On the modular forms of weight 1/2 over algebraic number fields (Q1981604)

This paper involves the following problems: \begin{itemize} \item[(1)]Verify that the spaces of modular forms (or cusp forms) of weight \(1/2\) can be spanned by theta series; \item[(2)] How to determine the explicit basis; \item[(3)] More precisely, if given a fixed level and a character, what about the explicit basis? \end{itemize} Such problems may arise from Shimura's work [\textit{G. Shimura}, Ann. Math. (2) 97, 440--481 (1973; Zbl 0266.10022)]. In [Lect. Notes Math. 627, 27--67 (1977; Zbl 0371.10019)], \textit{J. P. Serre} and \textit{H. M.Stark} definitely solved the total problems over the rational number field. Then \textit{S. Gelbart} and \textit{I. Piatetski-Shapiro} [Invent.Math. 59, 145--188 (1980; Zbl 0426.10027)] solved the first problem for cusp forms over algebraic number fields by representation theory. Later, in [Duke Math. J. 55, 765--838 (1987; Zbl 0636.10024)], \textit{G. Shimura} solved the first problem for modular forms over totally real number fields. On the other hand, \textit{D. Gove} [Acta Arith. 64, No. 2, 101--123 (1993; Zbl 0787.11014)] solved the whole three problems for modular forms over imaginary quadratic number fields of class number one. Moreover, \textit{S. Achimescu} and \textit{A. Saha} [J. Number Theory 128, No. 12, 3037--3062 (2008; Zbl 1204.11083)] solved the first two problems for modular forms over totally real number fields of narrow class number one. According to the induction, based on Serre and Stark's paper, the authors solve the total problems for modular forms over arbitrary algebraic number fields of narrow class number one. This article is professional, and it uses some precise arguments related to zero points and poles of \(L\)-functions. Reviewer's remark: From representation theory to treating theta series and modular forms of weight \(1/2\), one can see [\textit{G. Lion} and \textit{M. Vergne}, The Weil representation, Maslov index and Theta series, Boston, etc.: Birkhäuser (1980; Zbl 0444.22005); \textit{J. Bernstein (ed.)} and \textit{S. Gelbart (ed.)}, Lectures presented at the Hebrew University of Jerusalem, Jerusalem, Israel, March 12--16, 2001. Boston, MA: Birkhäuser (2003; Zbl 1112.11028); \textit{R. Howe}, Proc. Symp. Pure Math. 33, 275--285 (1979; Zbl 0423.22016); \textit{S. Rallis} and \textit{G. Schiffmann}, Am. J. Math. 100, 1049--1122 (1978; Zbl 0402.10029)], etc.