## The mystery of the primes

No, this is not the sequel to Marcus du Sautoy’s book, but rather a curious observation regarding George Schaeffer’s tables of “ethereal” weight one Katz modular eigenforms (which you can find starting on p.64 here). Let $N$ be a positive integer, let $\chi$ be an odd quadratic character of conductor dividing $N$, and let $p$ be an a prime not dividing $N$. Recall that the reduction map between spaces of Katz modular forms:

$M_1(\Gamma_1(N), \chi, \mathbf{Z}_p) \rightarrow M_1(\Gamma_1(N), \chi, \mathbf{F}_p)$

is not surjective in general, although it will be surjective for all but finitely many $p$. For what pairs $(N,p)$ is the map not surjective? As originally observed by Mestre (and predicted by Serre), such pairs do exist. One way to think of the primes which arise in this way are as the primes dividing the torsion subgroup of $H^1(X_H(N),\omega)$, where $H \subset (\mathbf{Z}/N \mathbf{Z})^{\times}$ is the subgroup of squares, and $X_H(N)$ is the corresponding modular curve (as a stack, if necessary) over $\mathbf{Z}[1/N]$. The reduction mod-p map is Hecke equivariant; let $\mathfrak{m}$ denote a maximal ideal of $\mathbf{T}$ in the support of the cokernel. Associated to $\mathfrak{m}$ is a Galois representation:

$\overline{\rho} = \overline{\rho}_{\mathfrak{m}}: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\overline{\mathbf{F}}_p)$

which is unramified at primes not dividing $N$ (including $p$). It is not necessarily the case that $\overline{\rho}$ does not lift to characteristic zero, although this is typically the case for the examples arising in the tables (and is always the case if the image of $\overline{\rho}$ contains $\mathrm{SL}_2(\mathbf{F}_q)$ for some $q > 5$). Not surprisingly, it turns out there are no such forms for small $N$. The reason is that the fixed field $K$ of the kernel of $\overline{\rho}$ would be a high degree field with a root discriminant which (for very small $N$) would violate the GRH discriminant bounds of Odlyzko, and for smallish $N$ would still give fields of unusually low root discriminant.

Of course, as $N$ increases, there do exist many such forms, sometimes in quite large characteristic. However, something peculiar happens in the range of the tables, namely, there is not a single example with $N$ prime. This leads to the (incredibly) vague question: can this be predicted in advance? If there is going to exist a $\mathrm{PGL}_2(\mathbf{F}_{199})$ representation unramified outside $N$ for small $N$, is it more likely that $N = 82$ (see here) rather than $N = 83$? One can try to use heuristics predicting the number of fields with certain ramification behavior, but these heuristics are much better behaved for fixed Galois groups $G = \mathrm{PGL}_2(\mathbf{F}_p)$ or $G = \mathrm{PSL}_2(\mathbf{F}_p)$ and increasing discriminant, not in the regime of fixed root discriminant and Galois group $G$ as above for varying $p$. Is there any conspiracy ruling out certain kinds of number fields with small root discriminant ramified at a single prime? For example, if you fix some arbitrary constant, say $M = 1000$, do there exist infinitely many primes $p$ such that there is a number field $K$ different from $\mathbf{Q}$ which is unramified away from $p$ and has root discriminant less that $M$?

These questions are hard to pin down, because they are really questions concerning the law of small numbers. Namely, they ask about the behavior/distribution of various quantities in the range before asymptotic behavior begins. Since the asymptotic behavior is (in these contexts) already mostly conjectural, it’s probably hard to say anything intelligent about these even more delicate questions. (Idle question: are there similar problems for which one does understand what happens before the asymptotic regime begins, even heuristically?)

Here’s one reason to consider these questions. Suppose one wants to compute “ethereal” Siegel modular forms. At what level does one first expect to find such forms? The numerics above suggest that it might be easier to find such forms at small composite levels rather than prime levels. Is that a reasonable inference?

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### 5 Responses to The mystery of the primes

1. Dick Gross says:

Frank,

I thought about this question when writing my paper on companion forms, and basically gave up. It’s predicting when a line bundle of small degree on a curve has a larger space of sections (mod p). That’s why Serre’s criterion is so subtle – this jump occurs precisely when you have an odd 2-dimensional modular representation which is unramified at p which does not come from a 2-dimensional representation over C.

Serre’s conjecture in weight 1 was what attracted me to the subject of companion forms in the first place, and I found it amusing that it was precisely the weight 1 situation that I couldn’t resolve completely. Fortunately, Robert Coleman understood what I was doing much better than I did, and finished it off. I should say that Mestre’s computations for p = 2 were more convincing than any of the proofs!

Dick

• Dear Dick,

To clarify something you said: even classical (odd) Artin representations can give rise to torsion classes, exactly when their mod-p reductions admit “extra” unramified p-adic deformations. For example, consider a modular representation:

$\rho: G_{\mathbf{Q}} \rightarrow S_3 \hookrightarrow \mathrm{GL}_2(\mathbf{C}).$

Let $K/\mathbf{Q}$ be (any of the) corresponding imaginary cubic fields inside the fixed field of the kernel of $\rho$. If $p>3$ is prime, then $\overline{\rho}$ admits a non-trivial unramified deformation to $\mathbf{F}_p[\epsilon]/\epsilon^2$ exactly when $p$ divides the class number of $K$. This deformation will be (by C-G) Katz modular but does not come from characteristic zero, so it will give rise to torsion in $H^1(X,\omega)$, or equivalently mod-p classes which don’t lift to characteristic zero. The smallest example (of the exact flavour above) occurs for the field

$K = \mathbf{Q}(\theta)/(\theta^3 - \theta^2 + 7 \theta - 6)$

of discriminant $-3 \cdot 521$, with class number $h_K = 5$.

• Dick Gross says:

Nice.

On the other hand, the cubic field K has a unit group of rank 1, so its class number will rarely be divisible by p — by the Cohen-Lenstra heuristic — and the existence of unramified deformations is still a sporadic phenomenon.

Dick

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