COMMON ZEROS OF THETA FUNCTIONS AND CENTRAL HECKE L-VALUES OF CM NUMBER FIELDS OF DEGREE 4

Abstract. In this note, we apply the method in [RVY] to construct a family of infinite many theta series over the Hilbert-Blumenthal modular surfaces with a com-mon zero. We also relate the nonvanishing 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 the Hilbert-Blumenthal modular surfaces. 0. Introduction. It is well-known that the theta function θ(z) = e(x2z) has no zeros in the upper half plane, where e(z) = e2piiz. It was proved in [RVY] that, for integers d ≡ 1 mod 4 and k ≥ 0, whether the theta function