Article
Mathematics
John Bergdall, Brandon Levin
Summary: This paper first determines rational Kisin modules associated with specific conditions on 2-dimensional crystalline representations, and then further identifies an integral Kisin module, used for calculating the semisimple reduction of the Galois representation. Within a specific range, it is found that the reduction is constant, thus improving upon a previous theorem by Berger, Li, and Zhu.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2022)
Article
Mathematics, Applied
Philip Dittmann, Borys Kadets
Summary: This article discusses the preimage tree T of a polynomial f under the action of the absolute Galois group of K. It introduces the concept of the arboreal Galois representation phi(f): Gal(K) -> Aut T. The conjecture that there exists a polynomial f for which phi(f) is surjective is shown to be false.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2022)
Article
Mathematics, Applied
Husileng Xiao
Summary: We first prove a version of Premet's conjecture for finite W-superalgebras associated with basic Lie superalgebras. Like in the case of W-algebras, this conjecture provides a classification of finite-dimensional simple modules of finite W-superalgebras. For basic type I Lie superalgebras, we classify the finite-dimensional simple supermodules with integral central character and develop an algorithm to compute their characters based on the g((0) over bar)-rough structure of g-modules.
SCIENCE CHINA-MATHEMATICS
(2023)
Article
Mathematics, Applied
Lian Duan
Summary: This paper proves that a 3-dimensional self-dual Galois representation constructed by van Geemen and Top is isomorphic to a quadratic twist of the symmetric square of the Tate module of an elliptic curve. The proof makes use of the refinement of the Faltings-Serre method to 3-dimensional l-adic self-dual representations with the ground field not equal to Q, utilizing Burnside groups.
MATHEMATICS OF COMPUTATION
(2021)
Article
Mathematics
Misja Fa Steinmetz
Summary: This article presents an explicit definition for the set of Serre weights associated with a mod p Galois representation. The authors prove that their definition is equivalent to previous definitions, and as a result, an explicit Serre's modularity conjecture for Hilbert modular forms over totally real number fields is obtained.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2022)
Article
Multidisciplinary Sciences
Mercy G. Amankwah, Daan Camps, E. Wes Bethel, Roel Van Beeumen, Talita Perciano
Summary: The study introduces a novel quantum pixel representation framework QPIXL, which can efficiently implement quantum circuits and significantly reduce gate complexity. The proposed method scales linearly in the number of pixels, does not use ancilla qubits, and comprises circuits solely of R-y gates.
SCIENTIFIC REPORTS
(2022)
Article
Mathematics
Jeffrey Manning, Jack Shotton
Summary: The study proves Ihara's lemma for the mod l cohomology of Shimura curves, based on an image hypothesis on the associated Galois representation. Utilizing the improved Taylor-Wiles method and the geometry of integral models of Shimura curves, the research avoids previous assumptions regarding l.
MATHEMATISCHE ANNALEN
(2021)
Article
Mathematics
Xiyuan Wang
Summary: We prove the weight elimination part of Serre's conjecture for rank two unitary groups in mod 2 Galois representations by modifying the results in [GLS14] and [GLS15].
MATHEMATICAL RESEARCH LETTERS
(2022)
Article
Mathematics
Xiyuan Wang
Summary: By modifying the results in [GLS14] and [GLS15], we prove the weight elimination part of Serre's conjecture for mod 2 Galois representations for rank two unitary groups.
MATHEMATICAL RESEARCH LETTERS
(2022)
Article
Mathematics, Applied
Andreas Maurischat
Summary: In the 90s, Pink gave a qualitative answer to the Mumford-Tate conjecture for Drinfeld modules. In contrast, we provide a family of uniformizable Anderson t-modules for which the Galois representations of their t-adic Tate modules have image that is far from being t-adically open in their motivic Galois groups. Nevertheless, the image is still Zariski-dense in the motivic Galois group, consistent with the Mumford-Tate conjecture.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2022)
Article
Mathematics, Applied
James Newton, Jack A. Thorne
Summary: In this paper, we prove the vanishing of the (Bloch-Kato) adjoint Selmer group of rho, under very mild hypotheses on rho. We obtain definitive results for the adjoint Selmer groups associated to non-CM Hilbert modular forms and elliptic curves over totally real fields.
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics
James Newton, Jack A. Thorne
Summary: The paper proves the automorphy of the symmetric power lifting under the assumption that the cuspidal Hecke eigenform does not have complex multiplication.
PUBLICATIONS MATHEMATIQUES DE L IHES
(2021)
Article
Mathematics
Fred Diamond, Payman L. Kassaei
Summary: This research proves that all mod p Hilbert modular forms can be generated via multiplication by generalized partial Hasse invariants from forms with specific weights. This result answers a previous question and generalizes previous findings for the non-ramified case. The study utilizes a different approach, using properties of the stratification at Iwahori level, which can be more readily extended to other Shimura varieties.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2023)
Article
Mathematics
Michael Harris, Chandrashekhar B. Khare, Jack A. Thorne
Summary: We construct a Langlands parameterization of supercuspidal representations of G2 over a p-adic field. For any finite extension K/Qp, we construct a bijection .Cg :A → g(G2, K) -> g →(G2, K) from the set of generic supercuspidal representations of G2(K) to the set of irreducible continuous homomorphisms p : WK > G2(C) with WK the Weil group of K. The construction of the map consists of assembling arguments from the literature and using a previously unpublished theorem of G. Savin on exceptional theta correspondences, included as an appendix. The proof that the map is a bijection relies on arithmetic properties and specifically utilizes automorphy lifting theorems, with application to Hundley and Liu's recent result on automorphic descent from GL(7) to G2.
ANNALES SCIENTIFIQUES DE L ECOLE NORMALE SUPERIEURE
(2023)
Article
Mathematics, Applied
Arno Kret, Sug Woo Shin
Summary: We have proven the existence of GSpin-valued Galois representations corresponding to cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. This confirms the Buzzard-Gee conjecture on the global Langlands correspondence in new cases. As an application, we have completed the argument by Gross and Savin to construct a rank 7 motive whose Galois group is of type G2 in the cohomology of Siegel modular varieties of genus 3. Under some additional local hypotheses, we have also shown automorphic multiplicity 1 as well as meromorphic continuation of the spin L-functions.
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
(2023)
