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Anilatmaja Aryasomayajula

Publications and source records attributed to Anilatmaja Aryasomayajula.

At least 19 recordsLinked to original sources
arXiv2026

$L^{\infty}$-norm bounds for Siegel--Jacobi cusp forms

In this article, we establish explicit and uniform $L^{\infty}$-norm bounds for $L^{2}$-normalized Siegel--Jacobi cusp forms of integral weight $k$ and index $m$ for the Siegel modular group $Γ_{0}=\mathrm{Sp}_{2g}(\mathbb{Z})$ for arbitrary genus $g\geq 1$. Using the generalization of the classical Eichler--Zagier theta decomposition to higher genus, any such Siegel--Jacobi cusp form can be written as a finite linear combination of Siegel cusp forms of half-integral weight $k-1/2$ multiplied by the higher-dimensional analogues of the classical Jacobi theta functions. By building upon the uniform $L^{\infty} $-norm bounds on average for Siegel cusp forms established by J.~Kramer and A.~Mandal~\cite{k1} via the associated Bergman kernels, we prove that for $k\in\mathbb{Z}_{\geq g+1}$, $m\in\mathbb{Z}_{\geq 1}$, and a given $ε>0$, the $L^{\infty}$-norm bound \begin{equation*} \Vertϕ\Vert_{L^{\infty}}=\sup_{(τ,z)\in\mathbb{H}_{g}\times\mathbb{C}^{g}}\Vertϕ(τ,z)\Vert_{\mathrm{Pet}}=O_ {Γ_{0},ε}\big(k^{(3g^{2}+5g)/8}\,m^{g^{2}+5g/4+ε}\big) \end{equation*} holds for any Siegel--Jacobi cusp form $ϕ$ that is $L^{2}$-normalized with respect to the Petersson inner product. These estimates provide the first explicit upper bounds in terms of both parameters $k$ and $m$ for arbitrary genus $g$.

math.NT↗
arXiv2025

Estimates of automorphic forms on $\mathrm{SU}(n,1)$

For $n\geq 2$, let $Γ\subset \mathrm{SU}((n,1),\mathcal{O}_{K})$ be a torsion-free, finite-index subgroup, where $\mathcal{O}_K$ denotes the ring of integers of a totally imaginary number field $K$ of degree $2$. Let $\mathbb{B}^n$ denote the $n$-dimensional complex ball endowed with the hyperbolic metric, and let $X_Γ:=Γ\backslash \mathbb{B}^n$ denote the quotient space. Furthermore, let $μ_{\mathrm{hyp}}^{\mathrm{vol}}$ denote the volume form associated to the hyperbolic metric. Let $Λ:=Ω_{\overline{X}_Γ}^{n}$ denote the line bundle, where $\overline{X}_Γ:=X_Γ\cup\lbrace \infty\rbrace$. For any $k\geq 1$, let $λ^{k}:=Λ^{\otimes k}\otimes O_{\overline{X}_Γ}((k-1)\infty)$. For any $k\geq 1$, the hyperbolic metric induces a point-wise metric on $H^{0}(\overline{X}_Γ,λ^{k})$. For any $k\geq 1$, let $\mathcal{B}_{X_Γ}^{λ^{k}}$ denote the Bergman kernel associated $H^{0}(\overline{X}_Γ,λ^{k})$. Then, for $k\gg1$, the first main result of the article, is the following estimate $$ \sup_{z\in \overline{X}_Γ}\big|\mathcal{B}_{X_Γ}^{λ^{k}}(z,z)\big|_{\mathrm{hyp}}=O_{X_Γ}(k^{n+1/2}).$$ For any $k\geq 1$, and $z\in X_Γ$, let $μ_{\mathrm{Ber},k}(z)$ denote the Bergman metric associated to the line bundle $λ^{ k}$, and let $μ_{\mathrm{ber},k}^{\mathrm{vol}}$ denote the associated volume form. Then, for $k\gg1$, the second main result of the article is the following estimate $$ \sup_{z\in \overline{X}_Γ}\bigg|\frac{μ_{\mathrm{Ber},k}^{\mathrm{vol}}(z)}{μ_{\mathrm{hyp}}^{\mathrm{vol}}(z)}\bigg|=O_{X_Γ}\big(k^{2(n-1)(n+1)+n+3} \big).$$ Our estimate for the Bergman metric completes our arguments and corrects our estimate from arXiv:2305.11609, for $n=1$.

