Restriction of Hecke eigenforms to horocycles

TL;DR

This paper establishes a precise upper bound on the L^2-norm of Hecke eigenforms when restricted to a horocycle, especially as the eigenform's weight increases.

Contribution

It provides a sharp upper bound on the restriction of Hecke eigenforms to horocycles, advancing understanding of their behavior at high weights.

Findings

Established a sharp upper bound on L^2-norms

Analyzed eigenform behavior as weight tends to infinity

Contributed to spectral theory of automorphic forms

Abstract

We prove a sharp upper bound on the $L^2$-norm of Hecke eigenforms restricted to a horocycle, as the weight tends to infinity.

Equations

$${\int }_X{y}^k|f(z){|}^2\frac{dxdy}{y^2}=1.$$

$${\int }_0^1{y}^k|f(z){|}^{2}dx\le C(\delta ),$$

$${\int }_0^1{y}^k|f(z){|}^{2}dx\to \frac{3}{\pi }.$$

$${\int }_0^1{y}^{k}f(x+iy)\overline{g(x+iy)}dx\to 0.$$

$$L(s,f)=\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)}{n^{s}}=\prod_{p}\Big{(}1-\frac{\alpha_{p}}{p^{s}}\Big{)}^{-1}\Big{(}1-\frac{\beta_{p}}{p^{s}}\Big{)}^{-1},$$

$$L(s,\operatorname{sym}^{2}f)=\zeta(2s)\sum_{n=1}^{\infty}\frac{\lambda_{f}(n^{2})}{n^{s}}=\prod_{p}\Big{(}1-\frac{\alpha_{p}^{2}}{p^{s}}\Big{)}^{-1}\Big{(}1-\frac{1}{p^{s}}\Big{)}^{-1}\Big{(}1-\frac{\beta_{p}^{2}}{p^{s}}\Big{)}^{-1}.$$

$$\Lambda (s,{\mathrm{sym}}^{2}f)={\Gamma }_{\mathbb{R}}(s+1){\Gamma }_{\mathbb{R}}(s+k-1){\Gamma }_{\mathbb{R}}(s+k)L(s,{\mathrm{sym}}^{2}f)=\Lambda (1-s,{\mathrm{sym}}^{2}f).$$

$$\mathcal{R}=\Big{\{}s=\sigma+it:\sigma\geq 1-\frac{c}{\log k(1+|t|)}\Big{\}}$$

$$|\{\rho =\beta +i\gamma :L(\rho ,{\mathrm{sym}}^{2}f)=0,\beta \ge \alpha ,|\gamma |\le T\}|\le C(T+1{)}^B{k}^{c(1-\alpha )}.$$

$${\int }_X{y}^k|f(z){|}^2\frac{dxdy}{y^2}=1.$$

$$f(z)=C_f\sum _{n=1}^{\infty }{\lambda }_f(n)(4\pi n{)}^{\frac{k-1}{2}}e(nz),$$

$$C_{f}=\Big{(}\frac{2\pi^{2}}{\Gamma(k)L(1,\operatorname{sym}^{2}f)}\Big{)}^{1/2}.$$

$$L(1,\operatorname{sym}^{2}f)\asymp\exp\Big{(}\sum_{p\leq k}\frac{\lambda_{f}(p^{2})}{p}\Big{)}.$$

$$\frac{1}{2\pi i}{\int }_{c-\infty }^{c+\infty }-\frac{L^{\prime }}{L}(s+\sigma ,{\mathrm{sym}}^{2}f)(s+1)\Gamma (s)x^{s}ds,$$

$$-\frac{L^{\prime }}{L}(s,{\mathrm{sym}}^{2}f)=\sum _{n=1}^{\infty }\frac{{\Lambda }_{{\mathrm{sym}}^{2}f}(n)}{n^s}$$

$${\Lambda }_{{\mathrm{sym}}^{2}f}(p^k)=({\alpha }_p^{2k}+1+{\beta }_p^{2k})\log p,$$

$$\frac{1}{2\pi i}\int_{(c)}(s+1)\Gamma(s)y^{s}ds=e^{-1/y}\Big{(}1+\frac{1}{y}\Big{)},$$

$$\frac{1}{2\pi i}{\int }_{(c)}-\frac{L^{\prime }}{L}(s+\sigma ,{\mathrm{sym}}^{2}f)(s+1)\Gamma (s)x^{s}ds$$

$$-\frac{L^{\prime }}{L}(\sigma ,{\mathrm{sym}}^{2}f)-\sum _{\rho }{x}^{\rho -\sigma }(\rho -\sigma +1)\Gamma (\rho -\sigma )+\frac{1}{2\pi i}{\int }_{(-3/2)}-\frac{L^{\prime }}{L}(s+\sigma ,{\mathrm{sym}}^{2}f)x^s(s+1)\Gamma (s)ds.$$

