## Gross Fugue

Here are some variations on the theme of the last post, which is also related to a problem of Dick Gross.

In this post, I want to discuss weight one modular forms where the level varies in the “vertical” aspect (that is, $N$ is a growing power of a fixed prime, rather than simply an increasing integer). First of all, consider the spaces

$S_1(\Gamma(M \cdot \ell^n),\mathbf{C})$

for fixed $M$ and growing $n$. For example, if $M = 1$, the corresponding Galois representations are associated to number fields unramified outside a single prime $\ell$. Given a cusp form $f$, the twists $f \otimes \chi$ by a finite order character of $\ell$-power order will also be modular (possibly with larger $n$), so all the finiteness statements below should be interpreted “up to twist.”

The first observation is that there exist only finitely many exceptional cusp forms (with projective image $A_4, S_4, A_5$) because, by a theorem of Hermite, there are only finitely many fields with a fixed Galois group unramified outside a fixed set of primes. (This echos a very general conjecture which says [very loosely] that if one fixes an infinitesimal character and varies the level in a $\ell$-adic tower, one should only see finitely many automorphic forms which do not arise via functoriality from constructions using discrete series.)

The second observation is that all the other cusp forms are easy to describe: they are induced from finite order characters of a fixed number of easily determined quadratic fields $K$.

So far so good. But what happens if one replaces $\mathbf{C}$ by $\mathbf{F}_p$, or more generally $\mathbf{Q}_p/\mathbf{Z}_p$? Here is the following optimistic guess:

Question: For a fixed prime $p \ne \ell$, are there only a finite number of non-liftable forms in the $p$-power tower?

Here we have to take the usual caveats — not only do we have to take into account twisting, but also the $\mathbf{GL}_2(\mathbf{Q}_{\ell})$-action (old forms).

This question is supposed to be a $\mathrm{GL}(2)$-analogue of Washington’s famous theorem on the $p$-part of the class group in the $\ell$-adic cyclotomic tower. We shall see that it is more than an analogy.

What will the source of torsion classes be?

1. One source are Galois representations: $\overline{\rho}: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\mathbf{F}_q)$ with big image that are unramified outside $\ell$ (with $q$ a power of $p$). Of course there are only finitely many such representations for any fixed $q$, but some heuristics I learnt from Akshay convince me that there should only be finitely many even if one varies $q$ over all powers of $p$ (taking into account twisting, of course).
2. Another source of torsion comes from deformations of big image Galois representations $\overline{\rho}$ as above, or from representations with projective image one of the exceptional groups. Since each unramified deformation ring will be finite, each $\overline{\rho}$ should only give rise to finitely many extra torsion classes.
3. A third source of torsion classes comes from reducible indecomposable representations. The residual representations $\overline{\rho}$ which arise in this way occur when $L(0,\chi)$ is divisible by $p$ for an odd character $\chi$ of finite order. In particular, there are only finitely many such representations which occur exactly if all but finitely many $L$-values $L(0,\chi)$ are prime to $p$, where $\chi$ is an odd character of conductor $M$ times a power of $\ell$. But this exactly the content from Washington’s Theorem (the oddness assumption is not, however, necessary).
4. The final class come from deformations of dihedral representations. If $\overline{\rho}$ is the induction of a character $\psi$ of $K/\mathbf{Q}$, then the tangent space to the unramified deformation ring of $\overline{\rho}$ gives rise to torsion classes when there are no everywhere unramified classes in $H^1(\mathbf{Q},\mathrm{Ind}(\psi/\psi^c))$ — the unramified dihedral representations in $H^1(\mathbf{Q},\eta_K)$ are seen globally. By inflation-restriction, this is equal to a certain invariant part of the class groups of the anti-cyclotomic tower. There are non-vanishing results concerning L-values of Hida that are relevant here, although I haven’t checked to see if they imply the finiteness statement or not.

The only way to start thinking about answering this question is to think in terms of the torsion in the cohomology of modular curves. But, I confess, I do not really have any ideas on how to prove it. (To be honest, I still find Washington’s proof very mysterious.)

On related matters, it would be nice if one could prove — say by analytic means — that $H^1(X_H(N),\omega)$ has torsion (prime to $N$) for all sufficiently large $N$. Taking $N$ to be a power of a prime, this would give a different construction of non-solvable Galois representations unramified outside a single prime (for all $\ell$) from the one suggested by Dick and carried out in for $\ell \in \{2,3,5,7\}$ by Dembélé and others. Moreover, although (as in those examples) it would involve the group $\mathrm{PSL}_2(\mathbf{F})$ as a simple factor, the residue characteristic would be different from $\ell$ rather than equal to $\ell$ in the previous constructions. (George Schaeffer told me he tried computing torsion coming from $X_H(343)$ but didn’t find any.) It might also (for suitable $H$) give a lower bound for $\pi_1(\mathcal{O}_{K})$ where $K = \mathbf{Q}(\sqrt{-D})$ which is better (at least for some primes) than one gets from class numbers.

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