The space of classical modular cuspforms of level one and weight 24 has dimension two — the smallest weight for which the dimension is not zero or one. What can we say about the Hecke algebra acting on this space without computing it?
Formally, the Hecke algebra is a rank two -algebra, which is either an order in the ring of integers of a real quadratic field, or a subring of Let’s investigate the completion of this algebra at various primes
Let’s first consider the prime The curve has genus two, and the corresponding Hecke algebra in weight two is where is the Golden Ratio. The prime does not split in this field, and hence modulo there is a pair of conjugate eigenforms with coefficients in Multiplying by the Hasse invariant, we see that this eigenform also occurs at level one and weight 24 over It follows that:
In particular, for some square-free integer
Now let us consider primes Any Galois representation modulo such a prime will occur — possibly up to twist — in lower weight. Yet all the spaces in lower weight have dimension at most one, and hence it follows that the residue fields of are all of the form Suppose further that Then, using theta operators, we may find two distinct eigenforms in weight 24, from which it follows that has two distinct residue fields of characterstic and so, for we have:
One expects at level one that always generates the Hecke field. This is still a conjecture, but we may deduce this unconditionally in weight 24 because the dimension of the cuspforms is two, and so this follows automatically from the Sturm bound! Hence we may write:
where Even better, using Hatada's Theorem — giving congruences for and for eigenforms of level one modulo and respectively — we may write
where This gives an upper bound on in light of the Deligne bound More precisely, we obtain the bound and hence that
Let's now think more carefully about and For these primes, there will be a unique Coleman family of slope for and for I can't quite see a pure thought way of proving this, but at least this would be a consequence of the strong form of the GM-conjecture as predicted by Buzzard. So we should expect that, in these cases
In addition to congruences for small primes, there will also be congruences between the unique cusp form with an Eisenstein series modulo the numerator of which is
I claim that these primes will also have to split in For example, it is impossible for to be divisible by because that would violate the inequality on above, and hence it follows that must also split in The same argument works for having ruled out some very small To summarize, we have the following:
The primes split in but does not split, and Moreover, we expect that and also split.
This is enough to determine completely up to 72 possibilities, and 9 with the unproven assumption at and On the other hand, all of these are quite large (the smallest are and respectively), which forces to be very small. But we also have the congruence
For the remaining we can determine, with satisfying the required inequality, whether there exists such a congruence with A simple check shows that is a unique solution (with the assumption on two or three or not), and hence, by (something close to) pure thought, we have shown that and moreover (using Deligne's bound again) that
One can indeed check this is the case directly, if you like. Curiously enough, this Hecke eigenvalue is quite close to the Deligne bound — the probability it is (in absolute value) this big is, assuming a Sato-Tate distribution, slightly under 5%.
Extra Credit Problem: Hack Ken Ribet’s Yelp password by using the fact that 144169 is his favorite prime number.