math.CV↗
arXiv2025

Off-diagonal estimates of the Bergman kernel associated to Siegel varieties

For $g\geq 2$, let $Γ\subset\mathrm{Sp}(2g,\mathbb{R})$ be a discrete subgroup, which is either a cocompact subgroup or an arithmetic subgroup without torsion elements, and let $\mathbb{H}_{g}$ denote the Siegel upper half space of genus $g$. Let $X_Γ:=Γ\backslash\mathbb{H}_{g}$ denote the quotient space, which is a complex manifold of dimension $g(g+1)/2$. Let $Ω_{X_Γ}$ denote the cotangent bundle, and let $\ell:=\mathrm{det}(Ω_{X_Γ})$ denote the determinant line bundle of $Ω_{X_Γ}$. For any $Z,W\in X_Γ$, let $d_{\mathrm{S}}(Z,W)$ denote the geodesic distance between the points $Z$ and $W$ on $X_Γ$. \vspace{0.15cm}\noindent For any $k\geq 1$, let $H^{0}(X_Γ,\ell^{\otimes k})$ denote the complex vector space of global sections of the line bundle $\ell^{\otimes k}$, and let $\|\cdot\|_{k}$ denote the point-wise norm on $\ell^{\otimes k}$. Let $\mathcal{B}_{X_Γ}^{\ell^{ k}}$ denote the Bergman kernel associated to $H^{0}_{L^{2}}(X_Γ,\ell^{\otimes k})\subset H^{0}(X_Γ,\ell^{\otimes k})$, vector subspace of $L^2$ global sections. For any $k\gg 1$, and $Z,W\in X_Γ$ , we derive estimates of the Bergman kernel $\|\mathcal{B}_{X_Γ}^{\ell^{ k}}(Z,W)\|_{\ell^{k}}$, when $Γ$ is a cocompact subgroup and when $Γ$ is an arithmetic subgroup.

math.CV↗
arXiv2025

Sup-norm bounds for Jacobi cusp forms

In this article, we give $L^{\infty}$-norm bounds for the natural invariant norm of cusp forms of real weight $k$ and character $χ$ for any cofinite Fuchsian subgroup $Γ\subset\mathrm{SL}_{2}(\mathbb{R})$. Using the representation of Jacobi cusp forms of integral weight $k$ and index $m$ for the modular group $Γ_{0}=\mathrm{SL}_{2}(\mathbb{Z})$ as linear combinations of modular forms of weight $k-\frac{1}{2}$ for some congruence subgroup of $Γ_{0}$ (depending on $m$) and suitable Jacobi theta functions, we derive $L^{\infty}$-norm bounds for the natural invariant norm of these Jacobi cusp forms. More specifically, letting $J_{k,m}^{\mathrm{cusp}}(Γ_{0})$ denote the complex vector space of Jacobi cusp forms under consideration and $\Vert\cdot\Vert_{\mathrm{Pet}}$ the pointwise Petersson norm on $J_{k,m}^{\mathrm{cusp}}(Γ_ {0})$, we prove that for $k\in\mathbb{Z}_{\ge 5}$ and $m\in\mathbb{Z}_{\ge 1}$, and a given $ε>0$, the $L^{\infty}$-norm bound \begin{align*} \Vertϕ\Vert_{L^{\infty}}=\sup_{(τ,z)\in\mathbb{H}\times\mathbb{C}}\Vertϕ(τ,z)\Vert_{\mathrm{Pet}}=O_{Γ_{0},ε}\big(k\,m^{\frac {7}{4}+ε}\big) \end{align*} holds for any $ϕ\in J_{k,m}^{\mathrm{cusp}}(Γ_{0})$, which is $L^{2}$-normalized with respect to the Petersson inner product, where the implied constant depends on $Γ_{0}$ and the choice of $ε>0$.

math.NT↗
arXiv2024

Estimates and asymptotics of Teichmüller modular forms

In this article, we derive estimates of Teichmüller modular forms, and associated invariants. Let $\mathcal{M}_{g}$ denote the moduli space of compact hyperbolic Riemann surfaces of genus $g\geq 2$, and let $\overline{M}_{g}$ be the Deligne-Mumford compactification of $\mathcal{M}_{g}$, and we denote its boundary by $\partial\mathcal{M}_{g}$. Let $π:\mathcal{C}_{g}\longrightarrow\mathcal{M}_{g}$ be the universal surface. For any $n\geq 1$, let $Λ_{n}:=π_{\ast}(T_{v}\mathcal{C}_{g})^{n}$, where $T_{v}\mathcal{C}_{g}$ denotes the vertical holomorphic tangent bundle of the fibration $π$, and the fiber of $Λ_{n}$ over any $X\in\mathcal{M}_{g}$ is equal to $H^{0}(X,Ω_{X}^{\otimes n})$, the space of holomorphic differentials of degree-$n$, defined over the Riemann surface $X$. Let $λ_{n}:=\mathrm{det}(Λ_{n})$ denote the determinant line bundle of the vector bundle $Λ_{n}$, whose sections are known as Teichmüller modular forms. The complex vector space of Teichmüller modular forms is equipped with Quillen metric, which is denoted by $\|\cdot\|_{\mathrm{Qu}}$.

