Toby Gee
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View article: The Ramanujan and Sato–Tate Conjectures for Bianchi modular forms
The Ramanujan and Sato–Tate Conjectures for Bianchi modular forms Open
We prove the Ramanujan and Sato–Tate conjectures for Bianchi modular forms of weight at least $2$ . More generally, we prove these conjectures for all regular algebraic cuspidal automorphic representations of $\operatorname {\mathrm {GL}}_…
View article: Components of moduli stacks of two-dimensional Galois representations
Components of moduli stacks of two-dimensional Galois representations Open
In the article [CEGS20b], we introduced various moduli stacks of two-dimensional tamely potentially Barsotti–Tate representations of the absolute Galois group of a p -adic local field, as well as related moduli stacks of Breuil–Kisin modul…
View article: P589 Real-world data on upadacitinib in the treatment of inflammatory bowel disease: safe and highly effective with extremely positive patient feedback
P589 Real-world data on upadacitinib in the treatment of inflammatory bowel disease: safe and highly effective with extremely positive patient feedback Open
Background Upadacitinib is a selective Janus kinase inhibitor that has recently been approved in England for the treatment of ulcerative colitis (UC) and Crohn’s disease (CD) though real-world data is lacking. Furthermore, there is very li…
View article: Cuspidal cohomology classes for GL_n(Z)
Cuspidal cohomology classes for GL_n(Z) Open
We prove the existence of a cuspidal automorphic representation $π$ for $GL_{79}/\mathbf{Q}$ of level one and weight zero. We construct $π$ using symmetric power functoriality and a change of weight theorem, using Galois deformation theory…
View article: The Ramanujan and Sato-Tate Conjectures for Bianchi modular forms
The Ramanujan and Sato-Tate Conjectures for Bianchi modular forms Open
We prove the Ramanujan and Sato-Tate conjectures for Bianchi modular forms of weight at least 2. More generally, we prove these conjectures for all regular algebraic cuspidal automorphic representations of $\mathrm{GL}_2(\mathbf{A}_F)$ of …
View article: Mepolizumab for treatment of adolescents and adults with eosinophilic oesophagitis: a multicentre, randomised, double-blind, placebo-controlled clinical trial
Mepolizumab for treatment of adolescents and adults with eosinophilic oesophagitis: a multicentre, randomised, double-blind, placebo-controlled clinical trial Open
Objective We aimed to determine whether mepolizumab, an anti-IL-5 antibody, was more effective than placebo for improving dysphagia symptoms and decreasing oesophageal eosinophil counts in eosinophilic oesophagitis (EoE). Methods We conduc…
View article: Potential automorphy over CM fields
Potential automorphy over CM fields Open
Let F be a CM number field. We prove modularity lifting theorems for regular n-dimensional Galois representations over F without any self- duality condition. We deduce that all elliptic curves E over F are poten- tially modular, and furthe…
View article: Modularity lifting theorems
Modularity lifting theorems Open
Updated version of 2013 Arizona WInter School notes on modularity lifting theorems for for two-dimensional p-adic representations, using wherever possible arguments that go over to the n-dimensional (self-dual) case.
View article: An introduction to the categorical p-adic Langlands program
An introduction to the categorical p-adic Langlands program Open
We give an introduction to the "categorical" approach to the p-adic Langlands program, in both the "Banach" and "analytic" settings.
View article: Local geometry of moduli stacks of two-dimensional Galois representations
Local geometry of moduli stacks of two-dimensional Galois representations Open
We construct moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field, as well as their resolutions by moduli stacks of two-dimensional Breuil-Kisin modules with tame descent data. We stud…
View article: The geometric Breuil-Mézard conjecture for two-dimensional potentially Barsotti-Tate Galois representations
The geometric Breuil-Mézard conjecture for two-dimensional potentially Barsotti-Tate Galois representations Open
We establish a geometrisation of the Breuil-Mézard conjecture for potentially Barsotti-Tate representations, as well as of the weight part of Serre's conjecture, for moduli stacks of two-dimensional mod p representations of the absolute Ga…
View article: Components of moduli stacks of two-dimensional Galois representations
Components of moduli stacks of two-dimensional Galois representations Open
In a previous article we introduced various moduli stacks of two-dimensional tamely potentially Barsotti-Tate representations of the absolute Galois group of a p-adic local field, as well as related moduli stacks of Breuil-Kisin modules wi…
View article: Localization of smooth p-power torsion representations of GL_2(Q_p)
Localization of smooth p-power torsion representations of GL_2(Q_p) Open
We show that the category of smooth representations of GL_2(Q_p) on p-power torsion modules localizes over a certain projective scheme, and give some applications.
