## Hilbert Modular Forms of Partial Weight One, Part III

My student Richard Moy is graduating!

Richard’s work has already appeared on this blog before, where we discussed his joint work with Joel Specter showing that there existed non-CM Hilbert modular forms of partial weight one. Today I want to discuss a sequel of sorts to that paper, which also forms part of Richard’s thesis (I should note that he already has five publications and will have 7 or 8 papers by the time he graduates.) The starting observation is as follows. Fix a real quadratic field F. From the perspective of Galois representations, the Hilbert modular forms of partial weight one fall under the case $\ell_0 = 1$ in the notation of my paper with David Geraghty (this is in the context of coherent cohomology). To orient the reader, let us discuss three classes of such forms:

1. Hilbert modular forms of weight $[2k+1,1]$ for a real quadratic field $F.$
2. Regular algebraic cuspidal automorphic forms for $\mathrm{GL}(3)/\mathbf{Q}.$
3. Regular algebraic cuspidal automorphic forms for $\mathrm{GL}(2)/F$ for an imaginary quadratic field $F.$

Suppose one fixes a tame level $N$ and then looks at the space of such forms as the weights vary. In both of the latter cases, the problem has been raised (or even conjectured, for $N = 1$ and $\mathrm{GL}(3)$ by Ash and Pollack here), of whether all but finitely many such forms arise via functoriality from a smaller group. More explicitly, one can ask whether:

1. If $G = \mathrm{GL}(2)/F,$ then all but finitely many cuspidal regular algebraic forms of conductor $N$ either arise (up to twist) via base change from $\mathrm{GL}(2)/\mathbf{Q},$ or are induced from a quadratic CM extension $E/F.$
2. If $G = \mathrm{GL}(3)/\mathbf{Q},$ then all but finitely many cuspidal regular algebraic forms of conductor $N$ arise up to twist as the symmetric square of a form from $\mathrm{GL}(2)/\mathbf{Q}.$

Naturally enough, one can make the same conjecture whenever $\ell_0 > 0,$ appropriately formulated. There does not seem to be any case of this conjecture which is known, although there are analogous results (where one fixes the weight and varies the level) in both weight one (where it is almost trivial) and for imaginary quadratic fields (in the work of Calegari-Dunfield and Boston-Ellenberg). Still, the conjectures in varying weight seem pretty hard even for $N = 1.$ In that context, Richard proves the following nice complementary pair of theorems below. Let $F = \mathbf{Q}(\sqrt{7}).$ The field $F$ has narrow class number $2$ and there is a unique odd everywhere unramified quadratic character $\chi$ of $G_F$ with fixed field $E = F(\sqrt{-1}).$

Theorem I (Moy) Let $F$ and $\chi$ be as above. Every Hilbert modular form over $F$ of weight $[2k+1,1]$ and level $N = 1$ is CM, and in particular is induced from $E.$

Theorem II (Moy) Let $F$ and $\chi$ be as above. Let $M$ be a strongly compatible family of two dimensional Galois representations of $F$ with determinant $\chi,$ level $N = 1,$ and Hodge–Tate weights $[0,0]$ and $[k,-k].$ Then $M$ is induced from $E.$

Theorem I is almost an immediate consequence of Theorem II, with the caveat that one doesn’t quite have complete local-global compatibility for partial weight one modular forms (though results and methods of Luu, Jorza, and Newton get close). Theorem II on the other hand is a consequence of the following:

Theorem III (Moy) Let $F$ and $\chi$ be as above. Let

$\rho: G_F \rightarrow \mathrm{GL}_2(\overline{\mathbf{Q}}_3)$

be a continuous irreducible representation with determinant $\chi$ that is unramified at all finite places except for one prime $v|3.$ Then $\rho$ is induced from a character of $G_E.$

The argument in this case is (roughly) the following. Using a Tate-style argument (with discriminant bounds), one proves that the residual representation $\overline{\rho}$ must have semi-simplification $\chi \oplus 1.$ The restriction of $\rho$ to $G_E$ then has the property that its image is pro-3 and unramified outside the fixed prime $v|3.$ Yet one shows by a class field theory computation that the largest abelian 3-extension unramified outside $v|3$ is cyclic, which (by consideration of the Frattini quotient) immediately implies that the image of $\rho$ restricted to $G_E$ factors through a cyclic quotient as well, and one is done.

Note that to deduce Theorem I, one first has to prove (using a congruence argument) that at the other prime $w|3,$ either:

1. The representation $\rho$ is unramified at $w,$
2. The representation $\rho$ restricted to $D_w$ has unramified semi-simplification. In particular, the generalized eigenvalues of $\mathrm{Frob}_w$ for $\overline{\rho}$ are both the same.

To finish, one rules out the second possibility by computing all the modular residual representations explicitly by doing computations in low weight (this can ultimately be reduced to a computation on the definite quaternion side, although Richard had to write his own programs to do this since the current magma implementation required trivial character for non-parallel weight.)

It is true that these arguments will not suffice for the more general conjecture, but then, I haven’t seen a viable strategy to prove those conjectures either!

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