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There are three exceptional Galois groups $L_2(5)$, $L_2(7)$ and $L_2(11)$ . These are cited as one of Arnold's "trinities" and are connected with other trinities and the McKay Correspondence.

Ramanujan studied partition numbers and found congruence relations modulo powers of 5, 7 and 11. the recent dramatic breakthrough by Ken Ono throws some light on the reasons behind these congruences.

My question is whether there is any known connection between these two instances of the three primes 5, 7 and 11 appearing in these two places. I realise that these are just small numbers so it is not a great coincidence, but partitions are connected to other areas of mathematics so I wondered if some correspondence was known.

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What is an exceptional Galois group? –  Qiaochu Yuan Jan 26 '11 at 23:02
Very nice question! Do the results of Ono make it clear that 5, 7, 11 will be better behaved than, say, 13? –  André Henriques Jan 26 '11 at 23:19
@Andre: yes. See aimath.org/news/partition/folsom-kent-ono.pdf . (This result has gotten some amazing publicity. Ken Ono is pretty good at that, it seems.) –  Qiaochu Yuan Jan 27 '11 at 0:29
Regarding the "dramatic breakthrough": See FJ's comments following his answer as well as my answer here mathoverflow.net/questions/52935/… and the links therein, which indicate that the results of Folsom, Kent, and Ono are less surprising, given the current state of the theory of $p$-adic modular forms, than they might appear to those who don't know this theory. –  Emerton Jan 27 '11 at 4:15
So does Ono call up the news outlets himself, or what? –  David Hansen Jan 28 '11 at 21:13
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up vote 12 down vote accepted


$$ $$

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Would you care to expand your answer? Of course, it is up to voters to decide if this is sufficient... –  David Roberts Jan 27 '11 at 0:14
Dear FJ, Well explained! –  Emerton Jan 27 '11 at 4:12
Thanks for a great explanation. If no one else responds I can take that as a correct answer to whether there are "known" connections which is what I asked. I am not completely convinced that a connection is ruled out by your observations, especially since this trinity is related to other exceptional structures including some that may not completely disconnected from modular forms. I'll wait a day or so to see if anyone else has a different answer. –  Philip Gibbs Jan 27 '11 at 8:57
Is it the case that what makes $p=5,7,11$ special is that $p+1$ divides $24$ but $p$ doesn't? or is this just another coincident property? –  Philip Gibbs Jan 27 '11 at 18:20
Not counting $p=23$ of course, oops. –  Philip Gibbs Jan 27 '11 at 18:24
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