Ribet's Converse to Herbrand: Part I
02 Mar 2017Tomorrow I’m giving the STAGE talk on Ribet’s converse to Herbrand’s theorem, after I’ll try and post more notes, but for now here’s a little intro to get us thinking about the problem.
Ribet's converse to Herbrand
We are interested in the class groups of cyclotomic fields \begin{equation*} h_p = h_{\mathbf{Q}(\mu_p)}\text{.} \end{equation*} Lets list the first few of these
| \(p\) | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 |
| \(h_p\) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 8 | 9 | 37 | 121 | 211 | 695 | 4889 | 41241 | 76301 | 853513 |
| \(p|h_p\) | no | no | no | no | no | no | no | no | no | no | no | yes | no | no | no | no | yes | no | yes |
Definition Regular primes
We'll call primes for which \(p\nmid h_p\) regular primes. Otherwise irregular primes.
Why is this important from a number theory perspective?
Theorem Kummer 1850
Fermat's last theorem is true for regular prime exponents.
It's hard to tell when a prime is a regular prime, you'd have to compute the class group.
Definition Bernoulli numbers
The Bernoulli numbers are the sequence of integers given by the exponential generating function \begin{equation*} \frac{x}{e^x - 1} + \frac x2 - 1 = \sum_{n\ge 2}^\infty B_k\frac{x^k}{k!}\text{.} \end{equation*}
These have a number of cool properties, such as:
Theorem Kummer's congruence
If \(h\equiv k \pmod {p-1}\) then \begin{equation*} \frac{B_k}{k}\equiv \frac{B_h}{h} \pmod{p}\text{.} \end{equation*}
But most important for us is the relation to class numbers:
Theorem Kummer's Criterion
\(p\) is a irregular prime if and only if there exists some \(2\le k \le p-3\text{,}\) even with \(p\) dividing the numerator of \(B_k\text{.}\)
This is a great theorem relating class numbers to the Bernoulli numbers, but can we do better? What if I know a specific \(k\) so that \(p|B_k\text{,}\) can I say anything more specific about the class group? Yes; there is a strengthening of this theorem due in this form to Herbrand (in one direction) and Ribet (later, in the other direction).
First we need to recall the mod \(p\) cyclotomic character \(\chi\colon \operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \mathbf{F}_p^*\) defined by \begin{equation*} \zeta_p^{\chi(\sigma)} = \sigma (\zeta_p)\text{.} \end{equation*}
Theorem Herbrand-Ribet
Write \(C = \operatorname{cl}(\mathbf{Q}(\mu_p))/\operatorname{cl}(\mathbf{Q}(\mu_p))^p\) this is an \(\mathbf{F}_p\) Galois representation which decomposes as a sum of eigenspaces \begin{equation*} C = \bigoplus_{i=0}^{p-1} C(\chi^i)\text{.} \end{equation*} Then for \(2\le k\le p-3\) even we have \begin{equation*} p| B_k \iff C(\chi^{1-k}) \ne 0\text{.} \end{equation*}
The \(\Leftarrow\) direction was proved by Herbrand in 1932. And the \(\Rightarrow \) direction by Ribet in 1974.
Now for completeness here is a table of factorisations of Bernoulli number numerators.
| \(k\text{:}\) | \(2\) | \(4\) | \(6\) | \(8\) | \(10\) | \(12\) | \(14\) | \(16\) | \(18\) | \(20\) | \(22\) | \(24\) | \(26\) | \(28\) | \(30\) | \(32\) | \(34\) | \(36\) | \(38\) | \(40\) | \(42\) | \(44\) | \(46\) | \(48\) | \(50\) | \(52\) | \(54\) | \(56\) | \(58\) |
| Numerator of \(B_{k}\text{:}\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(5\) | \(-1 \cdot 691\) | \(7\) | \(-1 \cdot 3617\) | \(43867\) | \(-1 \cdot 283 \cdot 617\) | \(11 \cdot 131 \cdot 593\) | \(-1 \cdot 103 \cdot 2294797\) | \(13 \cdot 657931\) | \(-1 \cdot 7 \cdot 9349 \cdot 362903\) | \(5 \cdot 1721 \cdot 1001259881\) | \(-1 \cdot 37 \cdot 683 \cdot 305065927\) | \(17 \cdot 151628697551\) | \(-1 \cdot 26315271553053477373\) | \(19 \cdot 154210205991661\) | \(-1 \cdot 137616929 \cdot 1897170067619\) | \(1520097643918070802691\) | \(-1 \cdot 11 \cdot 59 \cdot 8089 \cdot 2947939 \cdot 1798482437\) | \(23 \cdot 383799511 \cdot 67568238839737\) | \(-1 \cdot 653 \cdot 56039 \cdot 153289748932447906241\) | \(5^{2} \cdot 417202699 \cdot 47464429777438199\) | \(-1 \cdot 13 \cdot 577 \cdot 58741 \cdot 401029177 \cdot 4534045619429\) | \(39409 \cdot 660183281 \cdot 1120412849144121779\) | \(-1 \cdot 7 \cdot 113161 \cdot 163979 \cdot 19088082706840550550313\) | \(29 \cdot 67 \cdot 186707 \cdot 6235242049 \cdot 3734958336910412\) |