Subconvexity Bound for Hecke character L-Functions of Imaginary\n quadratic Number fields

2017/11/30 by Keshav Aggarwal, Aggarwal, Keshav
Arts and Humanities · Mathematics · #11L05 #11M41 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Historical Studies and Socio-cultural Analysis #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1712.00363

openalex publication_date 2017/11/30 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

Let K=\ℚ(\√(-D)) be an imaginary number field,\n(p)= mathfrakp mathfrakp' be a split odd prime and \ψ be a Hecke\ncharacter of conductor mathfrakp. Let L(s,\ψ) be the associated\nL-function. We prove the Burgess bound in t-aspect and a hybrid bound in\nconductor aspect, \L(1/2+it,
psi)
ll_{D,
varepsilon}\n(1+|t|)^{3/8+
varepsilon}p^1/8 for p\≪ t. In Appendix A,\nwe present the ideas for an elementary proof of Voronoi summation formula for\nholomorphic cusp forms with CM and squarefree level. This is done by exploiting\nthe lattice structure of ideals in number fields. Voronoi summation for such\ncusp forms is given by Kowalski, Michel and Vanderkam (2002). We hope that our\nmethod of proof can extend their Voronoi formula to any CM cusp form in\nS_k(\Γ₁(N)) and arbitrary additive twist. We encounter quadratic and\nquartic Gauss sums in the process. We shall present the calculations for the\ngeneral case in the next version of the paper.\n

Citations

• t-Aspect Subconvexity Bound for GL(2) L-functions

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