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Common zeros of theta functions and central Hecke L-values of CM number fields of degree 4

Author(s): Tonghai Yang
Journal: Proc. Amer. Math. Soc. 126 (1998), 999-1004.
MSC (1991): Primary 11F27, 11F67, 11M06
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Abstract: In this note, we apply the method of Rodriguez Villegas and Yang (1996) to construct a family of infinite many theta series over the Hilbert-Blumenthal modular surfaces with a common zero. We also relate the non-vanishing of the central L-values of certain Hecke characters of non-biquadratic CM number fields of degree 4 to the nonvanishing of theta functions at CM points in the Hilbert-Blumenthal modular surfaces.

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D. Rohrlich, Root numbers of Hecke $L$-functions of CM fields, Amer. J. Math. 104 (1982), 517-543. MR 83j:12011

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-, Galois conjugacy of unramified twists of Hecke characters, Duke Math J. 47 (1980), 695-703. MR 82a:12009

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F. Rodriguez Villegas and T. H. Yang, Central values of Hecke L-functions of CM number fields, submitted to Duke Math. J., 1996.

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G. Shimura, Introduction to the arithmetic theory of automorphic functions, vol. 11, Publ. Math. Soc. Japan, 1971. MR 47:3318

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K. Yamamura, The determination of the imaginary abelian number fields with class number one, J. Number Theory 62 (1994), 899-921. MR 94g:11096

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Additional Information:

Tonghai Yang
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: thyang@math.lsa.umich.edu

DOI: 10.1090/S0002-9939-98-04213-0
PII: S 0002-9939(98)04213-0
Keywords: Theta function, central Hecke L-value, ideal class number
Received by editor(s): September 27, 1996
Additional Notes: The author was partially supported by NSF grant DMS-9304580
Communicated by: William W. Adams
Copyright of article: Copyright 1998, American Mathematical Society

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