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On Dirichlet Series and Petersson Products for Siegel Modular Forms

Siegfried Böcherer, Francesco Ludovico Chiera (2008)

Annales de l’institut Fourier

We prove that the Dirichlet series of Rankin–Selberg type associated with any pair of (not necessarily cuspidal) Siegel modular forms of degree n and weight k n / 2 has meromorphic continuation to . Moreover, we show that the Petersson product of any pair of square–integrable modular forms of weight k n / 2 may be expressed in terms of the residue at s = k of the associated Dirichlet series.

On Minkowski units constructed by special values of Siegel modular functions

Takashi Fukuda, Keiichi Komatsu (2003)

Journal de théorie des nombres de Bordeaux

Using the special values of Siegel modular functions, we construct Minkowski units for the ray class field k 6 of ( e x p ( 2 π i / 5 ) ) modulo 6 . Our work is based on investigating the prime decomposition of the special values and describing explicitly the action of the Galois group G ( k 6 / ) for the special values. Futhermore we construct the full unit group of k 6 using modular and circular units under the GRH.

On special values of theta functions of genus two

Ehud De Shalit, Eyal Z. Goren (1997)

Annales de l'institut Fourier

We study a certain finitely generated multiplicative subgroup of the Hilbert class field of a quartic CM field. It consists of special values of certain theta functions of genus 2 and is analogous to the group of Siegel units. Questions of integrality of these specials values are related to the arithmetic of the Siegel moduli space.

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