|
This article is cited in 8 scientific papers (total in 8 papers)
A basis of eigenfunctions of Hecke operators in the theory of modular forms of genus $n$
S. A. Evdokimov
Abstract:
Let $\mathfrak M^n_k(\Gamma,\mu)$, where $n,k>0$ are integers, $\Gamma$ is some congruence subgroup of $\Gamma^n=\operatorname{Sp}_n(\mathbf Z)$ and $\mu\colon\Gamma\to\mathbf C^*$ is a congruence-character of $\Gamma$, be the space of all Siegel modular forms of genus $n$, weight $k$ and character $\mu$ with respect to $\Gamma$. In this paper, for a very broad class of congruence subgroups $\Gamma$ of $\Gamma^n$, including all those previously investigated and practically all those groups encountered in applications, the author constructs a sufficiently large commutative ring of Hecke operators, acting on $\mathfrak M^n_k(\Gamma,\mu)$, a canonical decomposition
\begin{equation}
\mathfrak M^n_k(\Gamma,\mu)=\bigoplus^n_{r=0}\mathfrak M^{n,r}_k(\Gamma,\mu)
\tag{1}
\end{equation}
and a canonical inner product $(\,{,}\,)_\Gamma$ on $\mathfrak M^n_k(\Gamma,\mu)$. It is shown that the Hecke operators preserve the canonical decomposition (1) and that they are normal with respect to the canonical inner product $(\,{,}\,)_\Gamma$.
Bibliography: 17 titles.
Received: 15.12.1980
Citation:
S. A. Evdokimov, “A basis of eigenfunctions of Hecke operators in the theory of modular forms of genus $n$”, Mat. Sb. (N.S.), 115(157):3(7) (1981), 337–363; Math. USSR-Sb., 43:3 (1982), 299–321
Linking options:
https://www.mathnet.ru/eng/sm2400https://doi.org/10.1070/SM1982v043n03ABEH002450 https://www.mathnet.ru/eng/sm/v157/i3/p337
|
Statistics & downloads: |
Abstract page: | 293 | Russian version PDF: | 90 | English version PDF: | 18 | References: | 56 |
|