The last seven words of Kedlaya-Medvedovsky

New paper by my student Noah Taylor! It addresses some conjectures raised by Kedlaya and Medvedovsky in this paper. Let \(\mathbf{T}\) denote the Hecke algebra acting on modular forms of weight two and prime level \(N\) generated by Hecke operators \(T_p\) for \(p\) prime to \(N\) and \(2\) (the so-called “anemic” Hecke algebra). If \(\mathfrak{m}\) is a maximal ideal of \(\mathbf{T}\) of residue characteristic two, and \(\mathbf{T}/\mathfrak{m} = k\), there exists a corresponding Galois representation:

\( \overline{\rho}: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\mathbf{T}/\mathfrak{m}) = \mathrm{GL}_2(k).\)

If \(S\) denotes the space of modular forms modulo \(2\), then certainly \(\mathrm{dim}_{k}(S[\mathfrak{m}]) \ge 1\). Since there can exist congruences between modular forms, it is certainly possible that the generalized \(\mathfrak{m}\)-eigenspace of \(S\) has dimension greater than one. Kedlaya and Medvedovsky observe that if one assumes that \(\overline{\rho}\) has (projectively) dihedral image, then one can systematically predict lower bounds for this generalized eigenspace contingent on various properties of \(\overline{\rho}\). They prove a number of such results, but they finish the paper with what amounts to six more conjectures. Actually, one of the conjectures splits into two completely different cases, and so I like to think of it as seven conjectures.

Before stating the conjectures, first note that \(\overline{\rho}\) (when projectively dihedral) is necessarily induced from the field \(\mathbf{Q}(\sqrt{\pm N})\). The corresponding representation may or may not be ordinary at the prime 2. Also, let \(h(N)\) denote the even part of the class number of \(\mathbf{Q}(\sqrt{N})\). Now we can state the conjectures, which are now all proved by Noah:

1. Suppose that \(N \equiv 1 \pmod 8\). If \(\mathfrak{m}\) is \(\mathbf{Q}(\sqrt{N})\)-dihedral, then the generalized \(\mathfrak{m}\)-eigenspace has dimension at least 4.
2. Suppose that \(N \equiv 1 \pmod 8\). If \(\mathfrak{m}\) is \(\mathbf{Q}(\sqrt{-N})\)-dihedral, then the generalized \(\mathfrak{m}\)-eigenspace has dimension at least \(h(-N)\).
3. Suppose that \(\mathfrak{m}\) is Eisenstein. Then the generalized \(\mathfrak{m}\)-eigenspace has dimension at least \((h(-N) – 2)/2\).
4. Suppose that \(N \equiv 5 \pmod 8\). If \(\mathfrak{m}\) is ordinary-\(\mathbf{Q}(\sqrt{N})\)-dihedral, then the generalized \(\mathfrak{m}\)-eigenspace has dimension at least 4.
5. Suppose that \(N \equiv 5 \pmod 8\). If \(\mathfrak{m}\) is \(\mathbf{Q}(\sqrt{-N})\)-dihedral, then the generalized \(\mathfrak{m}\)-eigenspace has dimension at least 2.
6. Suppose that \(N \equiv 3 \pmod 4\). If \(\mathfrak{m}\) is \(\mathbf{Q}(\sqrt{N})\)-dihedral, then the generalized \(\mathfrak{m}\)-eigenspace has dimension at least 2.
7. Suppose that \(N \equiv 3 \pmod 4\). If \(\mathfrak{m}\) is ordinary-\(\mathbf{Q}(\sqrt{-N})\)-dihedral, then the generalized \(\mathfrak{m}\)-eigenspace has dimension at least 2.

Noah uses quite a number of different arguments to prove this theorem. One basic idea is that the extra dimensions are related to deformations of \(\overline{\rho}\), but only in some of the proofs is this connection transparent. More directly, Noah exploits the following:

1. The existence of weight one dihedral representations. When \(\overline{\rho}\) is unramified at 2 it is natural to look to such forms. However, even when \(\overline{\rho}\) is ramified at two, the weight one forms, after giving rise via congruences to weight two forms, can often be level-lowered to level \(N\) using an argument similar to that employed by me and Matt in our paper on the modular degree of elliptic curves.
2. Known properties of the real points of the Jacobian \(J_0(N)\), in particular the connectedness of \(J_0(N)(\mathbf{R})\) for prime \(N\) as proved by Merel. This can be used to give a lower bound of \(2\) when \(\overline{\rho}\) is totally real. In order to get a better bound in the even case (if necessary) one has to combine this with other arguments.
3. The difference between the Hecke algebra \(\mathbf{T}\) and the Hecke algebra where the operator \(T_2\) is also included. If this Hecke algebra is strictly larger than \(\mathbf{T}\) after localization at \(\mathfrak{m}\), then one can show that the \(\mathfrak{m}\)-torsion of \(S\) has to be at least two, and moreover one can make this argument work nicely with some of the other methods for producing non-trivial lower bounds.

Concerning the third point: the difference between the Hecke algebra \(\mathbf{T}\) and the full Hecke algebra is the addition of the operators \(T_2\) and \(T_N\). Noah’s arguments crucially use this in the case of \(T_2\) but not of \(T_N\). But this is also explained in the paper: once you add the Hecke operator \(T_2\), it turns out that you have the full Hecke algebra! The fact that the Hecke algebra is integrally generated by \(T_p\) for \(p\) prime to the level is not true for general levels \(N\) but just happens to be true for \(N\) prime. It suffices to prove the result after localizing at any maximal ideal \(\mathfrak{m}\). Mazur proved it in the Eisenstein case by a somewhat subtle argument (it’s false in general for Eisenstein primes at non-prime level). Second, in the non-Eisenstein case, the argument uses the result that all irreducible representations modulo \(2\) are ramified at \(N\). If there were such a representation, it would be an absolutely irreducible and unramified away from \(2\), and Tate prove that no such representations exist!

Of course, apropos of the title, this post must finish with the following: