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 $k\to \infty$, 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 $k\in 2\mathbb{Z}$, 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 $n\ge 2$, and ${\mathcal{O}}_K$ is the ring of integers of $K$, and to the case of adelic modular forms.

arXiv comment9 pages. arXiv admin note: text overlap with arXiv:1507.00358