math.CV↗
arXiv2024

Estimates of Kähler metrics on noncompact finite volume hyperbolic Riemann surfaces, and their symmetric products

Let $X$ denote a noncompact finite volume hyperbolic Riemann surface of genus $g\geq 2$, with only one puncture at $i\infty$ (identifying $X$ with its universal cover $\mathbb{H}$). Let $\overline{X}:=X\cup\lbrace i\infty\rbrace$ denote the Satake compactification of $X$. Let $Ω_{\overline{X}}$ denote the cotangent bundle on $\overline{X}$. For $k\gg1$, we derive an estimate for $μ_{\overline{X}}^{\mathrm{Ber},k}$, the Bergman metric associated to the line bundle $\mathcal{L}^{k}:=Ω_{\overline{X}}\otimes \mathcal{O}_{\overline{X}}\big((k-1)\infty\big)$. For a given $d\geq 1$, the pull-back of the Fubini-Study metric on the Grassmannian, which we denote by $μ_{\mathrm{Sym}^d(\overline{X})}^{\mathrm{FS},k}$, defines a Kähler metric on $\mathrm{Sym}^d(\overline{X})$, the $d$-fold symmetric product of $\overline{X}$. Using our estimates of $μ_{\overline{X}}^{\mathrm{Ber},k}$, as an application, we derive an estimate for $μ_{\mathrm{Sym}^d(\overline{X}),\mathrm{vol}}^{\mathrm{FS},k}$, the volume form associated to the (1,1)-form $μ_{\mathrm{Sym}^d(\overline{X})}^{\mathrm{FS},k}$.

math.CV↗
arXiv2023

Estimates of Bergman Kernels and Bergman metric on compact Picard surfaces

Let $Γ\subset \mathrm{SU}((2,1),\mathbb{C})$ be a torsion-free cocompact subgroup. Let $\mathbb{B}^{2}$ denote the $2$-dimensional complex ball endowed with the hyperbolic metric $μ_{\mathrm{hyp}}$, and let $X_Γ:=Γ\backslash \mathbb{B}^{2}$ denote the quotient space, which is a compact complex manifold of dimension $2$. Let $Λ:= Ω_{X_Γ}^{2}$ denote the line bundle on $X_Γ$, whose sections are holomorphic $(2,0)$-forms. For any $k\geq 1$, the hyperbolic metric induces a point-wise metric on $H^{0}(X_Γ,Λ^{\otimes k })$, which we denote by $|\cdot|_{\mathrm{hyp}}$. For any $k\geq 1$, let $\mathcal{B}_Λ^{ k}$ denote the Bergman kernel of the complex vector space $H^{0}(X_Γ,Λ^{\otimes k })$. For any $k\geq 3$, and $z,w\in X_Γ$, the first main result of the article is an off-diagonal estimate of the Bergman kernel $ \mathcal{B}_Λ^{ k}$. For any $k\geq 1$, let $μ_{\mathrm{ber}}^{k}(z):=-\frac{i}{2π}\partial_{z}\partial_{\overline{z}}\log| \mathcal{B}_Λ^{ k}(z,z)|_{\mathrm{hyp}}$ denote the Bergman metric associated the line bundle $Λ^{\otimes k}$, and let $μ_{\mathrm{ber}}^{k,\mathrm{vol}}(z)$ denote the associated volume form. For $k\gg 1$ sufficiently large, and $ε>0$, the second main result of the article is the following estimate \begin{align*} \sup_{z\in X_Γ}\bigg|\frac{μ_{\mathrm{ber}}^{k,\mathrm{vol}}(z)}{μ_{\mathrm{hyp}}^{\mathrm{vol}}}\bigg|=O_{X_Γ,ε}\big(k^{4+ε}\big), \end{align*} where $μ_{\mathrm{hyp}}^{\mathrm{vol}}$ denotes the volume form associated to the hyperbolic metric $μ_{\mathrm{hyp}}$, and the implied constant depends on the Picard surface $X_Γ$, and on the choice of $ε>0$. \end{abstract}

