An explicit CM type norm formula and effective nonvanishing of class group L-functions for CM fields
- Creators
- Yang, Liyang
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 ζ_K(1/2)≪F log D_(K∕F) (note that GRH implies ζ_K(1/2) ≤ 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.
Additional Information
© 2020 Pacific Journal of Mathematics. Received: 21 May 2018; Revised: 5 March 2019; Accepted: 7 July 2019; Published: 18 January 2020. I am grateful to my advisor Dinakar Ramakrishnan for very helpful discussions and comments. I would like to thank Tonghai Yang for his careful reading and suggestions; and Jeffrey Hoffstein, Stéphane Louboutin for their comments. Sincere thanks are also due to Zavosh Amir-Khosravi for his kind help.Attached Files
Published - pjm-v304-n1-p12-p.pdf
Submitted - 1801.05562.pdf
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Additional details
- Eprint ID
- 101156
- Resolver ID
- CaltechAUTHORS:20200206-101320973
- Created
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2020-02-06Created from EPrint's datestamp field
- Updated
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2021-11-16Created from EPrint's last_modified field