\(p\)-adic local Langlands correspondence
Refined modular multiplicities
In their important paper ``Multiplicités modulaires et représentations de \(\mathrm{GL}_2(\mathbb{Z}_p)\) et de \(\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)\) en \(\ell=p\)'' [Duke Math. J. 115, No. 2, 205--310 (2002; Zbl 1042.11030)], the authors defined two kinds of multiplicities, an automorphic one and a Galois one, and conjectured that these two were equal. This conjecture was proved by \textit{M. Kisin} in [J. Am. Math. Soc. 22, No. 3, 641--690 (2009; Zbl 1251.11045)]. In the present paper, the authors give a refined version of their conjecture. Fix some representation \(\overline{\rho} : \mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p) \to \mathrm{GL}_2(k)\) where \(k\) is a finite field. The automorphic multiplicity is the number of times the ``Serre weights'' of \(\overline{\rho}\) appear in a certain representation. The Galois multiplicity of \(\overline{\rho}\) is the Hilbert-Samuel multiplicity of \(R/p\), where \(R\) is the ring corresponding to the universal deformation space (with many conditions) of \(\overline{\rho}\). The conjecture of Breuil-Mézard (proved by Kisin) says that the two multiplicities are equal.NEWLINENEWLINEThe refined version (Theorem 1.2 of the present paper) matches the irreducible components of \(R/p\) with the Serre weights of \(\overline{\rho}\). Each irreducible component and each corresponding Serre weight then has its own multiplicity, and these are then equal. Note that this is true only if \(\overline{\rho}\) is ``sufficiently generic''. The bijection can even be made explicit in certain cases (Theorem 1.3). The authors conjecture that their refined bijection holds for ``sufficiently generic'' representations of \(\mathrm{Gal}(\overline{\mathbb{Q}}_p/F)\), with \(F\) a finite unramified extension of \(\mathbb{Q}_p\) (Conjecture 1.4), and check this in certain cases (Theorem 1.5).NEWLINENEWLINEThe theorems and conjectures of this article have been generalized by \textit{M. Emerton} and \textit{T. Gee} in [J. Inst. Math. Jussieu 13, No. 1, 183--223 (2014; Zbl 1318.11061)].
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