Let be any prime. In this post, we show that there is an integer prime to such that has a torsion class of order . Almost equivalently, there exists a Katz modular form of level and weight one over which does not lift to characteristic zero. We shall give two different arguments. The first argument will have the virtue that the torsion class is non-trivial after localization at a maximal ideal which is new of level . The second argument, in contrast, will produce torsion classes at fairly explicit levels. Neither proof, unfortunately, implies the existence of interesting Galois representations unramified at with image containing . Rather, the classes will come from deformations of characteristic zero classes. (This post is an elaboration of my comment here.)
The first argument: Let be an imaginary cubic extension unramified outside with Galois closure with Galois group . There is a corresponding Galois representation:
This representation is modular. Suppose for convenience that . Associated to is an absolutely irreducible residual representation . Let denote the corresponding universal unramified deformation. The only characteristic zero deformations are dihedral. Let denote the corresponding universal unramified dihedral deformation ring. It’s easy to identify this ring explicitly; it is
where is the class group of the imaginary quadratic subfield of . The ring will fail to be -flat exactly when . Fortunately, this can be determined purely from the reduced tangent space of . Note that
where is the quadratic character of . The reduced tangent space of is the Bloch–Kato Selmer group , where denotes the subring of cohomology classes which are unramified everywhere. So it all comes down to finding so that is non-zero. However, an elementary argument using inflation-restriction shows that this is equivalent to showing that the class number of is divisible by . So we are done provided that we can find a suitable with class number divisible by . (I should mention, of course, that we are using the theorem that which was proved by David Geraghty and me.) The last step follows from the lemma below; the argument is essentially taken from this paper of Bilu–Luca.
Lemma: Fix a prime . There exists an imaginary cubic field of
discriminant prime to and class number divisible by .
Proof: Consider the field , where
and is an element of to be chosen later. Note that is manifestly a unit in . We may compute that
Since is trivial, it follows that for some ideal . We shall show that, for a suitably chosen , the element is non-trivial in the class group. If is trivial, then, up to a unit, is a th power. On the other hand, the rank of the unit group of is one, and is a unit. Hence, it suffices to choose a such that:
- generates a subgroup of of index prime to . Equivalently, is not a perfect th power in .
- None of the elements for is a perfect th power in .
- The polynomial defining is irreducible.
- The discriminant of is not a square.
- The discriminant of is prime to .
By working over the function field instead of , one finds that the first four conditions hold for all outside a thin set. (The discriminant is always negative, so the signature of the field is always .) On the other hand, the discriminant of the defining polynomial is , so if one (for example) takes to be an integer divisible by then the discriminant will be prime to . Note that the set of integers divisible by will contain elements not in any thin set, because the number of integral points of height at most in a thin set is .
Second Argument: Let , and let be the Hilbert class field of . There is a weight one modular form of level and quadratic character corresponding to the Galois representation:
Lemma: Let . Let be a prime such that . Equivalently, let be a prime which splits completely in . Then
is divisible by . More generally, for any prime , the quantity above is divisible by any prime divisor of , and for a prime is the coefficient of in .
Proof: This follows from “level–raising” in characteristic for weight one forms. Under the hypothesis that , we find that there is more cohomology (over ) in level than is accounted for by oldforms. Assuming that there is no torsion, this is inconsistent with the fact that there are no newforms in characteristic zero, because weight one forms cannot be Steinberg at any place. (The easiest way to see this is that the eigenvalue of would have to be non-integral — it also follows on the Galois side from local-global compatibility, but this is overkill.) Note that level–raising in this context does not follow from classical level–raising — for the details I refer to you my fifth lecture in Barbados on non-minimal modularity lifting theorems in weight one.