POINTWISE BOUNDS FOR EISENSTEIN SERIES ON G0(q)\SL2(R)

We construct pointwise bounds in the weight aspect for Eisenstein series on X-0(q) = Gamma(0)(q)\SL2(R), with squarefree level q, using a Sobolev technique. More specifically, we show that for an Eisenstein series E on X-0(q) of weight parameter n and type t, one has for all x is an element of X-0(q): |E(x,1/2 + it)| <<(epsilon) q(epsilon)(1 + |n|(1/2+epsilon) + |t|(1/2+epsilon) root y(x) + y(x)(-1), where y(x) is the Iwasawa y-coordinate of the point x.