Review of Buzzard-Gee

This is a review of the paper “Slopes of Modular Forms” submitted for publication in a Simons symposium proceedings volume.

tl;dr: This paper is a nice survey article on questions concerning the slopes of modular forms. Buzzard has given a (very explicit) conjecture which predicts the slopes of classical modular (\(p\)-stabilized) eigenforms of level prime to \(p,\) at least under a certain regularity hypothesis. One consequence is that, under favourable circumstances, all the slopes are integers. The current paper describes the link between this and related problems to the \(p\)-adic Langlands program, as well as raising several further intriguing questions concerning the distributions of these slopes. The paper is well written, and is a welcome addition to the literature. I strongly recommend that this paper be accepted.

Review: Buzzard’s slope conjectures live somewhere in the world between 19th and 21st century mathematics. Suppose that one considers the space of over-convergent cusp forms of level \(N = 1\) for \(p = 2.\) Then, using nothing more than classical identities between modular functions, one may prove that the smallest eigenvalue of the compact operator \(U_2\) is at most \(\|2^3\|_2 = 1/8.\) On the other hand, it is now a “folklore” conjecture (Conjecture 4.1.1 of the paper under review) that, if \(p\) is odd and \(f\) is a classical modular form of level \(\Gamma_0(N)\) prime to \(p\), then the residual representation:

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

is irreducible locally on the decomposition group at \(p\) whenever the valuation of \(a_p\) is not an integer. This problem seems to be a deep question in the p-adic Langlands program for \(\mathrm{GL}_2(\mathbf{Q}_p).\) The two cases where this is known, \(v(a_p) < 1\) and \(v(a_p)\) sufficiently large, both require machinery from p-adic Hodge theory — in the former case, one needs the full local Langlands correspondence for \(\mathrm{GL}_2(\mathbf{Q}_p).\)

Comments: Here are some comments given in some order that bears little relation to the actual paper.

• I don’t like the table on page 6, in particular, because certain ranges of numbers are bunched together, the output looks a little strange. Can one improve this in some way? Perhaps finish at \(3^{10}?\) Perhaps include only selected powers bigger than \(3^{10}?\) Perhaps normalize for the length of the range?
• Corollary 5.1.2. Do you want to speculate on what happens in the reducible case? In some sense, in Buzzard’s conjecture, one doesn’t see the fact that the residual representations are globally reducible or not. On the other hand, weird stuff certainly happens for \(p = 2,\) as previously mentioned here. What happens in the reducible case for \(p = 3?\)
• Conjecture 4.1.1 demands that \(k\) is even, but that is a consequence of the level being of the form \(\Gamma_0(N)\) — something which is noted immediately after the statement. So why include the condition in the statement of the conjecture? Also, perhaps it’s also worth remarking upon the case when \(a_p\) is a unit.
• The authors (in Remark 4.1.3) point to the origins of this Conjecture 4.1.1 to around 2005. However, I feel like I remember some discussion of this conjecture in the Durham symposium of 2004. There were certainly hints of this conjecture on «la serviette de Kisin», upon which Mark gave a heuristic local argument for why the Eigencurve was proper — although the argument was slightly dodgy in that it collapsed if the napkin was rotated 90 degrees. (Of course, Mark was proved right when Hansheng Diao and Ruochuan Liu did indeed prove this result using local methods here.) Also, isn’t Conjecture 4.1.1 a consequence of Buzzard’s original conjecture as modified by Lisa Clay? Somehow it seems to me that what Remark 4.1.3 is referring to is the idea that Conjecture 4.1.1 is a consequence of a purely local conjecture, and refers to the period (2005?) when Breuil was formulating the first versions of the p-adic Langlands program.
• For Conjecture 4.2.1, wouldn’t it make more sense to normalize the valuation in terms of the coefficient field \(\mathbf{Q}_p(\chi),\) so the statement once more becomes that \(a_p\) has integral valuation?
• Why is the condition on Buzzard’s conjecture different when \(p = 2?\) (I understand it has to be modified in order to have a chance of being true, but I am asking if there is any explanation for why this is necessary.)
• The authors remark (p.3) that it is not known whether there are infinitely many Buzzard irregular primes. Here is a short argument to prove that this is a consequence of standard conjectures of prime values of polynomials. We start with the observation that the first Buzzard irregular prime is \(p =59,\) and that the offending representation:

  \(\rho_{Q \Delta} : G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\mathbf{F}_{59})\)

  has exceptional image in the context of Serre and Swinnerton-Dyer (On l-adic representations and congruences for coefficients of modular forms, Antwerp III). Indeed, this particular example features prominently in that paper. I always thought this was not an entirely random coincidence, and since it seems relevant here, I thought I would finally bother to figure out what is going on. (For the next prime, \(p = 79,\) the corresponding representation has image containing \(\mathrm{SL}_2(\mathbf{F}_{79}),\) so it is somewhat of an accident.) The mod-59 representation above has projective image \(S_4.\) Now suppose that \(p \equiv 3 \mod 4\) is a prime such that \(H/\mathbf{Q}\) is an \(S_4\)-extension which is unramified away from \(p.\) Such a representation will give rise (following an argument of Tate) to a mod-\(p\) representation of \(\mathbf{Q}\) which is unramified away from \(p.\) The congruence condition on \(p\) implies that it will be odd, and hence modular, by Langlands-Tunnell. Now let us suppose, in addition, that \(4\) divides the ramification index \(e_p.\) Under this assumption, the representation cannot be locally reducible, because the ramification index of any power of the cyclotomic character divides \(p-1 \equiv 2 \mod 4.\) Hence, if there are infinitely many such fields, there are infinitely many \(\mathrm{SL}_2(\mathbf{Z})\)-irregular primes. Consider the following fields studied by Darrin Doud (here):

  \( \displaystyle{K = \mathbf{Q}[u]/f(u), \quad f(u) = (u + x)^4 – p^* u},\)

  \( \displaystyle{(-1)^{(p-1)/2} p = p^* =
  \frac{256 x^3 – y^2}{27}, \quad p^* \ne 1 + 4x}.\)

  Doud shows that \(K\) has discriminant \((p^*)^3\) and Galois group \(S_4,\) and it is easy to see that the splitting field \(H/\mathbf{Q}\) has all the required properties needed above as long as \(p \equiv 3 \mod 4.\) The formula relating tame ramification and the discriminant implies that \(e_p(K/\mathbf{Q}) = 4.\) Standard conjectures now predict that there are infinitely many such primes of this form — if \(y\) is odd, then \(p^* < 0.\) The first few such primes which are \(3 \mod 4\) are

  \(59, 107, 139, 283, \ldots \)

  which compares with the first few Buzzard irregular primes (taken from Buzzard's paper):

  \(59, 79, 107, 131, 139, 151, 173 \ldots \)

  On the other hand, an unconditional proof by these means seems out of reach, because any modular \(S_4\)-extension unramified outside \(p \equiv 3 \mod 4\) forces \(\mathbf{Q}(\sqrt{-p})\) to have class number divisible by \(3,\) and we don't know if there exist infinitely many such primes.