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Behavior of theta series of degree $n$ under modular substitutions
A. N. Andrianov,
G. N. Maloletkin
Abstract:
Let
$F$ be an integral, symmetric, positive definite matrix of order
$m\geqslant1$ with an even diagonal. For the theta series of
$F$ of degree
$n\geqslant1$
$$
\theta_F^{(n)}(Z)=\sum_x^F\exp(\pi i\operatorname{Tr}(^tXFXZ)),
$$
where
$X$ runs through all integral
$m\times n$ matrices and
$Z$ is a point of the Siegel upper halfplane of degree
$n$, the congruence subgroup of the group
$Sp_n(\mathbf Z)$ is found, with respect to which
$\theta_F^{(n)}(Z)$ is a Siegel modular form with a multiplicator system (the analog of the group
$\Gamma_0(q)$)). The analogous problem is solved for theta series of degree
$n$ with spherical functions. The appropriate multiplicator systems are computed for even
$m$.
Bibliography: 5 items.
UDC:
511.466+517.863
MSC: Primary
10A20,
10D05,
10G05; Secondary
10C05 Received: 18.02.1974