lambda)^2) (*)

where v_p is any p-adic valuation defined on \bar{Q}. This bound is independent of the weight k! However, by Gouvea conjecture on the distribution of the slopes of the modular forms, we expect $v_p(\lambda)/k$ is equidistributed with respect to the uniform measure on $[0,(1/p+1)].$ So, (*) gives $m_p(\lambda,k,N) \ll k^2$ for most of the Hecke eigenvalues which is unfortunately worse than the trivial bound $m_p(\lambda,k,N) \ll k$. In his recent post, Frank explains this and amazing p-adic version of the Weyl law:

https://galoisrepresentations.wordpress.com/2018/10/16/more-or-less-opaque/

2) Analytic normalization: Define \lambda_f(p):=a_f(p)/p^{(k-1)/2}. Hence

-2<\lambda_f(p)<2

In this normalization, there is not any better bound than Serre's bound O(k/log(k)) on the number of Hecke modular forms f where

\lambda_f(p)=\lambda.

So, as you pointed out: such a uniform bound on the multiplicity of an eigenvalue lambda isn’t known for any lambda other than 0.

In some sense my paper here does a version of this deformation-theoretic Chabauty: https://arxiv.org/abs/1809.03524

Take a curve over a finitely generated field k, and look at the deformation ring R of some residual representation of its geometric fundamental group (really one uses psuedorepresentations). Then a finite index subgroup of the Galois group of k acts on this ring; one of the goals of the paper is to show that there are finitely-many Q_p-points of the rigid generic fiber X of R with finite orbit. (In other words, that only finitely many representations extend to representations of a finite-index subgroup of the arithmetic fundamental group.)

If one fixes an element g of G_k, looking for g-fixed points is a Chabauty-type problem of this type; one wants to show that the intersection of the graph of g with the diagonal of X x X is transverse. This follows from Lafforgue + Weil II.

Then one has to bound the the size of a g-orbit independent of g, which is an argument of a rather different sort; it’s a deformation-ring version of dynamical Mordell-Lang.

I should have said — there’s a variant of the above (which I haven’t written down yet) which is closely related to honest Chabauty. If k is a number field, the Galois action on the deformation rings in question can be described “explicitly” in terms of p-adic iterated integrals!

One point where the analogy breaks down a little: Chabauty is really about (transverse) intersections of Qp manifolds, whereas modularity is more about intersections of rigid analytic spaces (so Cp rather than Qp), and these two types of spaces can have quite different behaviors.

I am intrigued by this comment but don’t really understand it!

Suppose you wanted to look at one dimensional p-adic representations of G_Q unramified outside p which are motivic of weight zero and are trivial mod p. So you can put on your Chabauty hat and think: I am intersecting a 1 dimensional space of global representations inside a 2 dimensional space of local representations with the 1 dimensional subspace of representations of Hodge-Tate weight zero, so the number of such characters should be finite. And this works for representations valued in Qp^* — the only representation is the trivial one (unless p=2 in which case there are three more). But it doesn’t work for Qbarp^* valued reps, where there are infinity many (all the finite order characters of p-power order). Note that I have fixed things mod p, so this is not because the residue field is now infinite. Rather, the difference is that the analytic function log(1+X) has finitely many zeroes in the open unit disc of Zp — which it must, since this ball is compact — but infinitely many zeros in the open unit disc of Zbar_p, which it can, because this ball is not compact. So in the modular case you have to think about dimensions more algebraically in terms of deformation rings rather than their Spfs.

Aha, very interesting!