On pullbacks of functions of lattice-index and an infinite family of Siegel cusp forms

Abstract

In the present note we consider the module E (n) (L) of elliptic functions of latticevalued index L and degree n. We introduce conditions of regularity and cuspidality based on Eichler and Zagier (The theory of Jacobi forms. Progress Math 55, Birkhäuser, Boston, Basel, Stuttgart, 1985) and Ziegler (Abh Math Sem Univ Hamburg 59:191-224, 1989) for such type of functions and determine the structure of the corresponding modules. The pullback operator induced by embeddings of lattices will turn out to be a homomorphism of free modules. Hence, linear algebra methods can be applied in order to determine the cases, in which a pullback operator induces an isomorphism between submodules of E (n) (L), that are generated by subspaces, which are invariant under the associated Weil representation. In the equi-rank case, the determinant of the pullback operator will turn out to be a Siegel modular form. We show that isomorphisms only exists if n = 1 and the determinant is a nonzero multiple of some certain power of Dedekind's η-function. As an application, we show that the embedding A 1 −→ E 7 gives rise to an infinite family of nontrivial Siegel cusp forms, satisfying a specific recurrence relation with respect to the Witt operator. This paper is a short version of some part of the author's 2015 thesis (Dieckmann, Pullback theory for functions of lattice-index and applications to Jacobi-and modular forms. Ph.D.-thesis, Aachen, 2015)-also some slight modifications and corrections have been carried out.