arXiv subjects

Hariom Sharma

Publications and source records attributed to Hariom Sharma.

10 recordsLinked to original sources

arXiv2026

On representations of GL(n) distinguished by GL(1)*GL(n-1) over a quaternion division algebra

Let $D$ be a quaternion division algebra over a non-Archimedean local field $F$ of characteristic zero, and let $G_n=GL_n(D)$. Let $H_{1,n-1}$ denote the subgroup of $G_n$ consisting of block-diagonal matrices of the form $diag(g_1,g_2)$, where $g_1\in G_1$ and $g_2\in G_{n-1}$. In this article, we formulate a conjectural classification of irreducible smooth $H_{1,n-1}$-distinguished representations of $G_n$ for $n>2$. We prove this conjecture in the cases $n=3$ and $n=4$. When $n=2$, the results are well known due to the contributions by various authors.

math.RT↗

arXiv2026

Asai Gamma Factors and Distinction in families

Let $F$ be a finite extension of $\mathbb{Q}_p$ and let $E$ be a quadratic extension of $F$. A representation $(\pi,V)$ of ${\rm GL}_n(E)$ is said to be ${\rm GL}_n(F)$-distinguished if there exists a non-zero linear functional $\phi$ on $V$ such that $\phi(\pi(h)v) = \phi(v)$ for all $h \in {\rm GL}_n(F)$ and $v \in V$. In this article, we study the notion of ${\rm GL}_n(F)$-distinguished representations for $R[{\rm GL}_n(E)]$ modules of Whittaker type, where $R$ is a Noetherian algebra over the ring of Witt vectors of $\overline{\mathbb{F}}_\ell$ with $\ell \ne p$. We first derive a functional equation, which gives the existence of the Asai $\gamma$-factors associated with $R[{\rm GL}_n(E)]$ modules of Whittaker type. We then provide a necessary condition for cuspidal $R[{\rm GL}_n(E)]$ modules of Whittaker type to be Whittaker ${\rm GL}_n(F)$-distinguished, expressed in terms of their Asai $\gamma$-factors.

math.RT↗

arXiv2026

Symplectic model for Zelevinsky modules of GL(n,D)

Let D be a quaternion division algebra over a non-archimedean local field F of characteristic zero. This article demonstrates the existence and uniqueness of the symplectic model for a family of Zelevinsky modules of GL(n, D) to a family of irreducible representations of GL(n, D). For this family of irreducible representations, we identify a necessary condition under which a symplectic model can exist. This work extends a result of Offen and Sayag beyond the case D = F.

math.RT↗

arXiv2024

Symplectic model for ladder and unitary representations

Let $D$ denote a quaternion division algebra over a non-archimedean local field $F$ with characteristic zero. Let $Sp_n(D)$ be the unique non-split inner form of the symplectic group $Sp_{2n}(F)$. An irreducible admissible representation $(π, V)$ of $GL_{n}(D)$ is said to have a symplectic model (or said to be $Sp_n(D)$-distinguished) if there exists a linear functional $ϕ$ on $V$ such that $ϕ(π(h)v) = ϕ(v)$ for all $v \in V$ and $h \in Sp_n(D)$. This article classifies those ladder representations of $GL_n(D)$ that possess a symplectic model (i.e., those representations that are $Sp_n(D)$-distinguished). Recently, Prasad conjectured that non-supercuspidal discrete series representations of $GL_n(D)$ do not admit a symplectic model. We confirm this for the Steinberg representations, which serve as canonical examples of discrete series representations. Furthermore, we demonstrate the hereditary nature of the symplectic model for induced representations derived from finite-length representations. In addition, we prove a part of Prasad's conjecture, which provides a family of irreducible unitary representations, all equipped with a symplectic model.

