Serre weights for rank two unitary groups

By Thomas Barnet-Lamb, T. Gee, David Geraghty. 2011

65 citations recorded by the source corpus.

We study the weight part of (a generalisation of) Serre’s conjecture for mod $$l$$ Galois representations associated to automorphic representations on rank two unitary groups for odd primes $$l$$. We propose a conjectural set of Serre weights, agreeing with all conjectures in the literature, and under a mild assumption on the image of the mod $$l$$ Galois representation we are able to show that any modular representation is modular of each conjectured weight. We make no assumptions on the ramification or inertial degrees of $$l$$. Our main innovations are the use of outer forms of $$\text{ GL}_2$$ to avoid the parity difficulties with inner forms, and the use of the lifting techniques and “change of weight” results introduced in Barnet-Lamb et al. [J Am Math Soc 24(2):411–469, 2011; Duke Math J 161(8):1521–1580, 2012; Potential automorphy and change of weight, Preprint, 2010].

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/42f658ac71287726c8fc69615648949726a143e5

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In Discrete Mathematics and Combinatorics

1. T. Gee
2. David Savitt
3. Thomas Barnet-Lamb
4. F. Herzig
5. David Geraghty
6. Tongyin Liu

In Geometry and Topology

1. T. Gee
2. David Savitt
3. David Geraghty
4. Thomas Barnet-Lamb
5. F. Herzig
6. M. Emerton
7. Tongyin Liu
8. M. Kisin

In Mathematical Physics

1. T. Gee
2. David Savitt
3. David Geraghty
4. Thomas Barnet-Lamb
5. F. Herzig
6. Tongyin Liu
7. M. Emerton
8. M. Kisin

Citations in the corpus

• General Serre weight conjectures (2015, methodology): “rms of weight 2 are more directly accessible than those of higher weight, this principle is often extremely important, cf. [Rib76]. (The same principle is also valuable for Hilbert modular forms, cf. [BLGG13].) This interplay did not feature in Serre’s formulation of his conjecture, but a reformulat”
• Congruences between Hilbert modular forms: constructing ordinary lifts (2010, methodology): “e image of G in PGL 2(F 3) is conjugate to PSL 2(F 3). • We have l = 5, and the image of G in PGL 2(F 5) is conjugate to PGL 2(F 5) or PSL 2(F 5). • G is adequate. Proof. This is Proposition A.2.1 of [BLGG11]. In the case that ρ(GF(ζ l)) is adequate, our main result follows exactly as in section 6 o”
• The buzzard-diamond-jarvis conjecture for unitary groups (2012, methodology): “In this section we explain the definition of the sets of weight W(r̄), and recall some results from [BLGG11].”