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

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

**Read or Download Automorphic Forms and Zeta Functions: Proceedings of the Conference in Memory of Tsuneo Arakawa Rikkyo University PDF**

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

**Sample text**

Case 1. G = Gi For U. -U. , G l t ,. = S 0 ( 2 ) , = {{x*,y*>z*) € R-3, f{x*,y*, z*) =x2t+ y*z* + k? = 0, y« > z»}, = {(a:»,j/»,i*) G R 3 ,/(a;»,y,,z*) = a:2, + y2 - z2 + k2 = 0,z* > 0}, where we introduced y* = y* + z*,z* = y* - z* to show the standard form of the upper half of the two-sheeted hyperboloid. Case 2. , GJu, = KJ = SO(2) x R, 5 = {(z*,y*,z»,p»,g«) € R ,/(z*,y*,z,,,p*,g») = 0,2/* > z»}, = {(a;*,y*,z*,p»,g*) 6 Rs,f(xt,y*,Zt,p*,q*) = 0 , 2 * > 0}, where /(z*,y*,z*,p*,g*) = m(a;2 + j/,z« + A;2) - 2p,g*a:» + p2,y» - Q2z» and f(x*,y*,z*,p*,q*) m(xl+y2-z2+k2)-2p*q*x*+(pl-q2)y*+(pl+q2)z:t.

Then WNo := L1A(WNO,WNO) is normalizing TQ(N) and 'h R W - * = 02 1£NO-U]-R with 'aN0A + NBbA + Here we put N' := Ni • N2. N0B\cqft}-. The Genus Version of the Basis Problem I We easily see that N1(R) = N1, N2(R) = No • N2l No(R) = 1. For the Eisenstein series this means E2k(Z,x,R,s) N0-k-2°E2k(Z^,R,s) |fc WNo = with V>(7) = x(WNojW^) = (XAT'XJVO)(7)- Now we fix a decomposition N = N1-N2 (N0 = 1) and a matrix TZ = 7ZN1}N2 AB C V = such that ( /02 - l a 12 0 2 n= I mod A2, 0-1 1 0 mod N\. Then TlTl(N) consists of all M £ Sp(2, Z) such that -C A M =I -D B mod iV2, c 3 c 4 d 3 ^4 mod JVi ci c 2 d\ d2 V a 3 «4 &3 &4 ) with AB CD M G Tl(N).

Then F has a Fourier expansion ir(ri1r2)= V3 5Z c(u>u)e ( 2 U + _ 6 ~ U ) T l + ( 2 u - — « (u,«)eA \V / V r2 i . Thus, easily we have (D2rF)(r) = 2 3 u 2 r c(u,i;)e(uT). 2. If F € A£+(2n), t/ien c(w, v) = 0 /or any u < n. Proof. We show this lemma by induction on n. If n = 0, this lemma is trivial. Now we assume that this lemma holds for n < r. Let F £ A~£+(2(r + 1)). From the assumption, c(u,v) = 0 for any u < r. If v > r, then 1 c(r,v)=c(e f- r + ^—v ] J = c(2r - v, - 3 r + 2w) = 0 because 2r — v < r.