Introductory Lectures on Siegel Modular Forms

Abstract

From their inception, Siegel modular forms have been studied extensively because of their significance in both automorphic functions in several complex variables and number theory. The comprehensive theory of automorphic forms to subgroups of algebraic groups and the arithmetical theory of modular forms illustrate these two aspects in an illuminating manner. The author's aim is to present a straightforward and easily accessible survey of the main ideas of the theory at an elementary level, providing a sound basis from which the reader can study advanced works and undertake original research. This book is based on lectures given by the author for a number of years and is intended for a one-semester graduate course, though it can also be used profitably for self-study. The only prerequisites are a basic knowledge of algebra, number theory and complex analysis.

On Fourier coefficients of Siegel modular forms of degree two with respect to congruence subgroups

“…This is a direct generalization of a result in Klingen's book [6], page 90. We briefly outline the argument.…”

“…On the last line of p.87, Klingen defines A n , which differs from our Γ (2) 1 (∞) by the factor 2. Therefore, g k n (z, t) defined on p.90 of [6] differs from our g N (Z, T ) by the factor 2. All other parts of the proof are the same as in [6].…”

“…We briefly outline the argument. We follow the argument in pp.76-90 of Klingen [6] with replacing Γ n and A n by Γ (2) 0 (N) and Γ (2) 1 (∞), respectively. The starting point, Proposition 1 ([6, p.76]), does not depend on N. Formulas (7), (8) in [6, p.78] are proved by the technique of decomposing the Siegel half space H n into copies (by the action of {±1}\Γ n ) of the fundamental domain F n .…”

Siegel Modular Forms and Their Applications

“…These Poincaré series behave well with respect to the Petersson product. We refer to [60], Ch. 6, or [8] for the general setting.…”

“…The reader may also consult the introductory book by Klingen [60]. Two other references to the literature are the two books [93,94] by Shimura.…”