Röhrig, Christina 
ORCID: 0000-0003-4820-7206
(2022).
Elliptic and Siegel Theta Series for Indefinite Quadratic Forms.
PhD thesis, Universität zu Köln.
Abstract
In this thesis, we embed four research articles on elliptic and Siegel theta series for indefinite quadratic forms. In the first two articles, we generalize results on elliptic theta series to Siegel theta series, i.e. theta series for arbitrary genus $n\in \N$. In the third and fourth article we consider elliptic theta series for quadratic forms of signature (m-1,1) in order to construct holomorphic and almost holomorphic modular forms. A useful tool in the theory of elliptic theta series is Vignéras' modularity criterion: considering functions with a certain growth condition, which satisfy a particular second order differential equation, one can construct modular theta series for indefinite quadratic forms. Furthermore, this result can be applied to check whether a given theta series transforms as a modular form. We show that for Siegel theta series we can define an analogous system of partial differential equations and thus obtain a straightforward generalization. If one does not consider positive definite quadratic forms but indefinite ones, the explicit construction of theta series that are holomorphic and modular, poses some difficulties. For elliptic theta series, there exist various approaches in which a modular (non-holomorphic) theta series is defined on the one hand and a holomorphic (but not modular) version on the other hand. We examine Zwegers' construction for quadratic forms of signature (m-1,1) and show that in a similar way non-holomorphic modular Siegel theta series for quadratic forms of this particular signature can be constructed. Moreover, we describe the holomorphic part of these Siegel theta series. In the last two articles, we generalize Zwegers' construction in a different direction by including homogeneous as well as spherical polynomials in the definition of the theta functions. The definition depends on two vectors c_1 and c_2, which are located in a special cone C_Q. In the first article on this matter, we consider the case that c_1 and c_2 are located in the inner part of C_Q and give a general construction of a holomorphic, an almost holomorphic, and a modular theta series. We state under which condition these versions agree and thus obtain almost holomorphic and holomorphic modular forms. This turns out to be a rich source of explicit examples, which we can often identify as eta products or eta quotients. In a sequel to this project, we also allow c_1 or c_2 to be located on the boundary of C_Q. This extension provides examples of a slightly different flavor. In particular, we can recover the modular transformation behavior of the Eisenstein series, of certain modular forms on $\Gamma_0(4)$ that appear during the investigation of sums of powers of quadratic polynomials of a fixed discriminant and of a Maass form of weight 3/2, whose holomorphic part is determined by the generating function of the Hurwitz class numbers H(8n+7).
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