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On Endomorphism Algebras of Gelfand-Graev Representations II (2023)
Journal Article
Li, T.-J., & Shotton, J. (2023). On Endomorphism Algebras of Gelfand-Graev Representations II. Bulletin of the London Mathematical Society, 55(6), 2876-2890. https://doi.org/10.1112/blms.12899

Let G be a connected reductive group defined over a finite field Fq of characteristic p, with Deligne–Lusztig dual G∗. We show that, over Z[1/pM] where M is the product of all bad primes for G, the endomorphism ring of a Gelfand–Graev representation... Read More about On Endomorphism Algebras of Gelfand-Graev Representations II.

Generic local deformation rings when l≠p (2022)
Journal Article
Shotton, J. (2022). Generic local deformation rings when l≠p. Compositio Mathematica, 158(4), 721-749. https://doi.org/10.1112/s0010437x22007461

We determine the local deformation rings of sufficiently generic mod l representations of the Galois group of a p-adic field, when l≠p, relating them to the space of q-power-stable semisimple conjugacy classes in the dual group. As a consequence, we... Read More about Generic local deformation rings when l≠p.

Ihara’s Lemma for Shimura curves over totally real fields via patching (2020)
Journal Article
Manning, J., & Shotton, J. (2021). Ihara’s Lemma for Shimura curves over totally real fields via patching. Mathematische Annalen, 379, 187-234. https://doi.org/10.1007/s00208-020-02048-8

We prove Ihara’s lemma for the mod l cohomology of Shimura curves, localized at a maximal ideal of the Hecke algebra, under a large image hypothesis on the associated Galois representation. This was proved by Diamond and Taylor, for Shimura curves ov... Read More about Ihara’s Lemma for Shimura curves over totally real fields via patching.

The Breuil–Mézard conjecture when l≠p (2017)
Journal Article
Shotton, J. (2018). The Breuil–Mézard conjecture when l≠p. Duke Mathematical Journal, 167(4), 603-678. https://doi.org/10.1215/00127094-2017-0039

Let l and p be primes, let F=Qp be a finite extension with absolute Galois group GF , let F be a finite field of characteristic l, and let W GF ! GLn.F/ be a continuous representation. Let R./ be the universal framed deformation ring for . If l D p,... Read More about The Breuil–Mézard conjecture when l≠p.

Local deformation rings for GL2 and a Breuil–Mézard conjecture when l≠p (2016)
Journal Article
Shotton, J. (2016). Local deformation rings for GL2 and a Breuil–Mézard conjecture when l≠p. Algebra & Number Theory, 10(7), 1437-1475. https://doi.org/10.2140/ant.2016.10.1437

We compute the deformation rings of two dimensional mod l rep- resentations of Gal(F/F) with fixed inertial type, for l an odd prime, p a prime distinct from l, and F/Qp a finite extension. We show that in this set- ting an analogue of the Breuil–M´e... Read More about Local deformation rings for GL2 and a Breuil–Mézard conjecture when l≠p.