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

Abstract

Let $\Gamma$ be a cocompact, discrete, and irreducible subgroup of $\mathrm{PSL}_{2}(\mathbb{R})^{n}$. Let $\nu$ be a unitary character of $\Gamma$. For $k\in1\slash 2\,\mathbb{Z}$, let $\sknu$ denote the complex vector space of cusp forms of weight-$\tk=\k$ and nebentypus $\nu^{2k}$ with respect to $\Gamma$. We assume that $\omega_{X,\nu}$, the line bundle of cusp forms of weight-$\tilde{1\slash 2}:=(1\slash 2,\ldots,1\slash2)$ with nebentypus $\nu$ 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 $\Gamma$. Furthermore, we extend these results to the case when $k\in2\mathbb{Z}$, and to the case when $\Gamma$ is commensurable with the Hilbert modular group $\Gamma_{K}:=\mathrm{PSL}_{2}(O_{K})$, 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.