Article
Mathematics
Karol Koziol, Stefano Morra
Summary: We prove a version of Serre's conjecture for the weight part of mod p Galois representations attached to automorphic forms on rank 2 unitary groups which are nonsplit at p.
ALGEBRA & NUMBER THEORY
(2022)
Article
Mathematics
Raphael Beuzart-Plessis
Summary: By utilizing a local functional equation for Asai gamma-factors, we establish an explicit Plancherel decomposition for GL(n)(F)\GL(n)(E) where E/F is a quadratic extension of local fields of characteristic zero. We provide two applications of this Plancherel formula: one to the global Ichino-Ikeda conjecture for unitary groups by completing a comparison left open by Zhang on local relative characters, and the other to the Hiraga-Ichino-Ikeda conjecture on formal degrees in the case of unitary groups.
INVENTIONES MATHEMATICAE
(2021)
Article
Physics, Mathematical
Xinhong Chen, Ming Lu, Weiqiang Wang
Summary: The paper discusses quantum symmetric pairs of Kac-Moody type and their generalizations, such as iota quantum groups and universal iquantum groups. It formulates and establishes Serre-Lusztig relations for iquantum groups in terms of idivided powers, which have applications to braid group symmetries on iota quantum groups.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2021)
Article
Mathematics
Xun Xie
Summary: In this paper, we prove Lusztig's conjectures P1-P15 for Coxeter groups with complete graphs, utilizing decreasing induction on a-values and a type of factorization formula for Kazhdan-Lusztig basis elements. Additionally, we provide a detailed description of the left, right, and two-sided cells. In the appendix, the same methods are used to prove P1-P15 for right-angled Coxeter groups.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics
Zhicheng Feng, Conghui Li, Jiping Zhang
Summary: This paper establishes the inductive Alperin weight condition for finite simple groups of Lie type A, contributing to the program to prove the Alperin weight conjecture by verifying the inductive condition for all finite simple groups.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics
Atsushi Ichino
Summary: We study the theta lifting for real unitary groups and determine the theta lifts of tempered representations. In particular, we demonstrate that the theta lifts of (limits of) discrete series representations can be expressed as cohomologically induced representations in the weakly fair range. This extends a result of J.-S. Li in the case of discrete series representations with sufficiently regular infinitesimal character, whose theta lifts can be expressed as cohomologically induced representations in the good range.
ADVANCES IN MATHEMATICS
(2022)
Article
Computer Science, Theory & Methods
Eiichi Bannai, Manabu Oura, Da Zhao
Summary: This study demonstrates that similar results can be obtained for the invariants of the complex Clifford group under certain conditions, and confirms a conjecture proposed by Zhu, Kueng, Grassl, and Gross.
DESIGNS CODES AND CRYPTOGRAPHY
(2021)
Article
Mathematics
Damiano Rossi
Summary: In this study, we prove new results in generalised Harish-Chandra theory by providing a description of the Brauer-Lusztig blocks using the p-adic cohomology of Deligne-Lusztig varieties. We then propose new conjectures for finite reductive groups by considering geometric analogues of the p-local structures. Our conjectures coincide with the counting conjectures for large primes, thanks to a connection established between p-structures and their geometric counterparts. Finally, we simplify our conjectures by reducing them to the verification of Clifford theoretic properties.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Zhicheng Feng, Conghui Li, Jiping Zhang
Summary: In this paper, the blockwise Alperin weight conjecture is proven for finite special linear and unitary groups, as well as for finite groups with abelian Sylow 3-subgroups. The inductive blockwise Alperin weight condition is also verified for certain cases of groups of type A. Additionally, a classification of the 2-blocks of special linear and unitary groups is provided.
JOURNAL OF ALGEBRA
(2023)
Article
Mathematics
Michael Magee, Doron Puder
Summary: This article investigates the expected value of the trace of a word in the free group and shows that it has a convergent Laurent expansion at n = infinity involving maps on surfaces and L2-Euler characteristics of mapping class groups. The results obtained generalize previous theorems and provide important corollaries and estimates.
MATHEMATISCHE ANNALEN
(2022)
Article
Mathematics
Jianwei Gao, Xun Xie
Summary: In this study, Lusztig's conjectures P1-P15 for hyperbolic Coxeter groups of rank 3 are proven, providing a description of the a-functions and Kazhdan-Lusztig cells for these Coxeter groups.
JOURNAL OF ALGEBRA
(2021)
Article
Mathematics
Wee Teck Gan, Benedict H. Gross, Dipendra Prasad
Summary: In a series of three previous papers, the author addressed restriction problems for classical groups using the local and global Langlands correspondence. The problems involved a pair of orthogonal, Hermitian, symplectic, or skew-Hermitian spaces W and V. This paper focuses on a twisted variant of these conjectures for the case when W is equal to V.
