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 hp=hQ(μp). 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 |
| hp | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 8 | 9 | 37 | 121 | 211 | 695 | 4889 | 41241 | 76301 | 853513 |
| p|hp | 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∤hp 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 xex−1+x2−1=∞∑n≥2Bkxkk!.
These have a number of cool properties, such as:
Theorem Kummer's congruence
If h≡k(modp−1) then Bkk≡Bhh(modp).
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≤k≤p−3, even with p dividing the numerator of Bk.
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|Bk, 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 χ:Gal(¯Q/Q)→F∗p defined by ζχ(σ)p=σ(ζp).
Theorem Herbrand-Ribet
Write C=cl(Q(μp))/cl(Q(μp))p this is an Fp Galois representation which decomposes as a sum of eigenspaces C=p−1⨁i=0C(χi). Then for 2≤k≤p−3 even we have p|Bk⟺C(χ1−k)≠0.
The ⇐ direction was proved by Herbrand in 1932. And the ⇒ direction by Ribet in 1974.
Now for completeness here is a table of factorisations of Bernoulli number numerators.
| k: | 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 Bk: | 1 | −1 | 1 | −1 | 5 | −1⋅691 | 7 | −1⋅3617 | 43867 | −1⋅283⋅617 | 11⋅131⋅593 | −1⋅103⋅2294797 | 13⋅657931 | −1⋅7⋅9349⋅362903 | 5⋅1721⋅1001259881 | −1⋅37⋅683⋅305065927 | 17⋅151628697551 | −1⋅26315271553053477373 | 19⋅154210205991661 | −1⋅137616929⋅1897170067619 | 1520097643918070802691 | −1⋅11⋅59⋅8089⋅2947939⋅1798482437 | 23⋅383799511⋅67568238839737 | −1⋅653⋅56039⋅153289748932447906241 | 52⋅417202699⋅47464429777438199 | −1⋅13⋅577⋅58741⋅401029177⋅4534045619429 | 39409⋅660183281⋅1120412849144121779 | −1⋅7⋅113161⋅163979⋅19088082706840550550313 | 29⋅67⋅186707⋅6235242049⋅3734958336910412 |