Extreme values of Hecke $L$-functions to angular characters

TL;DR

This paper investigates large values of Hecke L-functions associated with angular characters over imaginary quadratic fields, using the resonance method to identify significant logarithmic values along specific ranges.

Contribution

It introduces the application of the resonance method to this family of L-functions, utilizing the geometry of complex embeddings for classification and extraction of diagonal terms.

Findings

Identifies large values of ( frac{1}{2},\xi_{\u03bb})| with size at least (2 + o_K(1))(\u2206X / a0log a0X)^{1/2}

First application of the resonance method to this family of L-functions

Uses complex embedding geometry for classification and extraction of diagonal terms

Abstract

Let $K$ be an imaginary quadratic number field and let $L(s,\xi_{\ell})$ denote the Hecke $L$-function to an angular character $\xi_{\ell}$ with frequency $\ell$. We detect values of $\log |L(\tfrac12,\xi_{\ell})|$ with size at least $(\sqrt{2} + o_K(1))(\log X / \log \log X)^{1/2}$ along each dyadic range $X \leqslant \ell \leqslant 2X$. This result relies on the resonance method, which is applied for the first time to this family of $L$-functions, where the classification and extraction of diagonal terms depends on the geometry of the complex embedding of $K$.