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Sazzad Ali Biswas

Publications and source records attributed to Sazzad Ali Biswas.

10 recordsLinked to original sources
arXiv2024

Existence of non-abelian local constants, and their properties

In his Ph.D. thesis, John Tate attached the (abelian) local constants to the characters of a non-Archimedean local field of characteristic zero. Robert Langlands proved the existence theorem of a non-abelian local constant of a higher-dimensional complex local Galois representation. In 1990, Helmut Koch summarized Langlands' strategy for the existence of a non-abelian local constant (group theoretically). The Brauer induction formula plays a crucial role in Langlands' proof. Robert Boltje gives a canonical version of the Brauer induction formula. In this paper, we review Langlands' strategy using Boltje's canonical Brauer induction formula. We then review various properties of local constants, some applications, and open problems.

math.NT↗
arXiv2024

An extension of Deligne-Henniart's twisting formula and its applications

Let $F/\bbQ_p$ be a non-Archimedean local field, and $G_F$ be the absolute Galois group of $F$. Let $ρ_1$ and $ρ_2$ be two finite-dimensional complex representations of $G_F$. Let $ψ$ be a nontrivial additive character of $F$. Then, the question is: What is the twisting formula for the root number $W(ρ_1\otimesρ_2,ψ)$?} In general, the answer to this question is not yet known. However, if one of $ρ_i \quad(i=1,2)$ is one-dimensional with ``sufficiently'' large conductor, then in [13], Deligne gave a twisting formula for $W(ρ_1\otimesρ_2,ψ)$. Later, in [12], Deligne and Henniart gave a general twisting formula for a {\it zero}-dimensional virtual representation twisted by a finite-dimensional representation of $G_F$. In this paper, we first extend Deligne's twisting formula for U-isotropic Heisenberg representation of dimension prime $p$, then we further extend Deligne-Henniart's result. Finally, we provide two very important applications of our twisting formula: -- (i) invariant formula for the local root numbers for U-isotropic Heisenberg representations, and (ii) a converse theorem on the Galois side.

math.NT↗
arXiv2021

Epsilon factors of symplectic type characters in the wild case

By work of John Tate we can associate an epsilon factor with every multiplicative character of a local field. In this paper we determine the explicit signs of the epsilon factors for symplectic type characters of $K^\times$, where $K/F$ is a wildly ramified quadratic extension of a non-Archimedean local field $F$ of characteristic zero.

math.NT↗
arXiv2019

Lamprecht-Tate Formula

For multiplicative characters of a non-Archimedean local field, we have a formula for epsilon factors due to John Tate. Before Tate, Erich Lamprecht also gave a formula for local epsilon factors of linear characters. Then Tate generalizes the formula for epsilon factors. In this paper, we give a very short and neat proof of the Lamprecht-Tate formula. We also show that the famous twisting formula of Deligne is a special case of the Lamprecht-Tate formula.

math.NT↗
arXiv2017

Twisting formula of epsilon factors

For characters of a non-Archimedean local field we have explicit formula for epsilon factors. But in general, we do not have any generalized twisting formula of epsilon factors. In this paper we give a generalized twisting formula of epsilon factors via local Jacobi sums.

math.NT↗
arXiv2017

Computation of the Lambda function for a finite Galois extension

By Langlands and Deligne we know that the local constants are extendible functions. Therefore, to give an explicit formula of the local constant of an induced representation of a local Galois group of a non-Archimedean local field $F$ of characteristic zero, we have to compute the lambda function $λ_{K/F}$ for a finite extension $K/F$. In this paper, when a finite extension $K/F$ is Galois, we give a formula for $λ_{K/F}$.

math.NT↗
arXiv2017

Invariant formula of the determinant of a Heisenberg representation

In this paper we give an explicit formula of the determinant of a Heisenberg representation $ρ$ of a finite group $G$. Heisenberg representations are induced by 1-dimensional characters in multiple ways, but our formula will be independent of any particular choice of induction.

math.GR↗
arXiv2016

Local Constants for Galois Representations - Some Explicit Results

We can associate local constant to every continuous finite dimensional complex representation of the absolute Galois group $G_F$ of a non-archimedean local field $F/\mathbb{Q}_p$ by Deligne and Langlands. To give explicit formula of local constant of a representation, we need to compute $λ$-functions explicitly. In this thesis we compute $λ_{K/F}$ explicitly, where $K/F$ is a finite degree Galois extension of a non-archimedean local field $F$, except when $K/F$ is a wildly ramified quadratic extension with $F\ne\mathbb{Q}_2$. Then by using this $λ$-function computation, in general, we give an invariant formula of local constant of finite dimensional Heisenberg representations of the absolute Galois group $G_F$ of a non-archimedean local field $F$. But for explicit invariant formula of local constant for a Heisenberg representation, we should have information about the dimension of a Heisenberg representation and the arithmetic description of the determinant of a Heisenberg representation. In this thesis, we give explicit arithmetic description of the determinant of Heisenberg representation. We also construct all Heisenberg representations of dimensions prime to $p$, and study their various properties. By using $λ$-function computation and arithmetic description of the determinant of Heisenberg representations, we give an invariant formula of local constant for a Heisenberg representation of dimension prime to $p$.

math.NT↗
arXiv2015

Local Constants for Heisenberg Representations

We can attach a local constant to every finite dimensional continuous complex representation of a local Galois group of a non-archimedean local field $F/\mathbb{Q}_p$ by Deligne and Langlands. Tate \cite{JT1} gives an explicit formula for computing local constants for linear characters of $F^\times$, but there is no explicit formula of local constant for any arbitrary representation of a local Galois group. In this article we study Heisenberg representations of the absolute Galois group $G_F$ of $F$ and give invariant formulas of local constants for Heisenberg representations of dimension prime to $p$.

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