math.CV↗
arXiv2023

Estimates of Picard modular cusp forms

In this article, for $n\geq 2$, we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of $\mathrm{SU}\big((n,1),\mathbb{C}\big)$. The main result of the article is the following result. Let $Γ\subset \mathrm{SU}\big((2,1),\mathcal{O}_{K}\big)$ be a torsion-free subgroup of finite index, where $K$ is a totally imaginary field. Let $\mathcal{B}_Γ^{k}$ denote the Bergman kernel associated to the $\mathcal{S}_{k}(Γ)$, complex vector space of weight-$k$ cusp forms with respect to $Γ$. Let $\mathbb{B}^{2}$ denote the $2$-dimensional complex ball endowed with the hyperbolic metric, and let $X_Γ:=Γ\backslash \mathbb{B}^{2}$ denote the quotient space, which is a noncompact complex manifold of dimension $2$. Let $\big|\cdot\big|_{\mathrm{pet}}$ denote the point-wise Petersson norm on $\mathcal{S}_{k}(Γ)$. Then, for $k\geq 6$, we have the following estimate \begin{equation*} \sup_{z\in X_Γ}\big|\mathcal{B}_Γ^{k}(z)\big|_{\mathrm{pet}}=O_Γ\big(k^{\frac{5}{2}}\big), \end{equation*} where the implied constant depends only on $Γ$.

math.NT↗
arXiv2022

Estimates of cusp forms for certain cocompact arithmetic subgroups

In this article, we derive a sub convexity estimate of Hecke eigen cusp forms associated to certain cocompact arithmetic subgroups of SL(2,R). The main result can be considered as the holomorphic version of the estimate of Hecke eigen Maass forms, derived in a famous paper of Iwaniec and Sarnak. A stronger estimate was derived by Khayutin and Steiner in arXiv:2009.07194. However, techniques used in both the papers are very different.

math.NT↗
arXiv2019

Bergman kernel on Riemann surfaces and Kaehler metric on symmetric products

Let $X$ be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer $k\geq 2$, we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle $Ω^{\otimes k}_X$, where $Ø$ is the holomorphic cotangent bundle of $X$. Our first main result estimates the corresponding Bergman metric on $X$ in terms of the Poincaré metric. We then consider a certain natural embedding of the symmetric product of $X$ into a Grassmannian parametrizing subspaces of fixed dimension of the space of all global holomorphic sections of $Ω^{\otimes k}_X$. The Fubini-Study metric on the Grassmannian restricts to a Kähler metric on the symmetric product of $X$. The volume form for this restricted metric on the symmetric product is estimated in terms of the Bergman kernel of $Ω^{\otimes k}_X$ and the volume form for the orbifold Kähler form on the symmetric product given by the Poincaré metric on $X$.

math.CV↗
arXiv2019

Estimates of Fourier coefficients of integral and half-integral weight cusp forms associated to cofinite Fuchsian subgroups

Let $\G\subset \mathrm{SL}_{2}(\R)$ be a cofinite Fuchsian subgroup, and let $i\infty$ be a cusp of $\G$. For $k\in\Z_{\geq 0}$, let $\Sk$ denote the complex vector space of cusp forms of weight-$k$, with respect to the Fuchsian subgroup $\G$. Let $f\in\Sk$ be a cusp form of weight-$k$, which is normalized, with respect to the Petersson inner-product on $\Sk$. For any $n\in\Z_{\geq 1}$, let $a_{n}$ denote the $n$-th Fourier coefficient of $f$ at $i\infty$. Then, for any $k\in\Z_{\geq 5}$, we show that \begin{align*} \big|a_{n}\big|=O_{f,\G}\big(n^{\frac{k-1}{2}+\frac{2}{k}}\big), \end{align*} where the implied constant depends on the cusp form $f$, and the Fuchsian subgroup $\G$. The proof of the above estimate remains valid, for half-integral weight cusp forms of weight-$κ$ with $κ\in\big\lbrace \frac{1}{2}+m\big|\,m\in\Z_{\geq 4} \big\rbrace$, associated to the arithmetic subgroups $\G_{0}(4N)$, $\G_{1}(4N)$, and $\G(4N)$ (for $N\in\Z_{\geq 1}$).

math.NT↗
arXiv2018

Estimates of the Bergman kernel on a hyperbolic Riemann surface of finite volume-II

