Counting solutions to a_p = λ, Part II

This is a sequel to this post where the problem of counting eigenforms with \(a_p = \lambda\) and \(\lambda \ne 0\) was considered. Here we report on recent progress in the case \(\lambda = 0\).

It is a somewhat notorious conjecture attributed to Lehmer (who merely asked the question, naturally) that the coefficients of

\( \Delta = q \prod_{n=1}^{\infty} (1 – q^n)^{24} = \sum \tau(n) q^n = q-24q^2+252q^3+\ldots \)

never vanish. One problem with this conjecture is that there really isn’t any compelling reason it should be true except (basically) on probability grounds given the growth of the coefficients. As with a number of problems concerning “horizontal” questions about modular eigenforms (fixing the form and varying the prime \(p\)), it is often easier to consider the analogous “vertical” question where one fixes \(p\) and varies the weight. Namely: fix a tame level, say \(\Gamma = \Gamma_1(N)\), fix a \(p\) prime to the level, and then consider the eigenforms of level \(\Gamma\) and varying weight with \(a_p = 0\). Unlike in the case of Lehmer’s conjecture, is certainly can happen that \(a_p = 0\), for example:

1. If \(f\) is associated to a modular elliptic curve \(E\) with supersingular reduction at \(p \ge 5\).
2. If \(f\) has CM by an imaginary quadratic field \(K\) in which \(p\) is inert.

Let \(S(X)\) denote the number of cuspforms of level \(\Gamma\) and weight \( \le X\) such that \(a_p = 0\). Consider bounds on this function. The trivial bound, given by counting all cuspforms, is \(S(X) \ll X^2\). If you try to improve this bound using analytic techniques, namely via the trace formula, you can only do very slightly better, say \(S(X) \ll X^2/\log(X)\). The problem is that the condition \(a_p = 0\) within the space of all spherical representations \(\pi_p\) which could possibly occur has measure zero, so any trace formula approach will have to use a test function where \(a_p\) has support in some non-trivial interval \([-\varepsilon,\varepsilon]\) depending on the weight. This is the same problem (more or less) which prevents the analytic approach from giving optimal bounds to the number of weight one modular forms (where now the measure zero condition is being imposed at the infinite prime instead of at the prime \(p\)). One approach to improving these bounds is to use non-commutative Iwasawa theoretic methods as employed in my paper with Matt and then used by Simon Marshall to give the first non-trivial bounds for spaces of modular forms for \(\mathrm{GL}(2)\) over imaginary quadratic fields of fixed level and varying weight. This approach should lead, in principle, to a power saving over the trivial bound.

On the other hand. the best possible bound on \(S(X)\) will have the shape \(S(X) \ll X\), because the number of CM forms of each weight will be bounded independently of the weight, and there is no reason to imagine that the other exceptions will contribute anything of this order. Indeed, in my previous post, I conjectured that there should only be finitely many such forms of fixed level which are not CM as the weight varies.

When I last visited Madison in 2018, Naser Sardari was working on this problem, and in a preprint from late 2018, he proved exactly a bound of the optimal shape \(S(X) \ll X\), with the slight caveat that one should restrict to even weights. Quomodocumque blogged about it here.

Just a few weeks ago, Naser was in town in Chicago, and we got to talking about this problem again. Happily, we were able to come up with one more extra ingredient to push the original result to a (close to) optimal conclusion, and prove the aforementioned conjecture:

Theorem: (C, Sardari). Fix a prime \(p > 2\) and a tame level \(\Gamma_1(N)\). Then there are only finitely many eigenforms of level \(\Gamma\) and even weight with \(a_p = 0\) which are not CM.

This establishes a vertical version of Lehmer’s conjecture, up to a congruence on the weight, which arises for a technical reason discussed more below.

The first main idea of the proof is as follows. The \(p\)-adic Galois representation associated to \(\rho_f\) for a modular form can be very complicated viewed as a representation of the Galois group of \(\mathbf{Q}_p\). However, if \(a_p = 0\), then the local representation has a very special form: it is induced from an unramified extension \(K/\mathbf{Q}_p\). Breuil gave a precise formula for the representation, but a fairly soft argument shows that it is induced — 2-dimensional irreducible crystalline representations over \(\mathbf{Q}_p\) are determined by \(a_p\), and twisting by an unramified character fixes both the determinant and the condition \(a_p = 0\), hence \(V = V \otimes \eta_K\) is induced. That means that one can capture the locus of such representations by a local deformation condition. It is not the case that locally induced implies globally induced, as can be seen from the example of supersingular elliptic curves. This is related to the fact that the map

