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Subhajit Jana

Publications and source records attributed to Subhajit Jana.

At least 19 recordsLinked to original sources
arXiv2026

Regularized spectral expansion of a Rankin--Selberg period and moments

We establish a regularized spectral decomposition of the squared magnitude of a maximal degenerate Eisenstein series as a Schwartz distribution on $\mathrm{GL}_n$. Although the original problem is on $\mathrm{GL}_n$, its spectral components are described entirely by the automorphic spectrum of $\mathrm{GL}_2$. As an application, we prove a partial reciprocity formula relating the second moment of $\mathrm{GL}_n\times\mathrm{GL}_n$ Rankin--Selberg central $L$-values and a mixed moment of $L$-values of $\mathrm{GL}_2$. We also prove, assuming the generalized Ramanujan conjecture, an asymptotic formula for the second moment of the above Rankin--Selberg $L$-functions over a conductor-aspect family with an error term of square-root-cancellation strength.

math.NT↗
arXiv2026

Moments of $L$-functions via a relative trace formula

We prove an asymptotic formula for the second moment of the $\mathrm{GL}(n)\times\mathrm{GL}(n+1)$ Rankin--Selberg central $L$-values $L(1/2,Π\otimesπ)$, where $π$ is a fixed cuspidal representation of $\mathrm{GL}(n)$ that is tempered and unramified at every place, while $Π$ varies over a family of automorphic representations of $\mathrm{PGL}(n+1)$ ordered by (archimedean or non-archimedean) conductor. As another application of our method, we prove the existence of infinitely many cuspidal representations $Π$ of $\mathrm{PGL}(n+1)$ such that $L(1/2,Π\otimesπ_1)$ and $L(1/2,Π\otimesπ_2)$ do not vanish simultaneously where $π_1$ and $π_2$ are cuspidal representations of $\mathrm{GL}(n)$ that are unramified and tempered at every place and have trivial central characters.

math.NT↗
arXiv2025

Local integral transforms and global spectral decomposition

We establish an explicit global spectral decomposition of shifted convolution sums and the second moment of automorphic $L$-functions for Maass forms with explicit integral transforms as well as explicit inversion formulae over every local field.

math.NT↗
arXiv2025

On the spectral aspect density hypothesis and application

We prove that the density of non-tempered (at any $p$-adic place) cuspidal representations for $\mathrm{GL}_n(\mathbb{Z})$, varying over a family of representations ordered by their infinitesimal characters, is small -- confirming Sarnak's density hypothesis in this set-up. Among other ingredients, the proof uses tools from microlocal analysis for Lie group representations as developed by Nelson and Venkatesh. As an application, we prove that the Diophantine exponent of the $\mathrm{SL}_n(\mathbb{Z}[1/p])$-action on $\mathrm{SL}_n(\mathbb{R})/\mathrm{SO}_n(\mathbb{R})$ is \emph{optimal} -- resolving a conjecture of Ghosh, Gorodnik, and Nevo.

math.NT↗
arXiv2025

A Note On Rainbow 4-Term Arithmetic Progression

Let [n]=\{1,\,2,...,\,n\} be colored in k colors. A rainbow AP(k) in [n] is a k term arithmetic progression whose elements have diferent colors. Conlon, Jungic and Radoicic [10] had shown that there exists an equinumerous 4-coloring of [4n] which happens to be rainbow AP(4) free, when n is even and subsequently Haghighi and Nowbandegani [7] shown that such a coloring of [4n] also exists when n>1 is odd. Based on their construction, we shown that a balanced 4-coloring of [n] ( i.e. size of each color class is at least \left\lfloor n/4\right\rfloor ) actually exists for all natural number n. Further we established that for nonnegative integers k\geq3 and n>1, every balanced k-coloring of [kn+r] with 0\leq r<k-1, contains a rainbow AP(k) if and only if k=3. In this paper we also have discussed about rainbow free equinumerous 4-coloring of \mathbb{Z}_{n}.

math.CO↗
arXiv2024

Spectral reciprocity for $\mathrm{GL}(n)$ and simultaneous non-vanishing of central $L$-values

Let $F$ be a totally real number field and $n\ge 3$. Let $Π$ and $π$ be cuspidal automorphic representations for $\mathrm{PGL}_{n+1}(F)$ and $\mathrm{PGL}_{n-1}(F)$, respectively, that are unramified and tempered at all finite places. We prove simultaneous non-vanishing of the Rankin--Selberg $L$-values $L(1/2,Π\otimes\widetildeσ)$ and $L(1/2,σ\otimes\widetildeπ)$ for certain sequences of $σ$ varying over cuspidal automorphic representations for $\mathrm{PGL}_n(F)$ with conductor tending to infinity in the level aspect and bearing certain local conditions. Along the way, we also prove a reciprocity formula for the average of the product of Rankin--Selberg $L$-functions $L(1/2,Π\otimes\widetildeσ)L(1/2,σ\otimes\widetildeπ)$ over a conductor aspect family of $σ$.

