We show that the central value of class group
-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
(note that
GRH implies
).
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
