COMPANION FORMS FOR UNITARY AND SYMPLECTIC GROUPS

By T. Gee, David Geraghty. 2010

54 citations recorded by the source corpus.

We prove a companion forms theorem for ordinary n-dimensional automorphic Galois representations, by use of automorphy lifting theorems developed by the second author, and a technique for deducing companion forms theorems due to the first author. We deduce results about the possible Serre weights of mod l Galois representations corresponding to automorphic representations on unitary groups. We then use functoriality to prove similar results for automorphic representations of GSp4 over totally real fields.

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

View Paper: https://www.semanticscholar.org/paper/0956902e84574ccc05ab3cc7c0491efe32e27f66

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In Discrete Mathematics and Combinatorics

1. T. Gee
2. Thomas Barnet-Lamb
3. F. Herzig
4. David Geraghty
5. M. Emerton
6. J. Thorne
7. Richard Taylor
8. David Savitt

In Geometry and Topology

1. T. Gee
2. David Geraghty
3. M. Emerton
4. J. Thorne
5. Thomas Barnet-Lamb
6. F. Herzig
7. Richard Taylor
8. Frank Calegari

In Mathematical Physics

1. T. Gee
2. David Geraghty
3. M. Emerton
4. J. Thorne
5. Thomas Barnet-Lamb
6. F. Herzig
7. Frank Calegari
8. Richard Taylor

Citations in the corpus

• On the automorphy of l-adic Galois representations with small residual image With an appendix by Robert Guralnick, Florian Herzig, Richard Taylor and Jack Thorne (2011, background): “n M/M+, {R1 ev}v∈SM,v∤l∪{R △,ar Λve }v∈S M,v|l∪R ur ev1 . By construction, there is a map Runiv SM λ →Runiv S , given by restriction of the universal deformation, and the argument of Lemma 3.2.5 of [GG] shows that it is finite. Thus it suffices to show that Runiv SM λ is a finite O-algebra. Let χλM : Λ”
• The Sato-Tate conjecture for Hilbert modular forms (2009, unknown): “5 of [GG09]), and hence it suffices to show that R SL,r′ is finite over O.”
• Serre weights for rank two unitary groups (2011, methodology): “The arguments of [GG09], as refined in [Tho10] and [BLGGT10], allow us to replace this with an ordinary automorphic lift rl,ı(π ′ 1) which has the behaviour prescribed for r at places not dividing l.”
• Weight cycling and Serre-type conjectures for unitary groups (2011, result): “) We remark that other results have recently been obtained in [GG09] (in the ordinary case), and the forthcoming [BLGG11].”
• Automorphic lifts of prescribed types (2008, result): “R = T theorems have also been proved in the ordinary case by Geraghty [16], and the analogues of the results of this section are proved in [ 17 ].”
• Potential automorphy and change of weight (2010, background): “In section 3 we generalise a number of results from our earlier papers (particularly [GG09] and [BLGG09]) to work with essentially conjugate self-dual representations, rather than just conjugate self-dual representations; this is for the most part just a matter of carefully keeping track of twists.”