Martin Westerholt-Raum

Products of vector valued Eisenstein series

Abstract We prove that products of at most two vector valued Eisenstein series that originate in level 1 span all spaces of cusp forms for congruence subgroups. This can be viewed as an analogue in the level aspect to a result that goes back to Rankin, and Kohnen and Zagier, which focuses on the weight aspect. The main feature of the proof are vector valued Hecke operators. We recover several classical constructions from them, including classical Hecke operators, Atkin–Lehner involutions, and oldforms. As a corollary to our main theorem, we obtain a vanishing condition for modular forms reminiscent of period relations deduced by Kohnen and Zagier in the context their previously mentioned result.

Formal Fourier Jacobi expansions and special cycles of codimension two

We prove that formal Fourier Jacobi expansions of degree two are Siegel modular forms. As a corollary, we deduce modularity of the generating function of special cycles of codimension two, which were defined by Kudla. A second application is the proof of termination of an algorithm to compute Fourier expansions of arbitrary Siegel modular forms of degree two. Combining both results enables us to determine relations of special cycles in the second Chow group.We prove that formal Fourier Jacobi expansions of degree 2 are Siegel modular forms. As a corollary, we deduce modularity of the generating function of special cycles of codimension 2, which were defined by Kudla. A second application is the proof of termination of an algorithm to compute Fourier expansions of arbitrary Siegel modular forms of degree 2. Combining both results enables us to compute relations of special cycles in the second Chow group.