A geometric Serre weight conjecture for GL_3 in the partial weight one setting

We are concerned with a speculative geometric variant of Serre's weight conjecture for GL_3. Such conjectures predict the possible 'weights' of automorphic forms giving rise to a fixed Galois representation, which governs congruences between forms of different weights. In the GL_2 setting over certain degree 2 number fields, Diamond--Sasaki found a rich interplay between geometric modularity of 'partial weight one' and geometric modularity of several 'algebraic' weights, which should correspond to being modular of these weights in a classical, algebraic sense. These algebraic weights are found by applying geometric constructions: Hasse invariants and theta operators. On the Galois side, this phenomenon is paralleled by 'crystalline liftability' of certain Hodge-Tate weights. These results have recently been generalised to higher degree fields.

The main aim is to understand what the analogous results are in the case of GL_3 over the rational numbers. Some complications arise: it is harder to formulate a geometric variant of the conjecture, and it is known that the tight relationship between modularity and crystalline liftability no longer holds. We start by understanding the picture in terms of crystalline lifts (first for tame Galois representations, where it is expected the relationship with modularity still holds). We then hope to transfer the combinatorial results suggested by this into geometric statements on the relevant Emerton-Gee stack, which is essentially a moduli space of Galois representations. It is expected that a potential description of the general Serre weight conjectures can be given in such terms, using as input the (still conjectural) Breuil-Mezard cycles.

One hope is that we will be able to understand analogous statements on the automorphic side. It should be possible to show that geometric modularity of 'partial weight one' is related to modularity of certain other 'algebraic' weights, via Hasse invariants and theta operators.