Searcharxiv⌕ Search

arXiv subjects

Arnab Mitra

Publications and source records attributed to Arnab Mitra.

11 recordsLinked to original sources
arXiv2026

Moments and sign changes of symmetric power $L$-function coefficients over sums of squares

Let $f$ be a normalised Hecke eigenform of even integral weight for the full modular group $\mathrm{SL}(2,\mathbb{Z})$, let $L(s,\mathrm{sym}^{j}f)$ be the $j$th symmetric power $L$-function attached to $f$, and let $\lambda_{\mathrm{sym}^{j}f}(n)$ denote its $n$th Dirichlet coefficient. For each even integer $m$ with $2 \le m \le 12$, we establish upper bounds for the partial sums of $\lambda_{\mathrm{sym}^{j}f}(n)$ and asymptotic formulas for those of $\lambda_{\mathrm{sym}^{j}f}^{2}(n)$ taken over integers represented as a sum of $m$ squares. As an application, we obtain lower bounds for the number of sign changes of $\lambda_{\mathrm{sym}^{j}f}(n)$ along these sums of $m$ squares.

math.NT↗
arXiv2026

Shifted convolution sums of coefficients of symmetric power $L$-functions with $k$-full kernels over sums of squares in arithmetic progressions

Let $q$ be an integer and let $f$ be a normalised Hecke eigenform of integral weight for the full modular group. Let $L(s,\mathrm{sym}^j f)$ denote the $j$-th symmetric power $L$-function associated to $f$, and let $\lambda_{\mathrm{sym}^j f}(n)$ denote its $n$-th coefficient. We study the behaviour of the partial sum of $\lambda_{\mathrm{sym}^j f}(n)$, and of its second moment, taken over those sums of $m$ squares that are congruent to $1$ modulo $q$. As an application, we investigate the shifted convolution sum of $\lambda_{\mathrm{sym}^j f}(n)$ against a $k$-full kernel function, for any $k \geq 2$. We also study the number of sign changes of $\lambda_{\mathrm{sym}^j f}(n)$ twisted with a $k$-full kernel function, again over sums of $m$ squares. Throughout, $m$ is even with $m \in \{2,4,6,8,10,12\}$.

math.NT↗
arXiv2018

Models of representations and Langlands functoriality

In this article we explore the interplay between two generalizations of the Whittaker model, namely the Klyachko models and the degenerate Whittaker models, and two functorial constructions, namely base change and automorphic induction, for the class of unitarizable and ladder representations of the general linear groups.

math.RT↗
arXiv2018

On ${\rm Sp}$-distinguished representations of the quasi-split unitary groups

We study ${\rm Sp}_{2n}(F)$-distinction for representations of the quasi-split unitary group $U_{2n}(E/F)$ in $2n$ variables with respect to a quadratic extension $E/F$ of $p$-adic fields. A conjecture of Dijols and Prasad predicts that no tempered representation is distinguished. We verify this for a large family of representations in terms of the Moeglin-Tadic classification of the discrete series. We further study distinction for some families of non-tempered representations. In particular, we exhibit $L$-packets with no distinguished members that transfer under stable base change to ${\rm Sp}_{2n}(E)$-distinguished representations of ${\rm GL}_{2n}(E)$.

math.RT↗
arXiv2017

Constrained Receiver Scheduling in Flexible Time and Wavelength Division Multiplexed Optical

An increasing bandwidth demand has mandated a shift to the time and wavelength division multiplexing (TWDM) techniques in optical access networks (OAN). Typical TWDM scheduling schemes consider scheduling of the optical line terminal receiver only. In this paper we have identified an additional collision domain that is present in TWDM schemes that offer security, in addition to bandwidth flexibility. Scheduling of the identified collision domain is termed as group scheduling. We illustrate that consideration of receiver scheduling only (as done in typical TWDM schemes) severely affects their throughput when implemented on flexible and secure TWDM architectures. A novel media access control protocol has been proposed in this paper that considers the multiple collision domains. Through simulations, we are able to illustrate that the proposed scheme achieves a high throughput. A theoretical upper bound of throughput has also been derived to explain the simulation results. Complexity reduction of the proposed scheme has been illustrated, thereby making it an attractive proposal.

cs.NI↗
arXiv2017

Towards a Goldberg-Shahidi pairing for classical groups

Let G be either an orthogonal, a symplectic or a unitary group over a local field F and let P = MN be a maximal parabolic subgroup. Then the Levi subgroup M is the product of a group of the same type as G and a general linear group, acting on vector spaces X and W, respectively. In this paper we decompose the unipotent radical N of P under the adjoint action of M, assuming dim W less than or equal to dim X, excluding only the symplectic case with dim W odd. The result is a Weyl-type integration formula for N with applications to the theory of intertwining operators for parabolically induced representations of G. Namely, one obtains a bilinear pairing on matrix coefficients in the spirit of Goldberg-Shahidi, which detects the presence of poles of these operators at 0.

math.RT↗
arXiv2016

On two questions concerning representations distinguished by the Galois involution

Let E/F be a quadratic extension of non-archimedean local fields of characteristic 0. In this paper, we investigate two approaches which attempt to describe the smooth irreducible representations of GL(n,E) that are distinguished by its subgroup GL(n,F). One relates this class to representations which come as base change lifts from a quasi-split unitary group F, while another deals with a certain symmetry condition. By characterizing the union of images of the base change maps we show that these two approaches are closely related. Using this observation, we are able to prove a statement relating base change and distinction for ladder representations. We then produce a wide family of examples in which the symmetry condition does not impose GL(n,F)-distinction, and thus exhibit the limitations of these two approaches.

math.RT↗
arXiv2016

Klyachko models for ladder representations

We give a new proof of the existence of Klyachko models for unitary representations of ${\rm GL}_{n}(F)$ over a non-archimedean local field $F$. Our methods are purely local and are based on studying distinction within the class of ladder representations introduced by Lapid and Minguez. We classify those ladder representations that are distinguished with respect to Klyachko models. We prove the hereditary property of these models for induced representations from arbitrary finite length representations. Finally, in the other direction and in the context of admissible representations induced from ladder, we study the relation between distinction of the parabolic induction with respect to the symplectic groups and distinction of the inducing data.

math.RT↗
arXiv2016

On representations of ${\rm GL}_{2n}(F)$ with a symplectic period

The main aim of this paper is to classify the irreducible admissible representations of ${\rm GL}_{4}(F)$ and ${\rm GL}_{6}(F)$ for a nonarchimedean local field $F$, which bear a nontrivial linear form invariant under the groups ${\rm Sp}_{4}(F)$ and ${\rm Sp}_{6}(F)$ respectively. We propose a few conjectures for the general case.

math.RT↗
arXiv2013

An integration formula for unipotent radicals

Let P be a maximal parabolic of a classical group over a field F. Then the Levi subgroup M is isomorphic to the product of a classical group and a general linear group, acting on vector spaces X and W, respectively. In this paper we decompose the unipotent radical N of P under the adjoint action of M, assuming dim W is less than or equal to dim X and that dim W is even. When F is a local field, we obtain a Weyl-type integration formula for N.

math.RT↗