More on Lehmer’s Conjecture

Lehmer said it was a “natural question” whether there existed an integer such that \(\tau(n)=0\) or not. I’ve wondered a little bit recently about how reasonable this is. (See this post.) The historical context is presumably related to the fact that, by the multiplicativity of coefficients, the vanishing of \(\tau(p)\) for one prime guarantees that a positive proportion of other coefficients vanish. From the perspective of Galois representations, however, I’m a little confused as to whether we expect any sort of “automorphic” Lehmer’s conjecture to hold. To recall, we have

\(\Delta = q \prod_{n=1}^{\infty} (1 – q^n)^{24} = \sum_{n=1}^{\infty} \tau(n) q^n.\)

Deligne’s bound says that \(|\tau(p)| \le p^{11/2}\), so a probabilistic argument suggests that there should only be finitely many primes for which \(\tau(p)=0\). Since there aren’t any such primes in the first few thousand primes, that’s a fairly convincing heuristic for why it might be true. But it’s basically impossible to prove anything this way, so one might hope to formulate a more general conjecture which is true for a more systematic reason.

A first attempt might be to ask that \(a_p(f) \ne 0\) for any eigenform \(f\) of weight \(k \ge 3\) and level prime to \(p\) which is not CM. (When \(k = 2\), of course, there are plenty of modular elliptic curves without CM, and (thanks to Noam) there are plenty of primes \(p\) with \(a_p(f) =0\)). At first thought this seems a little strong; after all, if we just work in weight \(12\) (say) then we know that \(|a_2(f)| < 2^{11/2} < 46\), so surely if you take enough such forms you should find one with \(a_2(f) = 0\). However, this secretly assumes that there are many weight \(12\) forms with coefficients in \(\mathbf{Z}\), and it seems that there are only finitely many such forms. So, for most forms, the coefficients would lie in (presumably) larger and larger number fields, and there would be more possibilities for \(a_2(f)\). For those who did the computation and might be worried, note that the probabilistic heuristic above only applies when the weight \(k \ge 4\). On the other hand, an easy exercise shows that when the weight is odd and the coefficients are integral then the form has CM. The conjecture that there only finitely many non-CM forms with rational coefficients in large even weight is certainly made in this paper, although Dave seems to be hesitant on numerical grounds to make the conjecture for \(k = 4\). There seem to be enough forms of weight \(k = 4\) and integer coefficients that perhaps there exists a form of odd level with \(a_2(f)=0\). In fact, it should be easy to search for such forms if you can search the LMFDB with a fixed Hecke eigenvalue, which I remember John Voight demonstrating at the Simons Institute general meeting, but I couldn’t work out just know when writing this post. Ah, but I guess one can just search for forms with coefficients in \(\mathbf{Q}\) and just look at them by hand. It appears that there is a form

\(f = q + 4 q^3 – 8 q^4 – 5 q^5 – 22 q^7 – 11 q^9 \ldots \in S_4(\Gamma_0(95),\mathbf{Q})\)

with \(a_2(f) = 0\). Are there any examples in higher \(>4\) weight?

All of this becomes similarly confusing on the level of Galois representations. The modular forms with \(a_p(f) = 0\) have the special property that the local \(p\)-adic Galois representation \(\rho_f\) is induced from the unique unramified quadratic extension of \(\mathbf{Q}_p\). From this perspective, the Lehmer conjecture looks a little bit like Greenberg’s conjecture that an ordinary modular form is split if and only if it has complex multiplication. But whereas that conjecture (or at least a stronger version where one assumes such a splitting at all primes of the coefficient field) does follow from plausible conjectures about motives as explained by Matt. I wonder if Matt has any more opinions on what happens if one makes the assumption for only one prime of the coefficient field. Note that if you read Matt’s paper, you might be confused why you can’t also use Serre-Tate theory to prove that elliptic curves with \(a_p = 0\) have CM. I think Florian Herzig gives a nice explanation here of what is going on.

This is also related to the question raised in this post. While that conjecture is not unreasonable, it does skirt the border of conjectures which are actually false, for example, the conjecture that any exceptional splitting in a local Galois representation is caused by (more or less) some global splitting. After all, taken to it’s logical conclusion it would imply not only Lehmer’s conjecture but also (combined with Elkies’ theorem) that all elliptic curves are CM. Greenberg’s conjecture excludes the case of weight one forms, since certainly any form with finite image has many primes for which the local Galois representation is split but the global representation is not if the image is of exceptional type. One can still argue, however, that these forms are potentially CM. On the other hand, non-CM Hilbert modular forms of partial weight one, induced to \(\mathbf{Q}\), also admit some exceptional splitting on inertia. (Note that non-CM Hilbert modular forms actually exist, as follows from the computation of Moy and Specter described here). While these induced forms are not of regular weight (the HT weights are, up to twist \([0,2,2,4]\), the splitting of the local Galois representation is not explained by any correspondences over any finite extension.

I guess the summary is that all of this dicussion points to the fact that Lehmer’s conjecture is not true for any good reason beyond random probability grounds and so is kind of rubbish. Actually this reminds me of an experience one occasionally has after giving a seminar in which you feel like you proved a snazzy result and then the questions from the audience are somewhat deflating. Rest assured, this happens to the best of us — I remember watching a talk online where RLT was talking about his (joint) proof of the Sato-Tate conjecture for \(\Delta\), and the only question from the audience was does this have any implications for Lehmer’s conjecture?