Related papers: Langlands parameters for reductive groups over fin…

We consider various counting questions for irreducible binomials over finite fields. We use various results from analytic number theory to investigate these questions.

We classify Galois objects for the dual of a group algebra of a finite group over an arbitrary field.

We define a partition of a reductive group into finitely many subsets, refining the partition of the group into strata. We state some conjectural properties of these subsets (called substrata) and verify them in some examples.

Laurent Lafforgue has been awarded the Fields Medal for his proof of the Langlands correspondence for the full linear groups $\mathop{\rm GL}\nolimits_{r}$ ($r\geq 1$) over function fields. This article is a brief introduction to the…

Let $\mathbb{F}_q$ denote the finite fields with $q$ elements. The permutation behavior of several classes of infinite families of permutation polynomials over finite fields have been studied in recent years. In this paper, we continue with…

Let F be a non-archimedean local field. We establish the local Langlands correspondence for all inner forms of the group $SL_n (F)$. It takes the form of a bijection between, on the one hand, conjugacy classes of Langlands parameters for…

In this paper, we consider the construction of irreducible representations of finite pattern groups in terms of Panov's associative polarization, which is a finite-field analogue of Kirillov's orbital method. Using this construction, first,…

We begin by defining general hypergeometric functions over finite fields and obtaining a finite field analogue of a classical symmetry in their complex counterparts. We give a geometric proof for the symmetry by constructing isomorphisms…

Let G be an unramified reductive group over a non archimedian local field F. The so-called "Langlands Fundamental Lemma" is a family of conjectural identities between orbital integrals for G(F) and orbital integrals for endoscopic groups of…

We make a list of finite simple groups whose group rings over a given field are serial.

We obtain several structure results for a class of spherical subgroups of connected reductive complex algebraic groups that extends the class of strongly solvable spherical subgroups. Based on these results, we construct certain…

For any admissible pair of irreducible reduced crystallographic root systems, we present discrete orthogonality relations for a finite-dimensional system of Macdonald polynomials with parameters on the unit circle subject to a truncation…

We prove that the isomorphism problem for group algebras reduces to group algebras over finite extensions of the prime field. In particular, the modular isomorphism problem reduces to finite modular group algebras.

We prove that the Steinberg representation of a connected reductive group over an infinite field is irreducible. For finite fields, this is a classical theorem of Steinberg and Curtis.

I study how to apply relativistic quantum field theory to condensed matter systems. The motivation for this is examined and then two separate elements are considered. First we identify the precise relationship between relativistic and…

In this paper we give uniform annihilators for some relations of all systems of parameters in a local ring. Our results provide simple proofs for some classical results.

I prove lower bounds of some parameters of elliptic curve over finite field. There parameters are closely interrelated with cryptographic stability of elliptic curve.

In this paper we describe the unramified Langlands correspondence for two-dimensional local fields, we construct a categorical analogue of the unramified principal series representations and study its properties. The main tool for this…

Given a finite group $G$, we denote by $L(G)$ the subgroup lattice of $G$ and by ${\rm Isolated}(G)$ the set of isolated subgroups of $G$. In this note, we describe finite groups $G$ such that $|{\rm Isolated}(G)|=|L(G)|-k$, where…

We introduce a notion of refinements in the context of patching, in order to obtain new results about local-global principles and field invariants in the context of quadratic forms and central simple algebras. The fields we consider are…