COMPOSITIO MATHEMATICA
(2023)
Article
Endocrinology & Metabolism
Jennifer L. Miles-Chan, Laurie Isacco
Summary: Weight cycling, the repeated periods of weight loss and regain, is common in many populations. It is debated whether this practice increases future obesity risk, with evidence suggesting that those who were normal weight before cycling may be more affected. Athletes, particularly in weight-sensitive sports, often engage in weight cycling for competitive advantage, but studies on its long-term effects and cardiometabolic risks are limited.
Article
Mathematics, Applied
Pranab Sardar
Summary: This note aims to point out a mistake in the proof of Proposition 4.9 in Proc. Amer. Math. Soc. 146 (2018), 1859-1871, and discuss its consequences.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2022)
Article
Mathematics
Naomi Andrew, Armando Martino
Summary: We study the automorphism groups of free-by-cyclic groups and show that they are finitely generated in certain cases. By utilizing techniques such as actions on trees, relative hyperbolicity, and filtration of automorphisms, we establish an invariant tree for the group and reduce the initial problem to lower complexity groups. However, the challenge lies in finding a suitable invariant tree and demonstrating that the relevant groups are finitely generated.
JOURNAL OF ALGEBRA
(2022)
Article
Mathematics
Matthew Emerton, Toby Gee
ALGEBRA & NUMBER THEORY
(2015)
Article
Mathematics
Thomas Barnet-Lamb, Toby Gee, David Geraghty
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
(2018)
Article
Mathematics, Applied
Toby Gee, Florian Herzig, David Savitt
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
(2018)
Article
Mathematics
Rebecca Bellovin, Toby Gee
ALGEBRA & NUMBER THEORY
(2019)
Article
Mathematics
Toby Gee, James Newton
Summary: Under the assumption of the existence of p-adic Galois representations, we use Taylor-Wiles patching in the derived category to study the completed homology of locally symmetric spaces associated with GL(n) over a number field. By utilizing our construction and new results in non-commutative algebra, we establish that standard conjectures on completed homology imply 'big R = big U' theorems even in situations where the Zariski density of classical points cannot be relied upon. Furthermore, in the specific case where n = 2 and p splits completely in the number field, we establish a connection between our construction and the p-adic local Langlands correspondence for GL(2)(Q(p)).
JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU
(2022)
Article
Mathematics
Frank Calegari, Matthew Emerton, Toby Gee
Summary: Under the assumption of standard 'adequate image', this paper investigates the set of components of n-dimensional p-adic potentially semistable local Galois deformation rings that can be seen by potentially automorphic compatible systems of polarizable Galois representations over some CM field. The paper also improves the main potential automorphy result of Barnet-Lamb et al., replacing 'potentially diagonalizable' with 'potentially globally realizable', under the same assumption on n.
JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU
(2022)
Article
Mathematics
George Boxer, Frank Calegari, Toby Gee, Vincent Pilloni
Summary: This study demonstrates that abelian surfaces over totally real fields may be potentially modular, leading to the expected meromorphic continuation and functional equations of their Hasse-Weil zeta functions. Additionally, the modularity of numerous abelian surfaces over Q with EndCA=Z is shown. Furthermore, modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields are deduced.
PUBLICATIONS MATHEMATIQUES DE L IHES
(2021)
Article
Mathematics
Patrick B. Allen, Frank Calegari, Ana Caraiani, Toby Gee, David Helm, Bao V. Le Hung, James Newton, Peter Scholze, Richard Taylor, Jack A. Thorne
Summary: We prove modularity lifting theorems for regular n-dimensional Galois representations over a CM number field F without self-duality condition. As a result, we deduce that all elliptic curves over F are potentially modular and satisfy the Sato-Tate conjecture. Moreover, we also demonstrate the Ramanujan Conjecture for weight zero cuspidal automorphic representations for GL2(AF).
ANNALS OF MATHEMATICS
(2023)
Article
Mathematics
Matthew Emerton, Toby Gee
Summary: The paper provides criteria for certain morphisms to have scheme-theoretic image from an algebraic stack to a (not necessarily algebraic) stack. These criteria are then applied to show that certain natural moduli stacks of local Galois representations are algebraic (or Ind-algebraic) stacks.
ALGEBRAIC GEOMETRY
(2021)
Article
Mathematics
Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paskunas, Sug Woo Shin
COMPOSITIO MATHEMATICA
(2018)
Article
Mathematics
Frank Calegari, Matthew Emerton, Toby Gee, Lambros Mavrides
COMPOSITIO MATHEMATICA
(2017)
Correction
Mathematics
Frank Calegari, Toby Gee
ANNALES DE L INSTITUT FOURIER
(2017)
Proceedings Paper
Mathematics
Kevin Buzzard, Toby Gee
FAMILIES OF AUTOMORPHIC FORMS AND THE TRACE FORMULA
(2016)
Article
Mathematics
Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paskunas, Sug Woo Shin
CAMBRIDGE JOURNAL OF MATHEMATICS
(2016)
Article
Mathematics
Toby Gee, Florian Herzig, Tong Liu, David Savitt
DOCUMENTA MATHEMATICA
(2017)