Asymptotic behaviour of $K$-Bessel function in transition range

Question

It is known that the famous mistake of Iwaniec-Sarnak in their paper 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. Isn't the above asymptote correct?

2)It is mentioned in Sarnak's letter to Morawetz (equation 43) 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?

Answer 1

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.

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. Note that $\phi$ in that lemma satisfies ${\|\phi\|}_2=1$. Note also that (1) above was (*) in an earlier version of this post.