$$-\frac{L^{\prime }}{L}(s+\sigma ,{\mathrm{sym}}^{2}f)\ll \log (k(1+|t|)+\left|\frac{L^{\prime }}{L}(1-s-\sigma ,{\mathrm{sym}}^{2}f)\right|\ll \log (k(1+|t|)).$$

$$\sum_{n}\frac{\Lambda_{\operatorname{sym}^{2}f}(n)}{n^{\sigma}}e^{-n/x}\Big{(}1+\frac{n}{x}\Big{)}=-\frac{L^{\prime}}{L}(\sigma,\operatorname{sym}^{2}f)-\sum_{\rho}x^{\rho-\sigma}(\rho+1-\sigma)\Gamma(\rho-\sigma)+O(x^{-3/2}\log k).$$

$$|\rho -\sigma +1||\Gamma (\rho -\sigma )|\ll (\sigma -\beta {)}^{-1}{e}^{-n}.$$

$$\ll \sum _{n\le |\gamma |<n+1}\frac{x^{\beta -\sigma }}{\sigma -\beta }{e}^{-n}.$$

$$\ll e^{-n}\sum _{j=0}^{\log k}\frac{x^{1-\sigma -j/\log k}}{(j+1)/\log k}|\{\beta +i\gamma :1-(j+1)/\log k\le \beta <1-j/\log k,n\le |\gamma |<n+1\}.$$

$$\ll e^{-n}{x}^{1-\sigma }\log k\sum _{j=0}^{\log k}\frac{e^{-jA}}{j+1}(n+1{)}^B{k}^{c(j+1)/\log k}\ll (n+1{)}^B{e}^{-n}{x}^{1-\sigma }\log k,$$

$$\log L(1,\operatorname{sym}^{2}f)=\sum_{n=2}^{\infty}\frac{\Lambda_{\operatorname{sym}^{2}f}(n)}{n\log n}e^{-n/x}\Big{(}1+\frac{n}{x}\Big{)}+O(1)=\sum_{p\leq x}\frac{\lambda_{f}(p^{2})}{p}+O(1),$$

$$\sum_{p\leq x}\frac{1}{p}\Big{|}1-e^{-p/x}\Big{(}1+\frac{p}{x}\Big{)}\Big{|}+\sum_{p>x}\frac{1}{p}e^{-p/x}\Big{(}1+\frac{p}{x}\Big{)}=O(1).$$

$$L(1,\operatorname{sym}^{2}f)\asymp\exp\Big{(}\sum_{p\leq x}\frac{\lambda_{f}(p^{2})}{p}\Big{)}\asymp\exp\Big{(}\sum_{p\leq k}\frac{\lambda_{f}(p^{2})}{p}\Big{)},$$

$${\int }_0^1{y}^k|f(z){|}^{2}dx$$

$$\ll \frac{1}{\Gamma (k)L(1,{\mathrm{sym}}^{2}f)}\sum _{n=1}^{\infty }\frac{{\lambda }_f(n{)}^2}{n}(4\pi ny{)}^k{e}^{-4\pi ny}.$$

$$\frac{\xi^{k}e^{-\xi}}{\Gamma(k)}\asymp\sqrt{k}\Big{(}\frac{\xi}{k}\Big{)}^{k}e^{k-\xi}\ll\begin{cases}\sqrt{k}\exp(-(k-\xi)^{2}/(4k))&\text{if }\xi\leq 2k\\ \sqrt{k}(e/2)^{k-\xi}&\text{if }\xi>2k,\end{cases}$$

Full text

The second author is supported in part by a grant from the National Science Foundation, and a Simons Investigator award from the Simons Foundation

1. Introduction

A central problem in “quantum chaos” is to understand the limiting behavior of eigenfunctions. An important example that has attracted a lot of attention is that of Maass cusp forms with large Laplace eigenvalue on the modular surface $X=SL_{2}(\mathbb{Z})\backslash\mathbb{H}$. Let $\phi$ denote such a Maass form, with eigenvalue $\lambda$, and normalized to have $L^{2}$-norm $1$: that is, $\int_{X}|\phi(z)|^{2}\frac{dxdy}{y^{2}}=1$. Then the Quantum Unique Ergodicity (QUE) conjecture of Rudnick and Sarnak [15] states that the measure $\mu_{\phi}=|\phi(z)|^{2}\frac{dxdy}{y^{2}}$ tends to the uniform measure on $X$ as $\lambda\to\infty$. If $\phi$ is also assumed to be an eigenfunction of all the Hecke operators, then QUE holds by the work of Lindenstrauss [12], with a final step on escape of mass provided by Soundararajan [19]. Thus, the measure $\mu_{\phi}$ does not concentrate on subsets of $X$ with small measure, but is uniformly spread out. A finer problem is to understand how much the measure can concentrate on sub-manifolds; for example, on a geodesic, or a closed horocyle, or even at just a point (that is, bounding the $L^{\infty}$ norm). The letter of Sarnak to Reznikov [16] draws attention to such restriction problems, and these problems (and generalizations) have been studied extensively in recent years, see for example [1], [2], [4], [10], [11], [20], [21], [22].

