It is known that the famous mistake of Iwaniec-Sarnak in their [paper](http://www.jstor.org/stable/pdfplus/2118522.pdf?&acceptTC=true&jpdConfirm=true) of $L^\infty$ norm of eigenfunction of non-cocompact arithmetic surfaces in lemma (A1) is because of they did not consider the bump of $K$-Bessel function at transition range.


1. First, I do not understand which part of the proof (sketched below) is flawed.
An eigenfunction of $SL\_2(\mathbb{Z})\backslash\mathcal{H}^2$ with eigenvalue $1/4+t^2$ looks like $$\phi(z)=\sqrt{y}\sum\_{n\neq0}a\_nK\_{it}(2\pi|n|y)e(nx).$$ First they proved a bound of sum of Fourier coefficients (Ramanujan on average) which is $(t>0)$
$$
\sum\_{n\le N}|a\_n|^2\ll e^{\pi t}(t+N).
$$
Then, I guess that they are using a Cauchy-Schwarz inequality and an asymptote that for $t>2\pi|n|y$ $$e^{\pi t}|K\_{it}(2\pi|n|y)|^2\ll (t^2-|n|^2y^2)^{-1/2},$$to deduce final final bound. The above inequality can be found [here lemma 2.3 and 2.4](http://projecteuclid.org/download/pdf_1/euclid.em/1048610117). Isn't the above asymptote correct?


2)It is mentioned in [Sarnak's letter to Morawetz (equation 43)](http://publications.ias.edu/sites/default/files/Sarnak_Letter_to_Morawetz.pdf) that $K\_{it}$ has a bump of size $t^{1/6}$ near $y=t$. Do we know precise asymptote of $K\_{it}$ uniformly in $y$ and $t$ so that the above argument can be rectified?



Asymptotic behaviour of $K$-Bessel function in transition range
For a published account of the corrected proof, see Section 10 in Blomer-Holowinsky: Bounding sup-norms of cusp forms of large level, Invent. Math. 179 (2010), 645-681. See especially pages 679-680, where you can also find the precise asymptotics of $K\_{it}$ in the transitional range.


Actually, a few years ago, a colleague of mine asked me about the same thing. What follows is my email response in slightly edited form (I am too busy to re-read carefully):


There are several problems with Lemma A.1 and its proof. The problems arose from not using correctly the asymptotics of $K\_{ir}(y)$ in the transitional range $y\approx r$. For $|y-r|<r^{1/3}$ the function is about $e^{-\pi r/2} r^{-1/3}$ times a phase, while for $|y-r|>r^{1/3}$ it is about $e^{-\pi r/2} |y^2-r^2|^{-1/4}$ times a phase, with the additional remark that for $y$ large the exponential decay kicks in. As a consequence:


1. The display below (A.2) is wrong: $r^{-1}$ should be lowered to $r^{-4/3}$. This is not a big problem, since if we integrate from $r/2$ (instead from $r$), then the bound
is OK (I learned these things from papers by Strömbergsson). In other words, in the line before (A.3) one should set $N=r/(4\pi Y)$. Actually (A.3) is too weak for this
purpose, as explained below. Instead the following should be used, which is known by deeper methods:
$$ \sum\_{0<|n|<N} |\rho(n)|^2 \ll e^{\pi r} r^\epsilon N. \tag{1}$$
2. The last display on p.316 is wrong. In fact for fixed $b\geq 0$ one has
$$ \sum\_{n\neq 0} |n|^b |K\_{ir}(2\pi|n|y)|^2 \ll e^{-\pi r} (r/y)^b (r^{-2/3}+1/y), $$
and this is best possible for $2\pi y < r+O(1)$, e.g. the first term in the parantheses comes from $2\pi|n|y$ very close to $r$. This shows that the last display on p.316
should read
$$ |\phi(z)|^2 \ll Y^{-2} (Y+r) (y^{-1}r+Y)^2 (yr^{-2/3}+1). $$
For the choice $Y=r/(2\pi y)$ this gives
$$ |\phi(z)|^2 \ll (Y+r) (yr^{-2/3}+1), $$
which is too weak eventually. Using (1) above in place of (A.3), one gets
$$ |\phi(z)|^2 \ll r^\epsilon Y (yr^{-2/3}+1), $$
i.e.,
$$ \phi(z) \ll r^\epsilon (r^{1/6} + (r/y)^{1/2}) {\|\phi\|}\_2. \tag{2}$$
This is a bit weaker than Lemma A.1. In fact this is the correct version of Lemma A.1 since $\max\_{0\leq x\leq 1}|\phi(x+ir/(2\pi))|$ is about $r^{1/6}{\|\phi\|}\_2$.


Much the same correction is explained on pp.38-40 of [Sarnak's letter to Morawetz](http://publications.ias.edu/sites/default/files/Sarnak_Letter_to_Morawetz.pdf).


A bit more detail is given in the Blomer-Holowinsky paper, on p.679. Actually, the bound on top of p.680 is only justified for $t>1$ and $y>1$, but it is valid for all
$y>0$ and all forms (including forms violating the Selberg conjecture).


**Added.** For a generalization of (2) to spherical Hecke-Maass forms over an arbitrary number field, see Lemma 9 in [this paper](https://arxiv.org/abs/1605.09360). Note that $\phi$ in that lemma satisfies ${\|\phi\|}\_2=1$. Note also that (1) above was (\*) in an earlier version of this post.


