Pseudo-representations and the Eisenstein Ideal

Preston Wake is in town, and on Tuesday he gave a talk on his recent joint work with Carl Wang Erickson. Many years ago, Matt and I studied Mazur’s Eisenstein Ideal paper from the perspective of Galois deformation rings. Using some subterfuge (involving a choice of auxiliary ramification line at the prime \(N\) following an idea of Mark Dickinson), we proved an \(R = \mathbf{T}\) theorem. One satisfactory aspect (to us, at least) of our paper was that we were able to reconstruct from a purely Galois theoretic perspective some of the thorny geometric issues in Barry’s paper, particularly at the prime 2. Another problem of Barry’s that we studied was the question of determining for which N and p the cuspidal Hecke algebra was smooth (equivalently, whether the cuspidal Hecke algebra completed at a maximal Eisenstein ideal \(\mathfrak{m}\) of residue characteristic p was equal to Z_p). Our theorem showed this was equivalent to the existence of certain Galois deformations to GL_2(F[e]/e^3). Although we were able to give a precise account of what happens for p = 2, for larger p we could only prove the following:

Theorem Let p > 3 be prime, and let N = 1 mod p be prime. If the rank of the cuspidal Hecke algebra of level \(\Gamma_0(N)\) localized at the Eisenstein prime is greater than one, then

\(K = \mathbf{Q}(N^{1/p})\)

has non-cyclic p-class group.

Note that there is always trivial p-torsion class in the class group of K coming from the degree p extension inside the Nth roots of unity. In our paper, we speculated that this was actually an equivalence. To quote the relevant passage:

We expect (based on the numerical evidence) that the condition that the class group of K has p-rank [at least] two is equivalent to the existence of an appropriate group scheme, and thus to [the rank being greater than one].

Not a conjecture, fortunately, as it turns out to be false, already for p = 7 and N = 337. Oops! In fact, this had already been observed by Emmanuel Lecouturier here. Wake and Wang Erickson, however, give a complete characterization of when the rank is greater than one, namely

Theorem [Wake, Wang Erickson] Let \(a \in H^1(\mathbf{Z}[1/Np],\mathbf{F}_p(1))\) be the Kummer class corresponding to N. Let \(b \in H^1(\mathbf{Z}[1/Np],\mathbf{F}_p(-1))\) be the (unique up to scalar) non-trivial class which is unramified at p. Then the rank of the Hecke algebra is greater than one if and only if the cup product \(a \cup b\) vanishes.

They prove many other results in their paper as well. The main theoretical improvement of their method over the old paper was to work with pseudo-representations rather than representations. On the one hand, this requires some more technical machinery, in particular to properly define exactly what it means for a pseudo-representation to be finite flat. On the other hand, it avoids certain tricks that Matt and I had to make to account properly for the ramification at N as well as to make the deformation problem representable. Our methods would never work as soon as N is (edit: not) prime, whereas this is not true for the new results of W-WE. In particular, there is real hope that there method can be applied to much more general N.

Let me also note that Merel in the ’90s found a completely different geometric characterization of when the cuspidal Hecke algebra had rank bigger than one; explicitly, for p > 3 and N = 1 mod p, it is bigger than one when the slightly terrifying expression:

\(\displaystyle{\prod_{i=1}^{(N-1)/2} i^i}\)

is a pth power modulo N. So now there are a circle of theorems relating three things: vanishing of cup products, ranks of Eisenstein Hecke algebras, and Merel’s invariant above. It turns out that one can directly relate Merel’s invariant to the cup product using Stickelberger’s Theorem. On the other hand, Wake and Wang Erickson also have a nice interpretation of the expression above as it relates to Mazur-Tate derivatives (possibly this observation is due to Akshay), and they also prove some nice results in this direction. And I haven’t even mentioned their other results relating to higher ranks and higher Massey products, and many other things. Lecouturier’s paper is also a good read, and considers the problem from another perspective.

In Preston’s talk, he sketched the relatively easy implication that the vanishing of the cup product \(a \cup b\) above implies that the class group of Q(N^{1/p}) has non-cyclic p-part. The main point is that the vanishing of cup products is exactly what is required for a certain extension problem, and in particular the existence of a Galois representation of the form:

\(\left( \begin{matrix} 1 & a & c \\ 0 & \chi^{-1} & b \\0 & 0 & 1 \end{matrix} \right),\)

where \(\chi\) is the mod-p cyclotomic character. The class c gives the requisite extension (after some adjustment). Curiously enough, both the classes a and b exist for primes N = -1 mod p. On the other hand, the corresponding H^2 group vanishes in this case, and so the pairing is always zero. Hence one deduces the following very curious corollary:

Theorem: Let p > 3, and let N = – 1 mod p be prime. Then the class number of \(\mathbf{Q}(N^{1/p})\) is divisible by p.

Question: Is there a direct proof of this theorem? In particular, is there an easy way to contruct the relevant unramified extension of degree p for all such primes N? I offer a beer to the first satisfactory answer.