Siegel modular forms and theta series.

*(English)*Zbl 0681.10019
Theta functions, Proc. 35th Summer Res. Inst. Bowdoin Coll., Brunswick/ME 1987, Proc. Symp. Pure Math. 49, Pt. 2, 3-17 (1989).

[For the entire collection see Zbl 0672.00004.]

This is a well done survey about the question “Do all Siegel modular forms arise from theta series”? This vague question is precisely formulated in three problems, whose solutions are due to the author himself [Math. Z. 183, 21-46 (1983; Zbl 0497.10020)] and R. Weissauer [Stabile Modulformen und Eisensteinreihen (Lect. Notes Math. 1219) (1986; Zbl 0596.10023)].

Let \(M^ k_ n\) be the vector space of all Siegel modular forms of weight k and degree n and denote the subspace of cusp forms by \(S^ k_ n\). If 4 \(| k\), then \(\Theta\) (2k,n) stands for the subspace of \(M^ k_ n\), which is spanned by the theta series attached to all even, unimodular, positive-definite 2k\(\times 2k\) matrices. It follows from the theory of singular modular forms by Freitag and Resnikoff that \(M^ k_ n=\Theta (2k,n)\) holds for \(2k<n.\)

The main result is as follows: Given \(f\in S^ k_ n\), 4 \(| k\), which is a simultaneous Hecke eigenform, then \(f\in \Theta (2k,n)\) holds if and only if a certain condition of behavior of the automorphic L- function L(f,s) at \(s=k-n\) is satisfied. Since this condition is always satisfied for \(k>2n\), one concludes \(M^ k_ n=\Theta (2k,n)\quad for\quad k>2n,\quad 4 | k.\)

The main ingredient of the proof is the pullback of Klingen-Eisenstein series. This leads to a canonical way to express any \(f\in \Theta (2k,n)\) in terms of the involved theta-series. In a particular case this canonical form reduces to Siegel’s main theorem.

Finally the results are generalized in order to include theta series with harmonic coefficients.

This is a well done survey about the question “Do all Siegel modular forms arise from theta series”? This vague question is precisely formulated in three problems, whose solutions are due to the author himself [Math. Z. 183, 21-46 (1983; Zbl 0497.10020)] and R. Weissauer [Stabile Modulformen und Eisensteinreihen (Lect. Notes Math. 1219) (1986; Zbl 0596.10023)].

Let \(M^ k_ n\) be the vector space of all Siegel modular forms of weight k and degree n and denote the subspace of cusp forms by \(S^ k_ n\). If 4 \(| k\), then \(\Theta\) (2k,n) stands for the subspace of \(M^ k_ n\), which is spanned by the theta series attached to all even, unimodular, positive-definite 2k\(\times 2k\) matrices. It follows from the theory of singular modular forms by Freitag and Resnikoff that \(M^ k_ n=\Theta (2k,n)\) holds for \(2k<n.\)

The main result is as follows: Given \(f\in S^ k_ n\), 4 \(| k\), which is a simultaneous Hecke eigenform, then \(f\in \Theta (2k,n)\) holds if and only if a certain condition of behavior of the automorphic L- function L(f,s) at \(s=k-n\) is satisfied. Since this condition is always satisfied for \(k>2n\), one concludes \(M^ k_ n=\Theta (2k,n)\quad for\quad k>2n,\quad 4 | k.\)

The main ingredient of the proof is the pullback of Klingen-Eisenstein series. This leads to a canonical way to express any \(f\in \Theta (2k,n)\) in terms of the involved theta-series. In a particular case this canonical form reduces to Siegel’s main theorem.

Finally the results are generalized in order to include theta series with harmonic coefficients.

Reviewer: A.Krieg

##### MSC:

11F27 | Theta series; Weil representation; theta correspondences |