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Existence of Hilbert Cusp Forms with Non-vanishing L-values
Shingo Sugiyama, Masao Tsuzuki
Journal: Canadian Journal of Mathematics / Volume 68 / Issue 4 / 01 August 2016
Published online by Cambridge University Press: 20 November 2018, pp. 908-960
Print publication: 01 August 2016
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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.