The Sato-Tate conjecture for Hilbert modular forms

By Thomas Barnet-Lamb, T. Gee, David Geraghty. 2009

104 citations recorded by the source corpus.

We prove the Sato-Tate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the Sato-Tate conjecture for regular algebraic cuspidal automorphic representations of GL2(AF ), F a totally real field, which are not of CM type. The argument is based on the potential automorphy techniques developed by Taylor et. al., but makes use of automorphy lifting theorems over ramified fields, together with a “topological” argument with local deformation rings. In particular, we give a new proof of the conjecture for modular forms, which does not make use of potential automorphy theorems for non-ordinary n-dimensional Galois representations.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e24d7fae0866464e4c1c6e28fb8fd983d3659e42

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Citations in the corpus

Sato–Tate distributions and Galois endomorphism modules in genus 2 (2011, methodology): “ee, Geraghty, Harris, Shepherd-Barron, and Taylor. For example, Conjecture 1.1 is known (with ST A = SU(2)) whenever Ais an elliptic curve over a totally real eld without complex multiplication. (See [BLGG11] for an even stronger result covering Hilbert modular forms.) For g= 2, it is unclear at pre”
Congruences between Hilbert modular forms: constructing ordinary lifts (2010, methodology): “4 of [BLGG09], it follows that ρv ⊗ r′|GF3,ṽ ∼”
Potential automorphy and change of weight (2010, methodology): “In this section we present two automorphy lifting theorems which slightly generalize results of [BLGG09] and [Ger09].”
Patching and the p-adic local Langlands correspondence (2013, methodology): “ection 2.6 below for a more detailed discussion of this diﬀerence. Our construction uses the same general framework as in section 5 of [EG13] (which in turn is based on the approach taken in [CHT08], [BLGG11] and [Tho12]); we recall the key elements of this framework in the ﬁrst several subsections ”
Sato–Tate theorem for families and low-lying zeros of automorphic $$L$$L-functions (2012, background): “The automorphic Sato-Tate conjecture is known in the very limited cases of (the restriction of scalars of) GL1 and GL2 ([7], [8]).”
An algebraic Sato-Tate group and Sato-Tate conjecture (2011, result): “curve without complex multiplication, the Sato-Tate conjecture predicts equidistribution with respect to the image of the Haar measure on SU(2); this result is now known when F is a totally real ﬁeld [BLGG], building upon recent advances in the modularity of Galois representations led by R. Taylor. ”
THE BREUIL–MÉZARD CONJECTURE FOR POTENTIALLY BARSOTTI–TATE REPRESENTATIONS (2012, methodology): “+) Ie ev,and hence an element λa∈ (Z2 +) Ie p. 3. The patching argument 3.1. Hecke algebras. In this section we employ the Taylor–Wiles–Kisin patching method, following the approaches of [Kis09a] and [BLGG11] (which in turn follows [CHT08]). In particular, in the actual implementation of the patchin”
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): “This is deduced in [1] from the theorems of [22].”
A geometric perspective on the Breuil–Mézard conjecture (2011, background): “We closely follow the approaches of [29, 40] (which in turn follows [1, 14]).”
On the Sato-Tate conjecture for non-generic abelian surfaces (2013, methodology): “ore precisely, it tells us that the numbers θv = arccos(av/2q 1/2 v ) are distributed according to the measure 2 πsin 2θdθon [0,π]. This conjecture was recently settled in [HSBT] and [BGHT] (see also [BLGG]) when F is totally real as a result of progress on potential automorphy theorems for compatib”