In this article, we derive off-diagonal estimates of the Bergman kernel associated to the tensor-powers of the cotangent bundle defined on a hyperbolic Riemann surface of finite volume, when the distance between the points is less than injectivity radius. We then use these estimates to derive estimates of the Bergman kernel along the diagonal.

math.CV↗
arXiv2017

Estimates of automorphic forms over quaternion algebras

In this article, using methods from geometric analysis and theory of heat kernels, we derive qualitative estimates of automorphic cusp forms defined over quaternion algebras. Using which, we prove an average version of the holomorphic QUE conjecture. We then derive quantitative estimates of classical Hilbert modular cusp forms. This is a generalization of the results from [3] and [9] to higher dimensions.

math.NT↗
arXiv2016

On the Kaehler metrics over ${mathrm{Sym}^{d}(X)$

Let $X$ be a compact connected Riemann surface of genus $g$, with $g \geq 2$. For each $d <η(X)$, where $η(X)$ is the gonality of $X$, the symmetric product $\text{Sym}^d(X)$ embeds into $\text{Pic}^d(X)$ by sending an effective divisor of degree $d$ to the corresponding holomorphic line bundle. Therefore, the restriction of the flat Kähler metric on $\text{Pic}^d(X)$ is a Kähler metric on $\text{Sym}^d(X)$. We investigate this Kähler metric on $\text{Sym}^d(X)$. In particular, we estimate it's Bergman kernel. We also prove that any holomorphic automorphism of $\text{Sym}^d(X)$ is an isometry.

math.DG↗
arXiv2015

Estimates of Hilbert modular cusp forms of half-integral and integral weight

Let $Γ$ be a cocompact, discrete, and irreducible subgroup of $\mathrm{PSL}_{2}(\mathbb{R})^{n}$. Let $ν$ be a unitary character of $Γ$. For $k\in1\slash 2\,\mathbb{Z}$, let $\sknu$ denote the complex vector space of cusp forms of weight-$\tk=\k$ and nebentypus $ν^{2k}$ with respect to $Γ$. We assume that $ω_{X,ν}$, the line bundle of cusp forms of weight-$\tilde{1\slash 2}:=(1\slash 2,\ldots,1\slash2)$ with nebentypus $ν$ over $X$ exists. Let $\lbrace f_{1},\ldots,f_{j_{\tk}} \rbrace$ denote an orthonormal basis of $\sknu$. In this article, we show that as $k\rightarrow \infty$, the sum $\sum_{i=1}^{j_{\tk}}y^{k}|f_{i}(z)|^{2}$ is bounded by $O(k^{n})$, where the implied constant is independent of $Γ$. Furthermore, we extend these results to the case when $k\in2\mathbb{Z}$, and to the case when $Γ$ is commensurable with the Hilbert modular group $Γ_{K}:=\mathrm{PSL}_{2}(O_{K})$, where $K$ is a totally real number field of degree $n\geq 2$, and $\mathcal{O}_{K}$ is the ring of integers of $K$, and to the case of adelic modular forms.

math.NT↗
arXiv2015

Heat kernel approach for sup-norm bounds for cusp forms of integral and half integral weight

In this article, using the heat kernel approach from \cite{bouche}, we derive sup-norm bounds for cusp forms of integral and half integral weight. Let $Γ\subset \mathrm{PSL}_{2}(\mathbb{R})$ be a cocompact Fuchsian subgroup of first kind. For $k\in\frac{1}{2}\mathbb{Z}$ (or $k\in 2\mathbb{Z}$), let $S^{k}(Γ)$ denote the complex vector space of weight-$k$ cusp forms. Let $\lbrace f_{1},\ldots,f_{j_{k}} \rbrace$ denote an orthonormal basis of $S^{k}(Γ)$. In this article, we show that as $k\rightarrow \infty,$ the sup-norm for $\sum_{i=1}^{j_{k}}y^{k}|f_{i}(z)|^{2}$ is bounded by $O(k)$, where the implied constant is independent on $Γ$. Furthermore, using results from \cite{berman}, we extend these results to the case when $Γ$ is cofinite.

math.NT↗
arXiv2015

Heat kernels, Bergman kernels, and cusp forms

In this article, we describe a geometric method to study cusp forms, which relies on heat kernel and Bergman kernel analysis. This new approach of applying techniques coming from analytic geometry is based on the micro-local analysis of the heat kernel and the Bergman kernel in \cite{bouche} and \cite{berman}, respectively, using which we derive sup-norm bounds for cusp forms of integral weight, half-integral weight, and real weight associated to a Fuchsian subgroup of first kind.

math.NT↗