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

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• 作者：Anilatmaja Aryasomayajula
• 关键词：30F30 ; 30F35 ; 30F45 ; Cusp forms ; Heat kernels ; Bergman kernels
• 刊名：Archiv der Mathematik
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：106
• 期：2
• 页码：165-173
• 全文大小：468 KB
• 参考文献：1.Abbes A., Ullmo E.: Comparison des metriques d’Arakelov et de Poincare sur X 0(N). Duke Math. J. 80, 295–307 (1995)MATH MathSciNet CrossRef
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  8.J. Friedman, J. Jorgenson, and J. Kramer, Uniform sup-norm bounds on average for cusp forms of higher weights, arXiv preprint arXiv:​1305.​1348 (2013).
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  10.R. S. Steiner, Uniform bounds on sup-norms of holomorphic forms of real weight, Int. J. Number Theory, doi:10.​1142/​S179304211650071​8 .
  
• 作者单位：Anilatmaja Aryasomayajula (1)
  
  1. Department of Mathematics, University of Hyderabad, Prof. C. R. Rao Road, Gachibowli, Hyderabad, 500046, India
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8938

文摘

In this article, using the heat kernel approach from Bouche (Asymptotic results for Hermitian line bundles over complex manifolds: The heat kernel approach, Higher-dimensional complex varieties, pp 67–81, de Gruyter, Berlin, 1996), we derive sup-norm bounds for cusp forms of integral and half-integral weight. Let \({\Gamma\subset \mathrm{PSL}_{2}(\mathbb{R})}\) be a cocompact Fuchsian subgroup of first kind. For \({k \in \frac{1}{2} \mathbb{Z}}\) (or \({k \in 2\mathbb{Z}}\)), let \({S^{k}_{\nu}(\Gamma)}\) denote the complex vector space of cusp forms of weight-k and nebentypus \({\nu^{2k}}\) (\({\nu^{k\slash 2}}\), if \({k \in 2\mathbb{Z}}\)) with respect to \({\Gamma}\), where \({\nu}\) is a unitary character. Let \({\lbrace f_{1},\ldots,f_{j_{k}} \rbrace}\) denote an orthonormal basis of \({S^{k}_{\nu}(\Gamma)}\). 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 of \({\Gamma}\). Furthermore, using results from Berman (Math. Z. 248:325–344, 2004), we extend these results to the case when \({\Gamma}\) is cofinite. Keywords Cusp forms Heat kernels Bergman kernels