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

2018/01/17 by Liyang Yang, Yang, Liyang
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1801.05562

openalex publication_date 2018/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Central values of L-functions over CM fields
• Certain CM class fields with smaller generators
• Heegner points and Eisenstein series
• Hybrid subconvexity for class group L-functions and uniform sup norm bounds of Eisenstein series
• Non-vanishing of Class Group L-functions for Number Fields with a Small Regulator