By Siegfried Bocherer, Tomoyoshi Ibukiyama, Masanobu Kaneko, Fumihiro Sato

"This quantity incorporates a selection of articles awarded at a convention on Automorphic kinds and Zeta features in reminiscence of Tsuneo Arakawa, an eminent researcher in modular varieties in different variables and zeta features. The booklet starts with a assessment of his works, by means of sixteen articles via specialists within the fields. This selection of papers illustrates Arakawa's contributions and the present tendencies in modular kinds in numerous variables and comparable zeta services.

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**Extra info for Automorphic Forms and Zeta Functions: Proceedings of the Conference in Memory of Tsuneo Arakawa Rikkyo University**

**Example text**

Here we propose a somewhat brutal strategy: We write the genus Eisenstein series as a linear combination of Eisenstein series attached to the inequivalent cusps. Then one tries to compute the same kind of integral for each of these group-theoretical Eisenstein series individually (by considerations completely independent of theta series). This seems to be a somewhat painful task in general. Indeed it is the second main topic of this paper to work this out explicitly for the case n — 1, squarefree levels and all Eisenstein series.

It is also possible, to define in the same way as above Eisenstein series E%(Z, ip, R, s) for any R £ GSp+(n,Q) and any character if> of TQ (JV); occasionally we use such a more general Eisenstein series. 48 S. Bocherer For many purposes it is enough to know these Eisenstein series only up to the action of elements of the normalizer of TQ(N): Let W € GSp+(n, Q) normalize TQ(N), then we have, for R G Sp(n,Z) by a simple calculation EZ(Z,X,R,8)\kW = EZ(Z,rl>,RW,8), where the character V" of TQ (N) is defined by xiWjW-1).

P{2'Kyf^lw). We denote the C-vector space of all symmetric Hilbert modular forms of weight k by A+ = [F G Ak I F(TUT2) = Ffo,n)} and the C-vector space of all skew-symmetric Hilbert modular forms of weight k by A~k = { F G Ak | F ( T I , 7 5 ) = - F f a . T i ) } . If F ( T I , T 2 ) G Afc, then + ^ F(W)+F{M) F(r1,r2)-F(r2,r1)e fc 2 K 2 Hence we have Ak = Ak ® Ak , where the symbol © means the direct sum as C-vector spaces. Put A++ :={F A+ I F(TI,T2) {F&At At A~k G + ••={F€A~k Ak-:={FeAk = F(sTUe'T2)} , F(T1,T2) = F(T1,r2) = -F(£T1,S'T2)}, F(eT1,e'T2)}, -F(eTUe,T2)}.