13
$\begingroup$

Is anyone aware of the history of the following conjecture on the $p$-divisibility of (the numerators) of Bernoulli numbers?

CONJECTURE: For $p$ an odd prime, and $k$ even with $2 \leq k \leq p-3$, $p$ cannot divide both $B_k$ and $B_{p-k+1}$.

This conjecture is mentioned in Ohta's paper: Companion Forms and the structure of $p$-adic Hecke algebras. (see Example 3.3.4 on p. 25)

As far as I am aware, this is the only instance of the conjecture in literature. I believe I have a proof using companion forms and Ribet's proof of Herbrand-Ribet. I am currently writing it up and want to give proper references. Thanks

$\endgroup$
7
  • 3
    $\begingroup$ You might try emailing Ken Ribet. He will almost certainly answer you, and he's tremendously knowledgeable. $\endgroup$ Apr 12, 2010 at 17:12
  • $\begingroup$ I think that the paper of Frank Calegari on Eisentein primes, in Compositio, also discusses this conjecture. (This is a somewhat vague memory, so I apologize in advance if I'm wrong.) $\endgroup$
    – Emerton
    Apr 12, 2010 at 17:37
  • $\begingroup$ And if you do email Ribet, you might want to mention this MO question, I think he has registered here already (see user 5131). $\endgroup$ Apr 12, 2010 at 17:47
  • $\begingroup$ By the way, this would be a very nice result, considering its application to the structure of Hecke algebras and to Iwasawa theory. $\endgroup$
    – Olivier
    Apr 13, 2010 at 8:30
  • 2
    $\begingroup$ @unramified: Have you finally heard from Ribet ? And have you written up your proof of Ohta's conjecture ? $\endgroup$ Mar 14, 2012 at 9:10

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.