The Langlands program is a vast web of deep conjectures regarding the correspondences between representations of reductive groups over local and global fields and the representations of the Galois groups of those fields, as well as the functorial liftings of representations of different groups. The classical Langlands program is concerned with complex representations, that is, the representations in vector spaces over the field \(\mathbb{C}\) of complex numbers. There are various extensions of the classical correspondence, which consider reductive and Galois groups over different fields and representations in vector spaces over an appropriate field. One of the more mysterious among these is the mod \(p\) Langlands program, where \(p\) is a prime number. See [C. Breuil, ICM 2010, 203–230 (2011; Zbl 1368.11123)] for a nice survey of the mod \(p\) Langlands correspondence. The paper under review considers the mod \(p\) local Langlands correspondence for the unitary groups of rank two.
The mod \(p\) local Langlands correspondence is well understood only in the case of the general linear group \(\mathrm{GL}_2\) over the field \(\mathbb{Q}_p\) of \(p\)-adic numbers due to works [P. Colmez, Astérisque 330, 281–509 (2010; Zbl 1218.11107); P. Colmez et al., Camb. J. Math. 2, No. 1, 1–47 (2014; Zbl 1312.11090); M. Emerton, “Local-global compatibility in the \(p\)-adic Langlands programme for \(\mathrm{GL}_{2/\mathbb{Q}}\)”, Preprint, https://math.uchicago.edu/~emerton/pdffiles/lg.pdf]. Let \(\overline{\mathbb{Q}}_p\) be a fixed algebraic closure of \(\mathbb{Q}_p\), and let \(\overline{\mathbb{F}}_p\) denote its residue field. Then, the mod \(p\) local Langlands correspondence for \(\mathrm{GL}_2\) over \(\mathbb{Q}_p\) is the correspondence between smooth admissible representations of \(\mathrm{GL}_2(\mathbb{Q}_p)\) in vector spaces over \(\overline{\mathbb{F}}_p\) and the continuous representations of the Galois group of the extension \(\overline{\mathbb{Q}}_p / \mathbb{Q}_p\) in two-dimensional vector spaces over \(\overline{\mathbb{F}}_p\). The correspondence is compatible with the classical Langlands correspondence, i.e., the correspondence of representations in vector spaces over \(\mathbb{C}\), it should be realized in the torsion cohomology of the associated Shimura variety, and satisfy the local-global compatibility.
For any other reductive group, even for the general linear group \(\mathrm{GL}_n\) with \(n>2\), and over any local field other than \(\mathbb{Q}_p\), the local mod \(p\) correspondence is still highly conjectural. Nevertheless, the expected consequences of the mod \(p\) Langlands correspondence are already studied. In particular, this paper considers the weight part of Serre’s conjecture, which is a statement about the Serre weights of the Galois parameter of a representation.
This paper considers the case of the unramified unitary group \(\mathbf{U}_2\) in two variables associated to an unramified quadratic extension \(K_2/K\) of \(p\)-adic fields, where \(p\) is an odd prime number. It is a quasi-split group. In this case, K. Buzzard and T. Gee [London Mathematical Society Lecture Note Series 414, 135–187 (2014; Zbl 1377.11067)] defined the so called \(C\)-group \({}^C\mathbf{U}_2\), which is expected to play the role of the Langlands dual group in the mod \(p\) Langlands correspondence. The local \(L\)-parameter in this setting is a continuous homomorphism from the Galois group of the extension \(\mathbb{Q}_p/K\) to the group \({}^C\mathbf{U}_2(\overline{\mathbb{F}}_p)\), compatible with the projection of the latter to the Galois group of the extension \(K_2/K\). The task of establishing the mod \(p\) local Langlands correspondence would be a construction of an \(L\)-packet of smooth admissible representations of \(\mathbf{U}_2(K)\) in a vector space over \(\overline{\mathbb F}_p\) associated to each \(L\)-parameter. However, this final goal is still out of reach.
