# Pullback theory for functions of lattice-index and applications to Jacobi and modular forms

• Pullback-Theorie für Funktionen von gitterwertigem Index und Anwendungen auf Jacobi- und Modulformen

In this thesis we examine holomorphic functions that satisfy a certain invariance property of elliptic type with respect to a lattice $\underline{L}$. The main focus is laid on the study of structure-preserving maps arising from embeddings of lattices, called pullback operators. Chapter one serves to introduce the basic concepts and terms in the theory of lattices. The theory for embeddings of lattices is developed. As an example, some irreducible root lattices, namely $\underline{E_8},\underline{E_7}, \underline{E_6}, \underline{D_4}, \underline{A_2}$ and $\underline{A_1}$, are constructed in realization that will be utilized later. In chapter two we introduce elliptic functions of lattice-index. In order to have a notion of boundedness for this class of functions, we introduce conditions of regularity and cuspidality. The metaplectic group reflects the modular action on the space $\mathcal E^{(n)}(\underline{L})$ of elliptic functions of degree n and index $\underline{L}$. This action induces a certain representation, called the Weil representation, on the space of Jacobi theta functions associated to the lattice $\underline{L}$. We end the chapter by calculating certain determinant characters of Weil representations of degree 1 associated to some distinguished lattices of low rank. The detailed analysis of pullback operators between elliptic functions of lattice-index is dealt within chapter three. These maps preserve regularity and cuspidality. The question that arises naturally in this context is in what cases the pullback operator turns out to be an isomorphism. To this end, we take an algebraic point of view and consider the pullback operator as a homomorphism of free modules. We apply methods of linear algebra and consider its representation matrix, called automorphic transfer, and its determinant, provided existence. The latter will turn out as a Siegel modular form. We obtain a sufficient and necessary criterion for the existence of such isomorphisms. Suprisingly, those can occur in the case $n=1$ only. As a by-product of the theory developed, we derive the existence of an infinite family of non-trivial Siegel cusp forms satisfying a remarkable recurrence identity. Finally, we calculate the automorphic transfer and its determinant in the case $n=1$ explicitly for certain lattices of low rank. In chapter four we utilize these results in order to construct explicit isomorphisms of spaces of Jacobi forms, whose existence was partly known before. Here we provide a method to lift Jacobi forms of lower rank index to higher rank index by using matrix-vector multiplication with respect to the underlying space of vector valued modular forms. From chapter five on we draw attention to modular forms and discuss a problem initiated by G. Köhler concerning embeddings of paramodular groups into modular groups over orders. We extend his methods to noncommutative orders and develop a notion of equivalence to separate substantially different embeddings. We determine the action of the maximal discrete extension of the paramodular group on the set of modular embeddings. At the end we develop a pullback theory that turns modular into paramodular forms and which is compatible with the equivalence relation. The sixth and final chapter is conducted by the question to what extent the modular embedding is determined by the family of paramodular forms induced by it. Already in the case $n=2$ the equivalence relation is too restrictive. In order to handle this, we develop a notion of equivalence in the extended sense. By approaching the initial question on the basis of hermitian and quaternionic Jacobi forms we can solve the problem at least for certain orders and under suitable assumptions on the polarization.