On Transformation Laws for Theta Functions

Classification of the space spanned by theta series and applications

“…a) Equation (13) provides an isomorphism between the space of skewholomorphic Jacobi forms of weight k and index M and the space of vector-valued Siegel modular forms of half-integral weight satisfying (i), (ii), and the additional growth condition for n = 1 (see also Hayashida [12] for the case l = M = 1).…”

“…Various types of theta functions can be regarded as specializations of the symplectic theta function ϑ(τ ) = θ 1, 0 (τ, 0) (i.e., l = 1) as first observed by Eichler [7], and then also by many others (see for example [1], [2], [15], [10], and [13]). This yields an elegant way to prove transformation laws for such theta functions.…”

Jacobi Theta Functions over Number Fields

“…Friedberg [3] and Richter [8] examine the transformation properties of # À Z; À r s Á ; w; f Á under modular transformations, and Richter [8] shows the following theorem:…”

“…Friedberg [3] and Richter [8] prove transformation laws for # À Z; À r s Á ; w; f Á , a modified version of the usual symplectic theta function. We proceed as in [8] and regard certain coefficients of  ðKÞ Q;R;w ð(; zÞ as specializations of #ðZ; À 0 0 Á ; w; f Á . As an immediate consequence, we obtain the transformation law of  ðKÞ Q;R;w ð(; zÞ under modular transformations.…”

Арифметические Гипергеометрические Ряды

“…Для этого можно использовать тета-ряды, соответствующие неопределенным квадратичным формам [114], или ряды Эй-зенштейна [91; гл. 18].…”