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 be a positive integer, let be an odd quadratic character of conductor dividing , and let be an a prime not dividing . Recall that the reduction map between spaces of Katz modular forms:
is not surjective in general, although it will be surjective for all but finitely many . For what pairs 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 , where is the subgroup of squares, and is the corresponding modular curve (as a stack, if necessary) over . The reduction mod-p map is Hecke equivariant; let denote a maximal ideal of in the support of the cokernel. Associated to is a Galois representation:
which is unramified at primes not dividing (including ). It is not necessarily the case that 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 contains for some ). Not surprisingly, it turns out there are no such forms for small . The reason is that the fixed field of the kernel of would be a high degree field with a root discriminant which (for very small ) would violate the GRH discriminant bounds of Odlyzko, and for smallish would still give fields of unusually low root discriminant.
Of course, as 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 prime. This leads to the (incredibly) vague question: can this be predicted in advance? If there is going to exist a representation unramified outside for small , is it more likely that (see here) rather than ? 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 or and increasing discriminant, not in the regime of fixed root discriminant and Galois group as above for varying . 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 , do there exist infinitely many primes such that there is a number field different from which is unramified away from and has root discriminant less that ?
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?