I have three students graduating this year: Shiva Chidambaram, Eric Stubley, and Noah Taylor. In light of the last post, I should give them a boost by reminding you of their (numerous) results which have been discussed on this blog. You can read about Shiva’s work here, here, and here, about Eric’s work here and here, and Noah’s work here and here. Alternatively, you can always click on the work of my students link.
But even this link is not complete! Here’s a result from Noah’s thesis which I haven’t discussed before:
Let \(N\) be prime, and let \(\mathbf{T}\) denote the \(\mathbf{Z}_2\)-Hecke algebra generated by \(T_l\) for \(l\) prime to \(2\), and let \(\widetilde{\mathbf{T}}\) denote the Hecke alegbra where \(T_2\) is also included. These Hecke algebras are famously not the same in general. For example, when \(N = 23\), the space of cusp forms is \(2\)-dimensional and has a pair of conjugate cusp forms as follows:
\(\displaystyle{q – \frac{\sqrt{5}+1}{2} q^2 + \sqrt{5} q^3 + \frac{\sqrt{5} – 1}{2} q^4 – (1 + \sqrt{5})q^5 + \ldots}\)
So \(\mathbf{T} = \mathbf{Z}[\sqrt{5}]\) whereas \(\widetilde{\mathbf{T}} = \displaystyle{\mathbf{Z} \left[ \frac{\sqrt{5}+1}{2} \right]}\). Noah gives a formula for the index:
Theorem: Let \(N\) be prime. Then the index \([\widetilde{\mathbf{T}}:\mathbf{T}]\) is given by the order of the space
\(S_1(\Gamma_0(N),\mathbf{F}_2)\)
of Katz modular forms of weight one and level \(\Gamma_0(N)\).
In particular, the index at level \(23\) is coming from the fact that there is a classical weight one form of this level. From this one sees that the index is non-trivial for all primes \(N \equiv 3 \bmod 4\) except for \(N = 3,7,11,19,43,67\) and \(163\). For primes \(N \equiv 1 \bmod 4\), on the other hand, I might guess that there would be a positive density of primes for which either the index was trivial or non-trivial. The question more or less hinges on the expected number of \(\mathbf{SL}_2(\mathbf{F}_{2^n})\) representations of \(\mathbf{Q}\) (with \(n \ge 2\)) which become unramified at all finite places over \(\mathbf{Q}(\sqrt{N})\).