Meromorphic continuation of Poincaré series
In Section 1.6 we have sketched the proof of the meromorphic continuation in two variables of the Eisenstein series in the modular case. Most of this proof we have carried out for the general situation. Now we come to the last step, indicated in 1.6.5. This step amounts to adjusting and gluing the families of automorphic forms obtained in the previous chapter. In 1.6.5 we normalized one Fourier term of \(\widetilde E\)(r, s) by prescribing the value 1 at a. We have brought this term into the form \(\mathop \mu \nolimits_r^0\)(r, s) + (constant) · \(\mathop \mu \nolimits_r^0\)(r, —s) by multiplying it with a suitable meromorphic factor. We have used a functional analytic argument to show that this factor does not vanish identically.
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