Trace operator and theta series
The authors investigate the space \(M_n^k (M,\chi)\) of Siegel modular forms for \(\Gamma_0^{(n)}(M)\) and appropriate character \(\chi\) and the subspace \(\theta^{(n)} (V,M)\) spanned by the theta series of degree \(n\) of lattices \(L\) of level dividing \(M\) in the quadratic space \((V,q)\). The main question is whether \[ \theta^{(n)} (V,M)\cap M_n^k (N,\chi)= \theta^{(n)} (V,N) \] holds for \(N|M\), where mostly \(M/N\) is a prime. The question is reformulated by means of the trace operator from \(\Gamma_0^{(n)}(M)\) into \(\Gamma_0^{(n)}(N)\): Can the image of a theta series of a lattice of level \(M\) be expressed as a linear combination of theta series of lattices of level \(N\) in the same space? The question is dealt with by two different methods. At first they deal with singular modular forms and use commutation relations with the Siegel \(\varphi\)-operator. The other approach starts from a direct calculation of the Fourier expansion. Both ways lead to the same results.
scientific article; zbMATH DE number 1347934
scientific article