The modular transformation behavior of theta series for indefinite quadratic forms is well understood in the case of elliptic modular forms due to Vign\'eras, who deduced that solving a differential equation of second order serves as a criterion for modularity. In this paper we will give a generalization of this result for Siegel theta series.
Siegel theta series for quadratic forms of signature $(m-1,1)$
Lifting from pairs of two elliptic modular forms to Siegel modular forms of half-integral weight of even degree
Some congruences for Siegel theta series
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