Timeline for Can the failure of the multiplicativity of Euler factors at bad primes be corrected?
Current License: CC BY-SA 2.5
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| Feb 10, 2010 at 10:24 | comment | added | JBorger | Thanks. Yes, I agree with the first paragraph of your edit, or at least that solving the question at the level of Galois reps would be sufficient. As for your suggestion in the second paragraph, I like the spirit, but it looks to me that even when M is the trivial 1-dim rep, you get some H^1 and no higher H^i, so the Euler factors can change even when there is good reduction. I guess that's not a contradiction yet, but I can't be too confident that we'd be moving closer to any ultimate understanding of zeta-functions. I'll think about it some more, though. | |
| Feb 10, 2010 at 7:32 | comment | added | Kevin Buzzard | James: I had some more comments to make but they wouldn't fit in the margin so I just appended them to my answer. You might want to clarify whether I have misunderstood things. | |
| Feb 10, 2010 at 7:31 | history | edited | Kevin Buzzard | CC BY-SA 2.5 |
added comments that should have been comments but there were too many of them :-/
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| Feb 10, 2010 at 1:09 | comment | added | JBorger | Let me add a few more words, though I fear this question is beyond salvaging. It would be great to find a ring map from K_0 to some ring of zeta-like objects in analytic number theory. I suspect that that's not possible, in which case it would be nice to have some general formalism that controls its failure, perhaps like how Grothendieck's abstract Riemann-Roch formalism controls the failure of push-forward to commute with the Chern character. | |
| Feb 9, 2010 at 21:28 | comment | added | JBorger | Thanks, Kevin. I had considered looking at the representation ring, but I guess I was hoping for something zeta-like. It would be nice to have some non-tautological positive statement about Euler factors at the bad primes, especially since we have a very nice statement at the good primes. Also thanks for the simpler example. I agree that any reasonable solution would have work on the level of motives rather than just on the level of the Grothendieck ring of varieties. | |
| Feb 9, 2010 at 17:32 | history | answered | Kevin Buzzard | CC BY-SA 2.5 |