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

TL;DR

This paper establishes subconvexity bounds for Hecke L-functions over imaginary quadratic fields, improving understanding of their growth and providing new methods for Voronoi summation in CM forms.

Contribution

It proves Burgess and hybrid bounds for Hecke L-functions in imaginary quadratic fields and introduces an elementary approach to Voronoi summation for CM cusp forms.

Findings

Proved Burgess bound in t-aspect for Hecke L-functions.

Established hybrid bounds in conductor aspect.

Presented a new elementary proof approach for Voronoi summation in CM forms.

Abstract

Let $K=\mathbb{Q}(\sqrt{-D})$ be an imaginary number field, $(p)=\mathfrak{p}\mathfrak{p}'$ be a split odd prime and $\psi$ be a Hecke character of conductor $\mathfrak{p}$. Let $L(s,\psi)$ be the associated $L$-function. We prove the Burgess bound in $t$-aspect and a hybrid bound in conductor aspect, \begin{equation*} L(1/2+it,\psi)\ll_{D,\varepsilon} (1+|t|)^{3/8+\varepsilon}p^{1/8} \end{equation*} for $p\ll t$. In Appendix A, we present the ideas for an elementary proof of Voronoi summation formula for holomorphic cusp forms with CM and squarefree level. This is done by exploiting the lattice structure of ideals in number fields. Voronoi summation for such cusp forms is given by Kowalski, Michel and Vanderkam (2002). We hope that our method of proof can extend their Voronoi formula to any CM cusp form in $S_k(\Gamma_1(N))$ and arbitrary additive twist. We encounter quadratic and quartic Gauss sums in the process. We shall present the calculations for the general case in the next version of the paper.