math.RT↗

arXiv2024

A primitive normal pair in a finite field with prescribed traces and norms

Given ${\mathbb{F}_{p^t}}$, a field with $p^t$ elements, where $p$ is a prime power, $t$ is a positive integer. Let $f(x)$ be a polynomial over $\mathbb{F}_{p^t}$ of degree $m$ with some restrictions. In this paper, we construct a sufficient condition on $(p,t)$ which guarantees the existence of a primitive normal pair $(ε,f(ε))$ such that $Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(ε)=a$, $Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(f(ε))=b$ and $N_{\mathbb{F}_{p^t}/\mathbb{F}_p}(ε)=c$, $N_{\mathbb{F}_{p^t}/\mathbb{F}_p}(f(ε))=d$ where $c,d\in\mathbb{F}_{p}$ are primitive elements and $a,b\in\mathbb{F}_{p}^*$. Furthermore, we demonstrate that, for $p=11^k;$ $k\geq1,$ $m=8$ and $t\geq 15$, there are only $4$ possible exceptions where such pairs may not exist.

math.NT↗

arXiv2024

Symplectic period for a representation of $GL_n(D)$

Let $D$ be a quaternion division algebra over a non-archimedean local field $K$ of characteristic zero, and let $Sp_n(D)$ be the unique non-split inner form of the symplectic group $Sp_{2n}(K)$. This paper classifies the irreducible admissible representations of $GL_{n}(D)$ with a symplectic period for $n = 3$ and $4$, i.e., those irreducible admissible representations $(π, V)$ of $GL_{n}(D)$ which have a linear functional $l$ on $V$ such that $l(π(h)v) = l(v)$ for all $v \in V$ and $h \in Sp_n(D)$. Our results also contain all unitary representations having a symplectic period, as stated in Prasad's conjecture.

math.RT↗

arXiv2024

Arithmetic progression in a finite field with prescribed norms

Given a prime power $q$ and a positive integer $n$, let $\mathbb{F}_{q^{n}}$ represents a finite extension of degree $n$ of the finite field ${\mathbb{F}_{q}}$. In this article, we investigate the existence of $m$ elements in arithmetic progression, where every element is primitive and at least one is normal with prescribed norms. Moreover, for $n\geq6,q=3^k,m=2$ we establish that there are only $10$ possible exceptions.

math.NT↗

arXiv2021

Existence of Primitive Normal Pairs with One Prescribed Trace over Finite Fields

Given $m, n, q\in \mathbb{N}$ such that $q$ is a prime power and $m\geq 3$, $a\in \mathbb{F}_q$, we establish a sufficient condition for the existence of primitive pair $(α, f(α))$ in $\mathbb{F}_{q^m}$ such that $α$ is normal over $\mathbb{F}_q$ and $\text{Tr}_{\mathbb{F}_{q^m}/\mathbb{F}_q}(α^{-1})=a$, where $f(x)\in \mathbb{F}_{q^m}(x)$ is a rational function of degree sum $n$. Further, when $n=2$ and $q=5^k$ for some $k\in \mathbb{N}$, such a pair definitely exists for all $(q, m)$ apart from at most $20$ choices.

math.NT↗

arXiv2020

Existence of Primitive Pairs with Prescribed Traces over Finite Fields

Let $F=\mathbb{F}_{q^m}$, $m>6$, $n$ a positive integer, and $f=p/q$ with $p$, $q$ co-prime irreducible polynomials in $F[x]$ and deg$(p)$ $+$ deg$(q)= n$. A sufficient condition has been obtained for the existence of primitive pairs $(α, f(α))$ in $F$ such that for any prescribed $a, b$ in $E=\mathbb{F}_q$, Tr$F/E (α) = a$ and Tr$F/E (α^{-1}) = b$. Further, for every positive integer $n$, such a pair definitely exists for large enough $(q,m)$. The case $n = 2$ is dealt separately and proved that such a pair exists for all $(q,m)$ apart from at most $64$ choices.

math.NT↗

arXiv2019

Primitive values of rational functions at primitive elements of a finite field

Given a prime power $q$ and an integer $n\geq2$, we establish a sufficient condition for the existence of a primitive pair $(α,f(α))$ where $α\in \mathbb{F}_q$ and $f(x) \in \mathbb{F}_q(x)$ is a rational function of degree $n$. (Here $f=f_1/f_2$, where $f_1, f_2$ are coprime polynomials of degree $n_1,n_2$, respectively, and $n_1+n_2=n$.) For any $n$, such a pair is guaranteed to exist for sufficiently large $q$. Indeed, when $n=2$, such a pair definitely does {\em not} exist only for 28 values of $q$ and possibly (but unlikely) only for at most $3911$ other values of $q$.

math.NT↗