math.NT↗
arXiv2024

Anomalous Lasing Behavior in a Nonlinear Plasmonic Random Laser

An unprecedented double-threshold lasing behavior has been observed in a plasmonic random laser composed of Au nanoislands decorated on vertically standing ZnO nanorods, infiltrated with dye-doped polymer matrix. The strong coupling of random laser modes to plasmonic nanocavities results in a dominant absorption of the random laser emission, leading to the first unusual lasing threshold. At higher pump fluences, the nonlinear optical behavior of the Au nanoislands induces a second lasing threshold. Various statistical tools have been employed to analyze the intensity fluctuations of the random laser modes, validating this unique lasing behavior.

physics.optics↗
arXiv2024

On Fourier asymptotics and effective equidistribution

We prove effective equidistribution of expanding horocycles in $\mathrm{SL}_2(\mathbb{Z})\backslash\mathrm{SL}_2(\mathbb{R})$ with respect to various classes of Borel probability measures on $\mathbb{R}$ having certain Fourier asymptotics. Our proof involves new techniques combining tools from automorphic forms and harmonic analysis. In particular, for any Borel probability measure $\mu$, satisfying $\sum_{\mathbb{Z}\ni|m|\leq X}|\widehat{\mu}(m)| = O\left(X^{1/2-\theta}\right)$ with $\theta>7/64,$ our result holds. This class of measures contains convolutions of $s$-Ahlfors regular measures for $s>39/64$, as well as a subclass of self-similar measures. Moreover, our result is sharp upon the Ramanujan--Petersson Conjecture (upon which the above $\theta$ can be chosen arbitrarily small): there are measures $\mu$ with $\widehat\mu(\xi) = O\left(|\xi|^{-1/2+\epsilon}\right)$ for which equidistribution fails.

math.DS↗
arXiv2024

On the local $L^2$-Bound of the Eisenstein Series

We study the growth of the local $L^2$-norms of the unitary Eisenstein series for reductive groups over number fields, in terms of their parameters. We derive a \emph{poly-logarithmic} bound on an average, for a large class of reductive groups. The method is based on Arthur's development of the spectral side of the trace formula, and ideas of Finis, Lapid, and Müller. As applications of our method, we prove the optimal lifting property for $\mathrm{SL}_n(\mathbb{Z}/q\mathbb{Z})$ for square-free $q$, as well as the Sarnak--Xue counting property for the principal congruence subgroup of $\mathrm{SL}_n(\mathbb{Z})$ of square-free level. This makes the recent results of Assing--Blomer unconditional.

math.NT↗
arXiv2024

Optimal Diophantine Exponents for $\mathrm{SL}(n)$

The \emph{Diophantine exponent} of an action of a group on a homogeneous space, as defined by Ghosh, Gorodnik, and Nevo, quantifies the complexity of approximating the points of the homogeneous space by the points on an orbit of the group. We show that the Diophantine exponent of the $\mathrm{SL}_n(\mathbb{Z}[1/p])$-action on the generalized upper half-space $\mathrm{SL}_n(\mathbb{R})/\mathrm{SO}_n(\mathbb{R})$, lies in $[1,1+O(1/n)]$, substantially improving upon Ghosh--Gorodnik--Nevo's method which gives the above range to be $[1,n-1]$. We also show that the exponent is \emph{optimal}, i.e.\ equals one, under the assumption of \emph{Sarnak's density hypothesis}. The result, in particular, shows that the optimality of Diophantine exponents can be obtained even when the \emph{temperedness} of the underlying representations, the crucial assumption in Ghosh--Gorodnik--Nevo's work, is not satisfied. The proof uses the spectral decomposition of the homogeneous space and bounds on the local $L^2$-norms of the Eisenstein series.

math.NT↗
arXiv2022

The second moment of $\mathrm{GL}(n)\times\mathrm{GL}(n)$ Rankin--Selberg $L$-functions

We prove an asymptotic expansion of the second moment of the central values of the $\mathrm{GL}(n)\times\mathrm{GL}(n)$ Rankin--Selberg $L$-functions $L(1/2,π\otimesπ_0)$, for a fixed cuspidal automorphic representation $π_0$, over the family of $π$ with analytic conductors bounded by a quantity which is tending off to infinity. Our proof uses the integral representations of the $L$-functions, period with regularized Eisenstein series, and the invariance properties of the analytic newvectors.

