A p-adic family of Klingen - Eisenstein series

The p-adic interpolation properties of Fourier coefficients of elliptic Eisenstein series are by now classical. These properties can be considered as the starting point and as an important tool in the theory of p-adic L-functions and p-adic families of modular forms. In the case of Siegel modular forms there are two types of Eisenstein series. A Siegel Eisenstein measure which comes from the Siegel Eisenstein series was recently constructed by A.A.Panchishkin [10]. Another Eisenstein series on the symplectic group are those of Klingen type. The Fourier expansion of these series is of definite interest. One can associate a Klingen Eisenstein series to an elliptic cusp form f . Then the Fourier coefficients of this series involve special values of certain Dirichlet series connected with f . Namely, the Rankin convolutions of f with theta series associated with positive definite quadratic forms. Though the explicit formulas for these Fourier coefficients [1], [2] are in general considerably complicated, we prove (without making use of these explicit formulas) that, after suitable normalization and regularization, they become p-adically smooth functions. It becomes natural to consider vector valued Siegel modular forms in this context as well. The purpose of this paper is to construct a p-adic measure coming from the Klingen Eisenstein series. Our main tools are the A.A. Panchishkin’s construction of Siegel Eisenstein measure [10]; the Böcherer Garrett pull-back formula [1], [3]; H. Hida’s theory of p-ordinary Λ-adic forms [4], [5]. The author is grateful to the Minerva Fellowship for the financial support and to the University of Mannheim for its hospitality. It is a pleasure for me to thank Prof. Dr. Böcherer for his patient and detailed explanations connected with the theory of Siegel modular forms. This work was inspired by the talk given by Prof. A.A. Panchishkin on Lundi Arithmétique entitled Familles p-adiques de formes modulaires et de représentations galoisiennes, held at Institut Henri Poincaré. The author is very grateful to Prof. J. Tilouine for the kind invitation to this conference. The influence of the results and ideas of [10] and [7] on this paper is evident, and I am very grateful to Prof. A.A. Panchishkin for transmitting me the manuscripts before publication as well as for helpful discussions. The author thanks the referee for thoughtful remarks. The contents of the paper and the main ideas of our construction are as follows. We introduce necessary notations on vector valued Siegel modular forms and state an appropriate version of the pull-back formula in the first section (Proposition 1 in the text). The second section is devoted to our main result. In [10], certain subseries of the Siegel Eisenstein series Gk,χ is used for the p-adic construction. Moreover, since one knows the explicit formulae for the Fourier coefficients of Siegel Eisenstein series (the construction