#### Vol. 304, No. 1, 2020

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An explicit CM type norm formula and effective nonvanishing of class group L-functions for CM fields

### Liyang Yang

Vol. 304 (2020), No. 1, 347–384
##### Abstract

We show that the central value of class group $L$-functions of general CM fields can be expressed in terms of derivatives of real-analytic Hilbert Eisenstein series at CM points. Using this in conjunction with an explicit CM type norm formula established in Section 3, following an idea of Iwaniec and Kowalski (2004), we obtain a conditional explicit lower bound for class numbers of CM fields under the assumption ${\zeta }_{K}\left(\frac{1}{2}\right){\ll }_{F}log{D}_{K∕F}$ (note that GRH implies ${\zeta }_{K}\left(\frac{1}{2}\right)\le 0$). 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. Moreover, combining the CM type norm formula with Barquero-Sanchez and Masri’s work (2016), we shall deduce an explicit Chowla–Selberg formula for all abelian CM fields.

##### Keywords
class numbers, CM points, class group L-functions, nonvanishing
##### Mathematical Subject Classification 2010
Primary: 11G15, 11G35, 11M20, 11R29, 11R42