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Existence of Hilbert Cusp Forms with Non-vanishing L-values

Published online by Cambridge University Press:  20 November 2018

Shingo Sugiyama  and
Masao Tsuzuki
Shingo Sugiyama
Affiliation:
Institute of Mathematics for Industry, Kyushu University, 744 Motooka Nishi-ku Fukuoka 819-0395, Japan e-mail: s-sugiyama@imi.kyushu-u.ac.jp
Masao Tsuzuki
Affiliation:
Department of Science and Technology, Sophia University, Kioi-cho 7-1 Chiyoda-ku Tokyo 102-8554, Japan e-mail: m-tsuduk@sophia.ac.jp
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Abstract

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We develop a derivative version of the relative trace formula on $\text{PGL}\left( 2 \right)$ studied in our previous work, and derive an asymptotic formula of an average of central values (derivatives) of automorphic $L$-functions for Hilbert cusp forms. As an application, we prove the existence of Hilbert cusp forms with non-vanishing central values (derivatives) such that the absolute degrees of their Hecke fields are arbitrarily large.

Keywords

11F6711F72automorphic representationsrelative trace formulascentral L–valuesderivatives of L–functions
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 68 , Issue 4 , 01 August 2016 , pp. 908 - 960
Copyright
Copyright © Canadian Mathematical Society 2016

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