An Explicit CM Type Norm Formula and Effective Nonvanishing of Class Group L-functions for CM Fields

TL;DR

This paper derives an explicit formula for CM field class group L-functions and establishes effective nonvanishing results, advancing understanding of class numbers and L-functions in algebraic number theory.

Contribution

It provides a new explicit formula for CM type norm and proves effective nonvanishing of class group L-functions, improving upon previous ineffective results.

Findings

Explicit CM type norm formula derived.

Conditional lower bounds for class numbers established.

Effective nonvanishing results for class group L-functions obtained.

Abstract

We show that the central value of class group L-functions of CM fields can be expressed in terms of derivatives of real-analytic Hilbert Eisenstein series at CM points. Then, following an idea of Iwaniec and Kowalski we obtain a conditional explicit lower bound of class numbers of CM fields under a weaker assumption. Some results in the proof lead to an effective nonvanishing result for class group L-functions of general CM fields, generalizing the only known ineffective results.