Locally induced representations

Today is a post about work of my student Chengyang Bao.

Recall that Lehmer’s conjecture asks whether \(\tau(p) \ne 0\) for all primes \(p\), where

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

is Ramanujan’s modular form. You might recall that Naser Talebizadeh Sardari and I studied a “vertical” version of Lehmer’s conjecture where instead of fixing a modular form, we fixed a prime \(p\) and a tame level \(N\) and showed that there were only finitely many normalized eigenforms \(f\) of level \(N\) and even weight \(k\) with \(a_p(f) = 0\) which were not CM. We exploited the fact that such forms give rise to Galois representations

\(\rho: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\overline{\mathbf{Q}}_p)\)

which are crystalline at \(p\) but also locally induced at \(p\) from the unique unramified quadratic extension \(K/\mathbf{Q}_p\). As explained in this post, it’s hard to see this method being able to say much more to this (for example, to say anything about Lehmer’s actual conjecture), since there do actually exist non-CM forms with \(a_p(f) = 0\).

In practice, we don’t even know in level \(N=1\) whether there exist infinitely many normalized eigenforms \(f\) with \(a_p(f) \equiv 0 \bmod p\). As mentioned in this post, one source of such representations comes from modular forms with exceptional image. For example, if \(f = \Delta E_4\) is the normalized eigenform of weight \(16\), then as first observed by Serre and Swinnerton-Dyer, the mod-\(59\) representation

\(\overline{\rho}_{f}: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\overline{\mathbf{F}}_p)\)

has projective image \(S_4\) coming from the splitting field of \(x^4 – x^3 – 7 x^2 + 11 x + 3\). But the local residual representation in this case is induced, which implies that \(a_{59} \equiv 0 \bmod 59\). As explained in that post, standard conjectures about primes predict that there should be infinitely many \(S_4\)-representations unramified outside a single prime \(p\) giving rise to modular Galois representations which will then come from level one modular forms \(f\) with \(a_p(f) \equiv 0 \bmod p\).

Chengyang’s work concerns examples precisely of this sort. From my work with Naser, we can deduce that there are at most finitely many \(f\) of level one with \(a_{59}(f) = 0\). Chengyang proves that there are no such forms. More precisely:

Theorem [Bao] Suppose that \(f\) is a modular form of level one, and suppose that \(a_p(f) = 0\). Then all of the residual mod-\(p\) representations \(\overline{\rho}\) associated to \(f\) have big image, that is, image containing \(\mathrm{SL}_2(\mathbf{F}_p)\).

In other words, none of the (presumably many) infinite examples of \(S_4\) representations giving rise to \(f\) of level one with \(a_p(f) \equiv 0 \bmod p\) can ever give an \(f\) with \(a_p(f) = 0\).

Chengyang also proves some further results about the deformations of representations with exceptional image. For example, for the mod-\(59\) representation above, the only deformations to characteristic zero unramified outside \(p=59\) which are locally induced are the representations which up to twist are the ones which up to twist coincide with the unique lift with finite image and order prime to \(p=59\).

In contrast, one might ask what happens for \(p=79\), the next case where there exists a form \(f\) of level one with \(a_p(f) \equiv 0 \bmod p\). I suspect that in this case a (possibly quite complicated computation) should show that there should be at most one form with \(a_p(f) = 0\), but that it might be quite difficult to prove using \(p\)-adic methods that there are no such forms. The problem will be that there will exist a deformation which will have infinite image and be locally induced, but now it will have generalized Hodge–Tate weights \([0,\kappa]\) for some \(p\)-adic number \(\kappa\) for which it will be very hard to show is not an integer. This is analogous to the family of Eisenstein series of level one with \(p = 37\). One knows that the \(p\)-adic zeta function will have a unique zero, but it is very hard to probe the arithmetic nature of that zero and to rule out it occurring at some arithmetic weight. To put this slightly differently, is there an integer \(k \ge 1\) such that

\(\zeta_{37}(-31 + 36 k) = 0?\)

Presumably not, but this seems extremely difficult; the difficulty of course is that there will be a solution with \(k \in \mathbf{Z}_{37}\).