\(\pi: R^{\mathrm{loc}} \rightarrow R^{\mathrm{glob}}\)

of (unrestricted at \(p\)) local to global deformation rings is not a surjection. On the other hand, we know in some generality that this is a finite map. This was explored in this post, and then more properly taken up in a paper I wrote with Patrick Allen. The argument to this point is now enough to prove the original result of Sardari. Let \(R^{\mathrm{loc,ind}}\) denote the local deformation ring of induced representations. If \(R = R^{\mathrm{glob}} \otimes_{R^{\mathrm{loc}}} R^{\mathrm{loc,ind}}\) denotes the global deformation ring of locally induced representations, we know that the forms with \(a_p = 0\) and a fixed weight are the points of this deformation ring which lie in the fibre over some fixed point in local deformation space. Hence the finiteness of \(\pi\) gives a uniform bound on the number of points in this fibre, and hence a uniform bound over the number of such modular forms in any fixed weight. BTW, for those wondering why there is a restriction on the parity of the weight, it is only really there to prevent the residual representation from being globally reducible, a setting in which one doesn’t quite yet know the finiteness of \(\pi\). (When the optimal \(R = \mathbf{T}\) theorems become available in the reducible case, our methods should apply without any restrictions.)

Now comes the second ingredient. In order to explain it, let me describe the ring \(R^{\mathrm{loc,ind}}\) in more detail, or at least the part coming from inertia. This local deformation ring is basically equal to the deformation ring of the trivial character of \(G_K\), and in particular the ring has the form

\( \mathbf{Z}_p[[ \mathcal{O}^{\times}_K(p)]] \)

where \(A(p)\) denotes the maximal pro-\(p\) subgroup of \(A\). This ring is isomorphic (at least for odd p) to the Iwasawa algebra \(\mathbf{Z}_p[[X,Y]]\) after (via the p-adic logarithm) fixing a choice of multiplicative basis for \(\mathcal{O}^{\times}_K(p)\). Imagine that some component of the global deformation ring (with a locally induced condition) has infinitely many points which correspond to classical non-CM modular forms of level prime to \(p\). The points in weight space correspond to the algebraic characters of the following form:

\( \mathcal{O}_K \rightarrow K^{\times}, \qquad z \mapsto z^n \)

We now exploit the following fact which might (at first) be surprising: any infinite collection of these weights are Zariski dense! To make things a little more concrete, suppose we choose a basis of \(\mathcal{O}^{\times}_K(p)\) of the form \(1 + p\) and \((1+p)^{\eta}\), for a suitable \(\eta \in \mathcal{O}_K\) which will not be in \(\mathbf{Z}_p\), for example, \(\sqrt{u}\) for some non-quadratic-residue. The corresponding points with respect to the usual Iwasawa parameters have the shape:

\( X \mapsto (1 + p)^{n}-1, \qquad Y \mapsto (1 + p)^{\eta n}-1.\)

Instead of proving here why these are Zariski dense, it might be more useful to explain a very close analogy that Naser brought up with Lang’s Conjecture: if you take an infinite set of pairs of points of the form \((\exp(x),\exp(\eta x)) \subset (\mathbf{C}^{\times})^2\), then they will be Zariski dense whenever \(\eta \notin \mathbf{Q}\). In other words, the group subvarieties of the formal torus going through \((X,Y)=(0,0)\) basically all have to be of the form \((1+X)^{\eta} = (1+Y)\) for \(\eta \in \mathbf{Z}_p\). (Coincidentally, the arithmetic applications of Lang’s conjecture was the subject of the recent Ahlfors lecture by Peter Sarnak which you can watch here. Our result is yet another application!)

Once your non-CM points are Zariski dense, you are home and hosed: using an idea due to Ghate-Vatsal, you now specialize at lots of points which are inductions of finite order characters. The corresponding Galois representations have finite image on inertia and so are classical by known results. But then (apart from finitely many exceptions) they have to all be CM, because they are classical weight one forms, and the image of inertia is sufficiently large to rule out them having exceptional image.

One might ask whether the results are effective. I’m not so sure because of the following issue. Suppose you take \(p = 79\) and level one (I’m not sure this case will exhibit the required behavior but it might.) Then you might be able to prove that the global locally-induced deformation ring is (now over all weights) \(\mathbf{Z}_p = \Lambda/\mathfrak{P}\). But it might be very hard to tell if that weight \(\mathfrak{P}\) corresponds to a classical weight or a random weight, simply because \(\mathbf{Z}\) is dense in \(\mathbf{Z}_p\). This is not unlike the problem of showing that the zeros of the Kubota–Leopoldt zeta function are not in arithmetic weights.