math.NT↗
arXiv2022

The Weyl bound for triple product L-functions

Let $π_1, π_2, π_3$ be three cuspidal automorphic representations for the group ${\rm SL}(2, \Bbb{Z})$, where $π_1$ and $π_2$ are fixed and $π_3$ has large conductor. We prove a subconvex bound for $L(1/2, π_1 \otimes π_2 \otimes π_3)$ of Weyl-type quality. Allowing $π_3$ to be an Eisenstein series we also obtain a Weyl-type subconvex bound for $L(1/2 + it, π_1 \otimes π_2)$.

math.NT↗
arXiv2021

One-pot Liquid-Phase Synthesis of MoS$_2$-WS$_2$ van der Waals Heterostructures for Broadband Photodetection

Two dimensional (2D) van der Waals heterostructures (vdWHs) have their unique potential in facilitating the stacking of layers of different 2D materials for optoelectronic devices with superior characteristics at a reduced cost. However, the fabrication of large area all-2D heterostructures is still challenging towards realizing practical devices. In the present work, we have demonstrated a rapid yet simple, impurity free and highly efficient sonication-assisted chemical exfoliation approach to synthesize hybrid vdWHs based on 2D molybdenum disulphide (MoS$_2$) and tungsten disulphide (WS$_2$), with high yield. Microscopic and spectroscopic studies have confirmed the successful exfoliation of layered 2D materials and formation of their hybrid heterostructure. The co-existence of 2D MoS2 and WS2 in the vdW hybrid is established by optical absorption and Raman shift measurements along with their chemical stiochiometry determined by X-ray photoelectron spectroscopy. The spectral response of the vdWH/Si (2D/3D) heterojunction photodetector fabricated using the as-synthesized material is found to show superior broadband photoresponse compared to that shown by the individual 2D MoS$_2$ and WS$_2$ based devices. The peak responsivity is found to be ~2.15 A/W at a wavelength of ~560 nm for an applied bias of -5 V. The ease of fabrication and superior performance of the chemically synthesized vdWH-based devices have revealed their potential use for large area optoelectronic applications on Si compatible CMOS platforms.

physics.app-ph↗
arXiv2021

Applications of analytic newvectors for $\mathrm{GL}(n)$

We provide a few natural applications of the analytic newvectors, initiated in \cite{JN} arXiv:1911.01880, to some analytic questions in automorphic forms for $\mathrm{PGL}_n(\mathbb{Z})$ with $n\ge 2$, in the archimedean analytic conductor aspect. We prove an orthogonality result of the Fourier coefficients, a density estimate of the non-tempered forms, an equidistribution result of the Satake parameters with respect to the Sato--Tate measure, and a second moment estimate of the central $L$-values as strong as Lindelöf on average. We also prove the random matrix prediction about the distribution of the low-lying zeros of automorphic $L$-function in the analytic conductor aspect. The new ideas of the proofs include the use of analytic newvectors to construct an approximate projector on the automorphic spectrum with bounded conductors and a soft local (both at finite and infinite places) analysis of the geometric side of the Kuznetsov trace formula.

math.NT↗
arXiv2021

Joint equidistribution on the product of the circle and the unit tangent bundle of the modular surface

We use spectral method to prove a joint equidistribution of primitive rational points and the same along expanding horocycle orbits in the products of the circle and the unit cotangent bundle of the modular surface. This result explicates the error bound in a recent work of Einsiedler, Luethi, and Shah \cite[Theorem $1.1$]{ELS}. The error is sharp upon the best known progress towards the Ramanujan conjecture at the finite places for the modular surface.

math.NT↗
arXiv2020

Analytic newvectors for $\mathrm{GL}_n(\mathbb{R})$

We relate the analytic conductor of a generic irreducible representation of $\mathrm{GL}_n(\mathbb{R})$ to the invariance properties of vectors in that representation. The relationship is an analytic archimedean analogue of some aspects of the classical non-archimedean newvector theory of Casselman and Jacquet--Piatetski-Shapiro--Shalika. We illustrate how this relationship may be applied in trace formulas to majorize sums over automorphic forms on $\mathrm{PGL}_n(\mathbb{Z}) \backslash \mathrm{PGL}_n(\mathbb{R})$ ordered by analytic conductor.

math.NT↗
arXiv2019

Abundance of Progression in commutative semigroup by elementary means

Furstenberg and Glasner proved that for an arbitrary k in N, any piecewise syndetic set contains k term arithmetic progression and such collection is also piecewise syndetic in Z. They used algebraic structure of beta N. The above result was extended for arbitrary semigroups by Bergelson and Hindman, again using the structure of Stone-Cech compactification of general semigroup. Beiglbock provided an elementary proof of the above result and asked whether the combinatorial argument in his proof can be enhanced in a way which makes it applicable to a more abstract setting. In this work we provide a positive answer to Beiglbock question for commutative semigroup.

math.CO↗