arXiv subjects

Jürg Kramer

Publications and source records attributed to Jürg Kramer.

7 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 $\Gamma_{0}=\mathrm{Sp}_{2g}(\mathbb{Z})$ 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 $\epsilon>0$, the $L^{\infty}$-norm bound \begin{equation*} \Vert\phi\Vert_{L^{\infty}}=\sup_{(\tau,z)\in\mathbb{H}_{g}\times\mathbb{C}^{g}}\Vert\phi(\tau,z)\Vert_{\mathrm{Pet}}=O_ {\Gamma_{0},\epsilon}\big(k^{(3g^{2}+5g)/8}\,m^{g^{2}+5g/4+\epsilon}\big) \end{equation*} holds for any Siegel--Jacobi cusp form $\phi$ 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

On the height of the universal abelian variety

In this paper we extend the arithmetic intersection theory of adelic divisors on quasiprojective varieties developed by X. Yuan and S. W. Zhang to cover certain adelic arithmetic divisors that are not nef nor integrable. The key concept used in this extension is the relative finite energy introduced by T. Darvas, E. Di Nezza, and C. H. Lu. As an application, we prove that the line bundle of Siegel--Jacobi forms on the universal abelian variety endowed with its invariant hermitian metric is not integrable but we compute its arithmetic self-intersection number using the new extension. The techniques developed in this paper can be applied in many other situations like mixed Shimura varieties or the moduli space of stable marked curves.

math.NT↗

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↗

arXiv2023

Uniform sup-norm bounds on average for Siegel cusp forms

Let $Γ\subsetneq \mathrm{Sp}_n(\mathbb{R})$ be an arithmetic subgroup of the symplectic group $\mathrm{Sp}_n(\mathbb{R})$ acting on the Siegel upper half-space $\mathbb{H}_n$ of degree $n$. Consider the $d$-dimensional space of Siegel cusp forms $\mathcal{S}_κ^n(Γ)$ of weight $κ$ for $Γ$ and let $\{f_j\}_{1\leq j\leq d}$ be a basis of $\mathcal{S}_κ^n(Γ)$ orthonormal with respect to the Petersson inner product. In this paper we show using the heat kernel method that the sup-norm of the quantity $S_κ^Γ(Z):=\sum_{j=1}^{d}\det (Y)^κ\vert{f_j(Z)}\vert^2\,(Z\in\mathbb{H}_n)$ is bounded above by $c_{n,Γ} κ^{n(n+1)/2}$ when $M:=Γ\backslash\mathbb{H}_n$ is compact and by $c_{n,Γ} κ^{3n(n+1)/4}$ when $M$ is non-compact of finite volume, where $c_{n,Γ}$ denotes a positive real constant depending only on the degree $n$ and the group $Γ$. Furthermore, we show that this bound is uniform in the sense that if we fix a group $Γ_0$ and take $Γ$ to be a subgroup of $Γ_0$ of finite index, then the constant $c_{n,Γ}$ in these bounds depends only on the degree $n$ and the fixed group $Γ_0$.

math.NT↗

arXiv2022

Relating Siegel cusp forms to Siegel-Maaß forms

In this paper we generalize a well-known isomorphism between the space of cusp forms of weight $k$ for a Fuchsian subgroup of the first kind $Γ\subset\mathrm{SL}_{2}(\mathbb{R})$ and the space of certain Maaß forms of weight $k$ for $Γ$ to an isomorphism between the space of Siegel cusp forms of weight $k$ for a subgroup $Γ\subset\mathrm{Sp}_{n}(\mathbb{R})$, which is commensurable with the Siegel modular group $\mathrm{Sp}_{n}(\mathbb{Z})$, and a suitable space of Siegel-Maaß forms of weight $k$ for $Γ$.

math.NT↗

arXiv2018

Effective sup-norm bounds on average for cusp forms of even weight

Let $Γ\subset\mathrm{PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting on the upper half-plane $\mathbb{H}$. Consider the $d_{2k}$-dimensional space of cusp forms $\mathcal{S}_{2k}^Γ$ of weight $2k$ for $Γ$, and let $\{f_{1},\ldots,f_{d_{2k}}\}$ be an orthonormal basis of $\mathcal{S}_{2k}^Γ$ with respect to the Petersson inner product. In this paper we will give effective upper and lower bounds for the supremum of the quantity $S_{2k}^Γ(z):=\sum_{j=1}^{d_{2k}}\vert f_{j}(z)\vert^{2}\,\mathrm{Im}(z)^{2k}$ as $z$ ranges through $\mathbb{H}$.

math.NT↗

arXiv2014

The singularities of the invariant metric on the line bundle of Jacobi forms

A theorem by Mumford implies that every automorphic line bundle on a pure open Shimura variety, equipped with an invariant smooth metric, can be uniquely extended as a line bundle on a toroidal compactification of the variety, in such a way that the metric acquires only logarithmic singularities. This result is the key of being able to compute arithmetic intersection numbers from these line bundles. Hence it is natural to ask whether Mumford's result remains valid for line bundles on mixed Shimura varieties. In this paper we examine the simplest case, namely the sheaf of Jacobi forms on the universal elliptic curve. We show that Mumford's result cannot be extended directly to this case and that a new interesting kind of singularities appears. By using the theory of b-divisors, we show that an analogue of Mumford's extension theorem can be obtained. We also show that this extension is meaningful because it satisfies Chern-Weil theory and a Hilbert-Samuel type of formula.

math.AG↗