Siegel Modular Forms by Ameya Pitale

Author:Ameya Pitale
Language: eng
Format: epub
ISBN: 9783030156756
Publisher: Springer International Publishing

Shimura proved the following special values theorem for elliptic modular forms.

Theorem 8.1

(Shimura [95]) Let f be a primitive holomorphic cusp form of weight and level N. Then there exist nonzero complex numbers for such that, for any Hecke character , and any integer m satisfying , we have

where is given by ; is the Gauss sum attached to , and is the finite part of the L-function of f twisted by .

There has been tremendous progress in recent years toward proving arithmeticity of special values of L-functions in various settings. At the same time, this is a very active area of current research with wide open problems yet to be solved. We would like to discuss this problem in details in the setting of special values of .

One approach to studying is to obtain an integral representation for it. This was done by Furusawa in [30]. The essential idea is to construct an Eisenstein series E(g, s; f) on a bigger group using the representation . This is a function on depending on a section f in an induced representation obtained from and s, a complex number. Let be any cusp form in . Define the integral

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