As mentioned in the comments to the last post, Kevin Buzzard and Alan Lauder have made an extensive computation of weight one modular forms in characteristic zero (see also here). Thinking about what that data might contain, I wondered about the following question: what are the images of the Galois representations associated to the weight one forms of type ?
Let us take a step back. Consider a projective representation
with image , and assume that it is odd. (That is, complex conjugation has order ) According to Tate, there exists a lift
This lift is unique up to twisting. Since the Schur multiplier of is , there is a unique minimal lift up to twist whose image is a central extension by a cyclic group of -power order. Note that is not trivial, since does not have any two-dimensional representations. If , then the determinant of the corresponding 2-dimensional representation of is trivial, which contradicts the assumption that is odd. (Equivalently, there is an obstruction at to lifting to the central extension by ) Hence divides What is the expected distribution of as one runs over all odd -extensions?
My first guess (without any prior thought or computation) was that this might obey some form of Cohen–Lenstra heuristic, suitably interpreted.
Note that the image of the determinant has order The corresponding determinant representation is a character of of -power order. Since has trivial class number, the order is equal to the maximal ramification degree of this representation over all primes
Over , Tate’s lifting theorem has the following stronger form: one may choose a lift of and insist that ; that is, they agree on inertia. This is essentially a consequence of the fact that has trivial class group. For convenience, suppose that is unramified at and Suppose that is ramified at There are three possibilities:
- The image of at a ramified prime is cyclic of order 2, 3, or 5.
- The image of at a ramified prime is
- The image of at a ramified prime is
For a fixed , let denote the Teichmuller lift of the mod- cyclotomic character. (Fix an isomorphism of with for all )
Let us consider the three cases in turn.
In the first case, the image factors through a cyclic quotient. One may thus take to be a direct sum which, on inertia, has the shape up to twist. By comparing this to the projective representation, we see that has order 2, 3, or 5, and so, after finding the twist such that the determinant has -power order, we see that or
In the second case, the lift on inertia is (up to twist) of the form:
for some Since the order of is , the order of the ratio is
which must be equal to 3. It follows that is even. Yet the determinant is equal to
Since is even, we see that, after twisting, we may take
Finally, in the third case, the lift is of the form:
for some We now find that the order of the ratio of these characters is
which must be equal to 2, and the determinant is If is even, then, as above, we may twist so that Hence, the only way that the image after minimal twist does not have is if we are in this third situation with , with odd, and then (after twisting) we find that is the largest power of dividing
(I confess that I originally forgot the fact that the third possibility could occur, and was only after noticing that this seemed to imply the inverse Galois problem was false thought a little bit more about the possibilities.)
Lemma Assume that is unramified at and and has projective image , and a lift with image with minimal kernel. Then order of is unless there exists a prime such that the image of the decomposition group at under is In this case, we have to be twice the largest power of dividing for all such primes
Let denote the corresponding power of
We see that is determined by purely local phenomena. This still doesn’t quite answer what the distribution of the extension will be. However, I imagine that Bhargava style heuristics should certainly be able to predict the ratio of with Does anyone have a sense of how easy this might be to prove? Or, much more modestly, how easy it would be to compute from these quite precise heuristics the exact predicted distribution of the central extensions of coming from weight one modular forms?
(I confess, it is not even obvious to me from this construction how to prove that all central extensions occur as Galois groups — but I presume this is known, and hopefully one of my readers can provide a reference.)
(According to KB, BTW, all the representations with have )