View article: Modularity lifting theorems
Modularity lifting theorems Open
Updated version of 2013 Arizona WInter School notes on modularity lifting theorems for for two-dimensional p-adic representations, using wherever possible arguments that go over to the n-dimensional (self-dual) case.
View article: Abelian surfaces over totally real fields are potentially modular
Abelian surfaces over totally real fields are potentially modular Open
We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse–Weil zeta functi…
View article: Moduli stacks of (phi,Gamma)-modules: a survey
Moduli stacks of (phi,Gamma)-modules: a survey Open
This survey article is based on 10 lectures that we gave at the September 2019 Hausdorff School in Bonn on "the Emerton--Gee stack and related topics", and the sections correspond to the original lectures. We have retained the somewhat inf…
View article: "Scheme-theoretic" images of morphisms of stacks
"Scheme-theoretic" images of morphisms of stacks Open
We give criteria for certain morphisms from an algebraic stack to a (not necessarily algebraic) stack to admit an (appropriately defined) scheme-theoretic image. We apply our criteria to show that certain natural moduli stacks of local Gal…
View article: “Scheme-theoretic images” of morphisms of stacks
“Scheme-theoretic images” of morphisms of stacks Open
We give criteria for certain morphisms from an algebraic stack to a (not necessarily algebraic) stack to admit an (appropriately defined) scheme-theoretic image. We apply our criteria to show that certain natural moduli stacks of local Gal…
View article: Patching and the completed homology of locally symmetric spaces
Patching and the completed homology of locally symmetric spaces Open
Under an assumption on the existence of p -adic Galois representations, we carry out Taylor–Wiles patching (in the derived category) for the completed homology of the locally symmetric spaces associated with GLn over a number field. We use…
View article: Moduli stacks of \\'etale (phi,Gamma)-modules and the existence of\n crystalline lifts
Moduli stacks of \\'etale (phi,Gamma)-modules and the existence of\n crystalline lifts Open
We construct stacks which algebraize Mazur's formal deformation rings of\nlocal Galois representations. More precisely, we construct Noetherian formal\nalgebraic stacks over Spf Zp which parameterize \\'etale (phi,Gamma)-modules;\nthe form…
View article: Moduli stacks of étale (phi,Gamma)-modules and the existence of crystalline lifts
Moduli stacks of étale (phi,Gamma)-modules and the existence of crystalline lifts Open
We construct stacks which algebraize Mazur's formal deformation rings of local Galois representations. More precisely, we construct Noetherian formal algebraic stacks over Spf Zp which parameterize étale (phi,Gamma)-modules; the formal com…
View article: Moduli stacks of two-dimensional Galois representations
Moduli stacks of two-dimensional Galois representations Open
We construct moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field, and relate their geometry to the weight part of Serre's conjecture for GL(2).
View article: G-valued local deformation rings and globallifts
G-valued local deformation rings and globallifts Open
We study G-valued Galois deformation rings with prescribed properties, where G is an arbitrary (not necessarily connected) reductive group over an extension of Zl for some prime l. In particular, for the Galois groups of p-adic local field…
View article: Potential automorphy over CM fields
Potential automorphy over CM fields Open
Let $F$ be a CM number field. We prove modularity lifting theorems for regular $n$-dimensional Galois representations over $F$ without any self-duality condition. We deduce that all elliptic curves $E$ over $F$ are potentially modular, and…