38 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS
(101) now follows with t = t1 + :Z.:::iEI μi. D
12.7. Conclusion of the proof of Theorem 12.4. Let f be an automorphic
form on Yp for fp. We apply Section 12.6 to G = P and P 0 , where the roles of
(^0) G, AG are played by Mp and Ap. There exists then t E X(A
0 ), in the domain of
convergence of E P,Po ( 1, s) such that
(105) lf(x)I -< Ep,p 0 (l, t)(x) (x E Yp ).
Therefore the series Ec,P(f,s) has the series Ec,P(EP,Pa(l,t),s) as a termwise
majorant.
Ep,p 0 (l, t) is a summation over fo\fp. The characters belongs to X(Ap),
hence is left-invariant under fp. Hence
(106)
Thus, Ec,p(f,s) is majorized termwise by Ec,p 0 (EP,Pa(l,t + s)), which can be
written as
(107) L L a('yx)t+s+PpiJ+PP
rp\r r o\rp
where p~ = Po n Mp' hence p Po +PP = p Pa. The majorant is therefore Ee ,Po ( 1, t +
s), and we are back to Section 12.5.
Acknowlegments. I am grateful to Ramin Takloo-Bighash and Akshay Venka-
tesh, who put in TeX my typewritten or handwritten notes, and raised questions
or pointed out inaccuracies which led to various improvements of the text.