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@PHDTHESIS{Dieckmann:479780,
author = {Dieckmann, Till},
othercontributors = {Krieg, Aloys and Heim, Bernhard},
title = {{P}ullback theory for functions of lattice-index and
applications to {J}acobi and modular forms},
school = {Aachen, Techn. Hochsch.},
type = {Dissertation},
address = {Aachen},
publisher = {Publikationsserver der RWTH Aachen University},
reportid = {RWTH-2015-03284},
pages = {IV, 156 S.},
year = {2015},
note = {Aachen, Techn. Hochsch., Diss., 2015},
abstract = {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.},
cin = {114110 / 110000},
ddc = {510},
cid = {$I:(DE-82)114110_20140620$ / $I:(DE-82)110000_20140620$},
typ = {PUB:(DE-HGF)11},
urn = {urn:nbn:de:hbz:82-rwth-2015-032847},
url = {https://publications.rwth-aachen.de/record/479780},
}
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