Potentially crystalline deformation rings and Serre weight conjectures: shapes and shadows

By Daniel Le, Bao V. Le Hung, B. Levin, Stefano Morra. Inventiones Mathematicae, 2015

47 citations recorded by the source corpus.

We prove the weight part of Serre’s conjecture in generic situations for forms of U(3) which are compact at infinity and split at places dividing p as conjectured by Herzig (Duke Math J 149(1):37–116, 2009). We also prove automorphy lifting theorems in dimension three. The key input is an explicit description of tamely potentially crystalline deformation rings with Hodge–Tate weights (2, 1, 0) for $$K/\mathbb {Q}_p$$K/Qp unramified combined with patching techniques. Our results show that the (geometric) Breuil–Mézard conjectures hold for these deformation rings.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/230c20ef84fc0eff8a3b39777d1b93c00b711847

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

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

In Mathematical Physics

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

Citations in the corpus

• General Serre weight conjectures (2015, background): “We had a narrow escape, then, when (prior to the completion of this paper) Le, Le Hung, Levin, and Morra [LLHLM15] produced counterexamples to the shadow weight conjecture for GL3 over Q (in the non-semisimple case), thus also disproving the crystalline lifts version of the weight part of Serre’s co”