## Schaefer and Stubley on Class Groups

I talked previously about work of Wake and Wang-Erickson on deformations of Eisenstein residual representations. In that post, I also mentioned a paper of Emmanuel Lecouturier who has also proved some very interesting theorems. Today, I wanted to talk about some complementary results by my student Eric Stubley in collaboration with Karl Schaefer (a student of Matthew Emerton). To duplicate slightly from that previous post, recall that Matt and I proved 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. Using work of Merel, one can dispense with the discussion of Hecke algebras and instead give an equivalent reformulation of the first condition, namely, $e > 1$ if and only if $M_1$ is a p-th power, where

$M_1 = \displaystyle{\prod_{k=1}^{p-1} (Mk)!^k \in \mathbf{F}^{\times}_N, \qquad M = \frac{N-1}{p}}$

We followed up this result with the comment:

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].

As noted previously, there are counter-examples, already for p = 7 and N = 337. However, there was still clearly some relationship between these quantities beyond the one-way implication above. In particular, the numerical evidence still stubbornly supported the hope that the converse may indeed be true for p = 5. This is the first theorem that Schaefer and Stubley prove. More precisely, they completely determine the rank of the class group of $\mathbf{Q}(N^{1/5})$ for primes N which are 1 mod 5.

Theorem [Schaefer, Stubley]: Let $N \equiv 1 \mod 5$ be prime. Then the 5-rank r_K of the class group of $K = \mathbf{Q}(N^{1/5})$ is either 1, 2, or 3. Moreover:

1. $r_K = 1$ if and only if the Merel invariant $M_1$ is not a perfect 5th power.
2. $r_K = 2$ if and only if $M_1$ is a perfect 5th power, and $\displaystyle{\alpha = \frac{\sqrt{5} - 1}{2}}$ is not a perfect 5th power modulo N.
3. $r_K = 3$ if and only if $M_1$ and $\alpha$ are both 5th powers modulo N.
4. This also answers a conjecture of Lecouturier. Their argument greatly clarified (to me) the exact relationship between the class group of K and a number of other related quantities in this picture. To recall, a third reformulation of whether the Hecke algebra has non-trivial deformations can be given (as in Wake–Wang-Erickson) by whether a certain pairing between specific classes $b$ and $c_{-1}$ in $H^1_{Np}(\mathbf{Q},\epsilon)$ and $H^1_{Np}(\mathbf{Q},\epsilon^{-1})$ vanish or not. The point is that the vanishing of a cup product ensures the existence of an extension

$\left( \begin{matrix} 1 & b & c_0 \\ 0 & \epsilon^{-1} & c_{-1} \\ 0 & 0 & 1 \end{matrix} \right)$

and one can show (after some massaging) that c_0 gives rise to something in the p-class group of K. Conversely, if one starts with a class in the p-class group of K, and then takes the Galois closure over Q, then (sometimes) one arrives with a Galois extension M/Q with a Galois representation to GL(3) of the above form. The problem is, in other circumstances, one arrives at a representation which has a much larger Galois group and a map to the Borel subgroup in higher dimension, which looks something like this:

$\displaystyle{ \left( \begin{matrix} 1 & \epsilon^{-1} \cdot b & \epsilon^{-2} \cdot b^2/2 & \epsilon^{-3} \cdot b^3/6 & & \ldots & c_{0} \\ 0 & \epsilon^{-1} & \epsilon^{-2} \cdot b & \epsilon^{-3} \cdot b^2/2 & & \ldots & c_{-1} \\ & & \ddots & & & \\ \ldots & & & & \epsilon^{1-m} & \epsilon^{-m} \cdot b & c_{1-m} \\ \ldots & & & & & \epsilon^{-m} & c_{-m} \\ \ldots & & & & & & 1 \end{matrix} \right)}$

Suppose one now tries to construct a representation of this form in order to find a non-trivial class in the p-class group of K. First, one can start by finding a suitable class $c_{-m} \in H^1_{Np}(\mathbf{Q},\epsilon^{-m})$ which cups trivially with $b.$ The vanishing of a generalized Merel invariant (under a regularity hypothesis) is exactly what guarantees the existence of such a suitable class $c_{-m},$ at least when m is odd. However, one is then faced with an increasing sequence of obstruction problems in order to climb the ladder and get all the way to the full representation of the form above. Here one has to deal with not only cup products, but also (implicitly) higher Massey products. Ultimately, the relation between the quantity $r_K$ and the deformation rings of Hecke algebras is most precise only when $p = 5$. It turns out that there is still something one can say for $p = 7,$ however. Consider the higher Merel invariant

$M_n = \displaystyle{\prod_{k=1}^{p-1} (Mk)!^{k^n} \in \mathbf{F}^{\times}_N, \qquad M = \frac{N-1}{p}}$

for odd values of n. Suppose that p is a regular prime. One can show that if $r_K \ge 2$, then at least one of these quantities M_n is a perfect pth power for an odd $n \le p-4.$ When p = 5, this is a weaker version of the theorem above. So an optimistic variation on the conjecture above is that $r_K \ge 2$ if and only if $M_n$ is a perfect pth power of for at least one odd $n \le p-4.$ The description of the relationship between these classes (which also come up in Lecouturier, they arise via an explicit analysis of Gauss sums and Stickelberger’s theorem) suggests that this conjecture is too optimistic in general, and indeed there are counter-examples for p = 11. But, Schaefer and Stubley do prove the following:

Theorem [Schaefer, Stubley]: Let p = 7, and let N = 1 mod p be prime. Then the 7-class group of $K = \mathbf{Q}(N^{1/p})$ has rank $r_K \ge 2$ if and only if either M_1 or M_3 is a perfect 7th power modulo N.

For example, consider the previous “counter-example” for N = 337 and p = 7. Here the non-trivial class group is explained by the fact that M_3 is a perfect 7th power modulo N.

One thing I especially like about this result is that there are three groups of people (Wake–Wang-Erickson, Lecouturier, and Schaefer–Stubley) are all working around a similar problem, but their results are complementary to each other. I believe that all five people will be at the upcoming IAS workshop, so I hope to hear more about this then.

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