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

2015/06/28 by Anilatmaja Aryasomayajula, Aryasomayajula, Anilatmaja · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1506.08497

6 pages

openalex publication_date 2015/06/28 · arxiv created 2015/07/03 · arxiv updated 2015/07/06 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

In this article, using the heat kernel approach from \citebouche, we derive sup-norm bounds for cusp forms of integral and half integral weight. Let Γ⊂ PSL₂(ℝ) be a cocompact Fuchsian subgroup of first kind. For k∈(1)/(2)ℤ (or k∈ 2ℤ), let S^k(Γ) denote the complex vector space of weight-k cusp forms. Let \lbrace f₁,…,f_j_k \rbrace denote an orthonormal basis of S^k(Γ). In this article, we show that as k→ ∞, the sup-norm for ∑_i=1^j_ky^k|f_i(z)|² is bounded by O(k), where the implied constant is independent on Γ. Furthermore, using results from \citeberman, we extend these results to the case when Γ is cofinite.

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• Heat kernels, Bergman kernels, and cusp forms

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