Therefore, the route taken by this paper is to study the problem in the global context, that is, by considering a unitary group over a number field for which the local problem appears at finite places, and exploit the local–global compatibility, under certain technical assumptions. More precisely, let \(F^+\) be a totally real finite extension of the field \(\mathbb{Q}\) of rational numbers, and let \(F/F^+\) be a \(\mathrm{CM}\) quadratic extension of \(F^+\). Let \(\mathbf{G}\) be a unitary group in two variables associated to the extension \(F/F^+\). In order to study the local problem in the global context of the group \(\mathbf{G}\), it is assumed that \(p\) is unramified in \(F\) and that every place of \(F^+\) lying above \(p\) is unramified and inert in \(F\), so that \(\mathbf{G}\) at places over \(p\) is not split, i.e., it is the unramified unitary group. This differs from the previous work [T. Gee et al., J. Am. Math. Soc. 27, No. 2, 389–435 (2014; Zbl 1288.11045); T. Barnet-Lamb et al., Math. Ann. 356, No. 4, 1551–1598 (2013; Zbl 1339.11064)] on unitary groups in the global context, in which the assumption is that the group is split at places above \(p\). Additional assumption on \(\mathbf{G}\) is that it is a definite unitary group at Archimedean places.
The global \(L\)-parameter in this setting is a continuous homomorphism from the Galois group of the extension \(\overline{\mathbb{Q}} / F^+\), where \(\overline{\mathbb{Q}}\) is a fixed algebraic closure of \(\mathbb{Q}\), to the \(C\)-group \({}^C \mathbf{U}_2(\overline{\mathbb{F}}_p)\). Let \(\overline{r}\) be such a global \(L\)-parameter. The assumption on the parameter is that \(\overline{r}\) is associated to a nonzero Hecke eigenclass in the mod \(p\) cohomology with infinite level at \(p\) of the unitary group \(\mathbf{G}\) as above. Furthermore, the parameter \(\overline{r}\) is assumed to be semisimple and sufficiently generic at all places above \(p\), unramified outside \(p\), and that its multiplier is a cyclotomic character.
The set of Serre weights in which the global \(L\)-parameter \(\overline{r}\) is modular is denoted by \(W_{\mathrm{mod}}(\overline{r})\). It is the set of representations of the product of \(\mathbf{U}_2(\mathcal{O}_{F_v^+})\) over all places \(v\) of \(F^+\) lying over \(p\), where \(\mathcal{O}_{F_v^+}\) is the ring of integers of \(F_v^+\), which appear in the socle of the Hecke isotypic component attached to the \(L\)-parameter \(\overline{r}\) in the mod \(p\) cohomology of \(\mathbf{G}\).
On the other hand, T. Gee et al. [J. Eur. Math. Soc. (JEMS) 20, No. 12, 2859–2949 (2018; Zbl 1456.11093)] made a representation theoretic construction of certain set \(W^?(\overline{r})\) of weights associated to the global \(L\)-parameter \(\overline{r}\). They conjectured that this set of Serre weights coincides with the set of Serre weights in which \(\overline{r}\) is modular. The main result of this paper is that, under all the assumptions on the global \(L\)-parameter \(\overline{r}\) explained above and certain additional technical assumptions that we omit in this review, the conjecture on Serre weights for the unitary group \(\mathbf{U}_2\) holds. In other words, \[ W^?(\overline{r})=W_{\mathrm{mod}}(\overline{r}). \]
The proof of the inclusion \(W^?(\overline{r})\supseteq W_{\mathrm{mod}}(\overline{r})\) uses a new argument which is based on the global base change from the unitary group to \(\mathrm{GL}_2\). Thus, the problem is reduced to the case of \(\mathrm{GL}_2\), which is already settled by T. Gee [Invent. Math. 184, No. 1, 1–46 (2011; Zbl 1280.11029)]. The base change argument can be viewed as the evidence for the mod \(p\) base change is the mod \(p\) Langlands program. The opposite inclusion \(W^?(\overline{r})\subseteq W_{\mathrm{mod}}(\overline{r})\) is obtained using a modification of the patching functor of A. Caraiani [Camb. J. Math. 4, No. 2, 197–287 (2016; Zbl 1403.11073)], strengthened Taylor-Wiles-Kisin conditions and the explicit description of \({}^C\mathbf{U}_2\)-valued local deformation rings.