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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 . (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… 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

1 Answer 1

up vote 14 down vote accepted


(Moderator's note: I am adding the explanation of the above answer, as provided in comments given by the poster whose account was deleted long ago, to make this a proper answer.)

Philosophically, it's a little hard to prove that something is not a coincidence, so you should take my answer to mean that I understand both and that there is no relation. Perhaps one way to think of it is as follows. The congruences proved by Ramanujan for $5$, $7$, and $11$ are very similar to each other, more so than the exceptional isomorphisms $A_5 \simeq L_2(5)$ and $L_3(2) \simeq L_2(7)$.

More generally, there is a unified reason why congruences for the partition function exist, they arise by projection onto some finite dimensional space of ordinary $p$-adic modular forms of weight $−1/2$. This space has dimension which grows roughly linear in $p$. For $p < 12$ this space is zero dimensional, hence Ramanujan's congruences. For $p$ in some further range, this space is one dimensional, explaining the congruences of Atkin. For larger $p$, this space is still finite dimensional but consists of eigenforms for the Hecke operators with different eigenvalues, so the implied congruences are not so transparent. (Nick Ramsey is writing an appendix to Folsom-Kent-Ono which will explain some of this.)

On the other hand, the exceptional isomorphisms are, well, exceptional. They differ in flavor from each other, and there is nothing that persists of this nature for larger $p$. As for what happens to $p=2$ and $p=3$, they are absent from the "exceptional" list because the groups are solvable, whereas in the case of partitions they are absent because the generating function for the partitions is $q^{1/24} \eta^{−1}$, where $\eta = q^{1/24}\prod_{n=1}^\infty (1−q^n)$ is modular form. That means one is really studying the function given by

$$1/\eta(24\tau) = \sum a_m q^m,$$

where $a_m = 0$ unless $m = −1+24n$, in which case $a_m$ is the number of partitions $p(n)$ of $n$. The methods above for $p=2,3$ give congruences for the coefficients of this series when $m$ is divisible by $2$ or $3$. Since $a_m=0$ in these cases, one does not deduce anything about congruences for the partition function $p(n)$. Putting this together, it seems that the connection between the fact that Ramanujan's congruences hold for the same set of primes as those for which $L_2(p)$ is simple is the law of small numbers.

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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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