Article
Mathematics
Peter Dillery
Summary: We extend the concept of rigid inner forms defined by Kaletha in [12] to local function field F in order to state the local Langlands conjectures for arbitrary connected reductive groups over F. To achieve this, we define a new cohomology set H1(& POUND;, Z & RARR; G) & SUB; H1fpqc(& POUND;, G) for a gerbe & POUND; attached to a certain class in H2fppf(F, u) for a canonically-defined profinite commutative group scheme u, leading to an analogue of the classical Tate-Nakayama duality theorem. We define a relative transfer factor for an endoscopic datum serving a connected reductive group G over F, and use rigid inner forms to extend this to an absolute transfer factor, enabling the statement of endoscopic conjectures relating stable virtual characters and s-stable virtual characters for a semisimple s associated to a tempered Langlands parameter.
ADVANCES IN MATHEMATICS
(2023)
Article
Multidisciplinary Sciences
Patrick B. Allen, Chandrashekhar B. Khare, Jack A. Thorne
Summary: The study focuses on an analog of Serre's modularity conjecture for projective representations over a totally real number field, with cases being proven when k = F-5.
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
(2021)
Article
Mathematics
Siyan Daniel Li-Huerta
Summary: In this paper, we present a new proof of the local Langlands correspondence for GLn over a local field of characteristic p > 0, by adapting methods introduced by Scholze (2013). We construct l-adic Galois representations associated with discrete automorphic representations over global function fields, and use them to establish a mapping from isomorphism classes of irreducible smooth representations of GLn(F) to isomorphism classes of n-dimensional semisimple continuous representations of the Weil group WF. This mapping, characterized by a local compatibility condition on traces of a specific test function, is shown to be equivalent to the usual local Langlands correspondence after disregarding the monodromy operator.
ALGEBRA & NUMBER THEORY
(2022)
Article
Mathematics, Applied
David Hansen, Tasho Kaletha, Jared Weinstein
Summary: This article discusses Kottwitz's conjecture and its extension to Scholze's spaces of local shtukas. We prove the extended conjecture using a new Lefschetz-Verdier trace formula and obtain some application results.
FORUM OF MATHEMATICS PI
(2022)
Article
Mathematics
Masao Oi, Kazuki Tokimoto
Summary: The paper demonstrates the consistency of Kaletha's recent construction and Harris-Taylor's local Langlands correspondence for regular supercuspidal representations in general linear groups. The crucial factors are the essentially tame local Langlands correspondence by Bushnell-Henniart and Tam's findings on Bushnell-Henniart's rectifiers. By combining these results, the comparison of Kaletha's and Tam's chi-data is simplified to a basic root-theoretic computation.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2021)
Article
Mathematics
Pavel Etingof, Edward Frenkel, David Kazhdan
Summary: We construct analogues of Hecke operators for G-bundles on a curve X over a local field F with parabolic structures at finitely many points. We conjecture that these operators define commuting compact normal operators on the Hilbert space of half-densities on the moduli space. In the case F = C, we also conjecture that their joint spectrum is naturally in correspondence with the set of LG-opers on X with real monodromy. Additionally, we conjecture an explicit formula relating the eigenvalues of the Hecke operators and the global differential operators, which follows from a system of differential equations satisfied by the Hecke operators assuming the compactness conjecture, and we prove this for G = PGLn.
DUKE MATHEMATICAL JOURNAL
(2023)
Article
Mathematics
Jack Shotton
Summary: This paper investigates the local deformation rings of mod l representations of the Galois group of a p-adic field, when l is not equal to p, and establishes a connection with the space of q-power-stable semisimple conjugacy classes in the dual group. As a consequence, the author provides a local proof of the l not equal p Breuil-Mezard conjecture in the tame case.
COMPOSITIO MATHEMATICA
(2022)
Article
Mathematics, Applied
Rui Chen, Jialiang Zou
Summary: By using the theta correspondence, a classification of irreducible representations of any even orthogonal group was obtained, which coincides with the local Langlands correspondence established by Arthur and precisely formulated by Atobe-Gan for quasi-split even orthogonal groups.
SELECTA MATHEMATICA-NEW SERIES
(2021)
Article
Mathematics
Jishnu Ray
Summary: This article investigates the locally analytic vectors of two-dimensional crystalline Galois representations. An explicit description is provided, showing the existence of rigid analytic vectors within the locally analytic representation. It is proven that these vectors are non-null, establishing their significance in the study of p-adic representations.
JOURNAL OF NUMBER THEORY
(2022)
Article
Mathematics
R. Kurinczuk, N. Matringe
Summary: The passage discusses the classification and construction of W-F-semisimple Deligne l-modular representations in a non-archimedean local field, as well as the extension of Artin-Deligne local constants in this setting. It also introduces a variant of the l-modular local Langlands correspondence with a preservation statement for pairs of generic representations.
JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU
(2021)
Article
Mathematics, Applied
Peter J. Cho
Summary: The paper shows that for an elliptic curve E, the average analytic rank of E over cyclic extensions of degree l over Q with l a prime not equal to 2 is at most 2+rQ(E), where rQ(E) is the analytic rank of the elliptic curve E over Q. This bound is independent of the degree l. Using a recent result of Bhargava, Taniguchi and Thorne, a non-trivial upper bound on the average analytic rank of E over S3-fields is obtained.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics
Tim Dokchitser, Vladimir Dokchitser, Celine Maistret, Adam Morgan
Summary: We studied hyperelliptic curves y(2) = f(x) over local fields of odd residue characteristic. We introduced the notion of a cluster picture associated to the curve, which describes the p-adic distances between the roots of f(x). We showed that this elementary combinatorial object encodes various important characteristics of the curve.
MATHEMATISCHE ANNALEN
(2023)
Article
Mathematics
Evis Ieronymou
Summary: We study evaluation maps determined by elements of the Brauer group of varieties over local fields and demonstrate the constancy of these maps in several interesting cases.
MATHEMATISCHE ANNALEN
(2022)
Article
Mathematics
Vlere Mehmeti
Summary: This paper explores higher-dimensional patching in the Berkovich setting by demonstrating its applicability around certain fibers of a relative Berkovich analytic curve, leading to a proof of a local-global principle over the field of overconvergent meromorphic functions on said fibers. Additionally, by establishing the algebraicity of these germs of meromorphic functions, the authors are able to extend their results to function fields of algebraic curves defined over a class of (not necessarily complete) ultrametric fields, thus generalizing prior findings by Mehmeti (Compos Math 155:2399-2438, 2019).
MATHEMATISCHE ANNALEN
(2022)
Article
Mathematics
Alex Cohen, Guy Moshkovitz
Summary: This study proves that the partition rank and the analytic rank of tensors are equal up to a constant in finite fields. The core of the proof lies in a technique for polynomial decomposition and finding rational points satisfying certain conditions. This research is of great importance to the field of additive combinatorics.
DUKE MATHEMATICAL JOURNAL
(2023)
