Automorphic lifts of prescribed types

By T. Gee. 2008

116 citations recorded by the source corpus.

We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hilbert modular forms of parallel weight 2. We discuss some conjectures on the weights of n-dimensional mod p Galois representations. Finally, we use recent work of Taylor to prove level raising and lowering results for n-dimensional automorphic Galois representations.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/29d894a45c67dfa4323e0ac4e9704014379fa64c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Level-raising and symmetric power functoriality, III (2017, background): “We note that an argument of Taylor (see [Tay08], [Gee11]) allows one to construct automorphic liftings of an irreducible residual Galois representation with essentially any prescribed local behaviour (for example, with Steinberg type ramification).”
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): “se l6=p. Let Lv be a ﬁnite extension ofQp. Letρ: GLv →GLn(O)beacontinuous representation. Let R denote the universal (unrestricted) lifting ring. Then R ⊗ O Ql is equidimensional of dimension n2 (see [Gee]). Let Cbe an irreducible component or a set of irreducible components. Then we will write RC v”
On the weights of mod p Hilbert modular forms (2006, background): “By Corollary 3.1.7 of [ 15 ] there is a modular lift ρ : GF → GL2(Qp) of ρ which is potentially Barsotti-Tate of type τ �� b,Jv for each v|p for which ρ|GFv is reducible, and of type τ � a,�v for each v|p for which ρ|GFv is irreducible.”
Congruences between Hilbert modular forms: constructing ordinary lifts (2010, methodology): “To see that we can do this, note that by Propositions 6.1.3 and 6.1.4, there is a finite solvable extension of totally real fields F+1 /F + and a regular algebraic ι-ordinary cuspidal automorphic representation π1 of GL2(AF+1 ) of weight 0 such that • π1 is unramified at all finite places.”
Elliptic curves over real quadratic fields are modular (2013, background): “[3] T. Barnet-Lamb, T. Gee and D. Geraghty, Congruences between Hilbert modular forms: constructing ordinary lifts, Duke Math.”
Potentially crystalline deformation rings and Serre weight conjectures: shapes and shadows (2015, unknown): “The precise relation between Serre weights of r and Hodge-Tate weights of crystalline lifts of ρ was first made precise in [Gee11] and the obvious weights are shown to be modular in [BLGG] using global methods (namely, modularity lifting techniques) under the assumption that r is modular of a lower…”
Deformations of polarized automorphic Galois representations and adjoint Selmer groups (2014, methodology): “Indeed, a result of Gee [Gee11, Theorem 2.1.6] gives the dimension of the generic fibre of this framed deformation ring, so it suffices to check when the tangent space, which is related to an H1, has strictly larger dimension, and this can be related to our genericity condition using local Euler cha”
Potential automorphy and change of weight (2010, background): “ill imply that all irreducible components of SpecR O,ρ[1/l] are generically formally smooth of dimension n 2. (The corresponding result for smooth points can be found in the proof of Theorem 2.1.6 of [Gee11] or in [Cho09], but our proof seems to be diﬀerent even in this special case. This case is al”
Level raising and symmetric power functoriality, II (2015, methodology): “If not, then applying [Gee11, Cor. 3.1.7] once more we can find an RAESDC automorphic representation π′′ which satisfies conditions (1)–(4) above and such that rι(π′′) ∼= rι(π′).”
THE BREUIL–MÉZARD CONJECTURE FOR POTENTIALLY BARSOTTI–TATE REPRESENTATIONS (2012, methodology): “In fact, examining the proof of Proposition 3.1.5 of [Gee11a], we see that it is enough to check that there is some finite solvable extension L/F of totally real fields with the property that L is linearly disjoint from F ker ρ (ζp) over F , and ρ|GL has a modular lift which is potentially…”
COMPANION FORMS FOR UNITARY AND SYMPLECTIC GROUPS (2010, background): “This is Theorem 2.0.6 of [Gee06].”
Weight cycling and Serre-type conjectures for unitary groups (2011, background): “There have been many new conjectures formulated (cf. [BDJ10], [Her09], [Sch08b], [Gee10a]) and several cases of these conjectures have been established (cf. [Gee10b], [GS10a]).”