This note is concerned with a related question for holomorphic modular forms for $SL_{2}({\mathbb{Z}})$ that are also eigenfunctions of all Hecke operators, when the weight $k$ becomes large. Let $f$ be a Hecke eigenform of weight $k$ on the modular surface $X$, with $L^{2}$-norm $1$: that is,

[TABLE]

To $f$, we associate the measure $\mu_{f}=y^{k}|f(z)|^{2}\frac{dxdy}{y^{2}}$. The analog here of QUE states that $\mu_{f}$ tends to the uniform measure $\frac{3}{\pi}\frac{dxdy}{y^{2}}$ as $k\to\infty$, and this is known to hold by the work of Holowinsky and Soundararajan [6]. As with Maass forms, one may now ask for finer restriction theorems for holomorphic Hecke eigenforms. We study the problem of bounding the $L^{2}$-norm of Hecke eigenforms on a fixed horocycle, and establish the following uniform bound.

Theorem 1**.**

Let $f$ be a Hecke eigenform of weight $k$ on $X=SL_{2}(\mathbb{Z})\backslash\mathbb{H}$ with $L^{2}$-norm normalized to be $1$. Let $\delta>0$ be fixed. Uniformly in the range $1/k\leq y\leq k^{1/2-\delta}$ we have

[TABLE]

for some constant $C(\delta)$.

Our result gives a uniform bound for the $L^{2}$-norm restricted to horocycles, answering a question from Sarnak [16]. In the Maass form situation, Ghosh, Reznikov and Sarnak [4] establish weaker restriction bounds (of size $\lambda^{\epsilon}$) for the corresponding problem, and Sarnak [16] notes that uniform boundedness there follows from the Ramanujan conjecture and a sub-convexity bound (in eigenvalue aspect) for the Rankin-Selberg $L$-function $L(s,\phi\times\phi)$. One might hope to strengthen and extend Theorem 1 in the following two ways. First, Young [22, Conjecture 1.4] has conjectured that for any fixed $y>0$, the restriction of $\mu_{f}$ to the horocycle $[0,1]+iy$ still tends to the uniform measure, as $k\to\infty$: in particular, as $k\to\infty$

[TABLE]

Second, one might expect that two different eigenforms $f$ and $g$ of weight $k$ are approximately orthogonal on the horocycle $[0,1]+iy$, so that (as $k\to\infty$)

[TABLE]

Our proof, which relies crucially on bounds for mean-values of non-negative multiplicative functions in short intervals, does not allow us to address these refined conjectures.

2. Preliminaries

Let $f$ be a Hecke eigenform of weight $k$ on $X=SL_{2}(\mathbb{Z})\backslash\mathbb{H}$. Write

[TABLE]

where $\lambda_{f}(n)$ are the Hecke eigenvalues for $f$, and $\alpha_{p}$, $\beta_{p}=\alpha_{p}^{-1}$ are the Satake parameters. Our $L$-function has been normalized such that the Deligne bound reads $|\lambda_{f}(n)|\leq d(n)$ (the divisor function), or equivalently that $|\alpha_{p}|=|\beta_{p}|=1$.

The symmetric square $L$-function $L(s,\operatorname{sym}^{2}f)$ is defined by

[TABLE]

From the work of Shimura [17] we know that $L(s,\operatorname{sym}^{2}f)$ has an analytic continuation to the entire complex plane, and satisfies a functional equation connecting $s$ and $1-s$: namely, with $\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2)$,

[TABLE]

Moreover, Gelbart and Jacquet [3] have shown that $L(s,\operatorname{sym}^{2}f)$ arises as the $L$-function of a cuspidal automorphic representation of $GL(3)$. Invoking the Rankin-Selberg $L$-function attached to $\operatorname{sym}^{2}f$, a standard argument establishes the classical zero-free region for $L(s,\operatorname{sym}^{2}f)$, with the possible exception of a real Landau-Siegel zero (see Theorem 5.42 of [8]). The work of Hoffstein and Lockhart [5] (especially the appendix by Goldfeld, Hoffstein and Lieman) has ruled out the existence of Landau-Siegel zeroes for this family. Thus, for a suitable constant $c>0$, the region

[TABLE]

does not contain any zeroes of $L(s,\operatorname{sym}^{2}f)$ for any Hecke eigenform $f$ of weight $k$.

Lastly, we shall need a “log-free” zero-density estimate for this family, which follows from the work of Kowalski and Michel (see [9], and also the recent works of Lemke Oliver and Thorner [14], and Motohashi [13]).

Lemma 2**.**

There exist absolute constants $B$, $C$, and $c$ such that for all $1/2\leq\alpha\leq 1$, and any $T$ we have

[TABLE]

The special value $L(1,\operatorname{sym}^{2}f)$ shows up naturally when comparing the $L^{2}$ normalization and Hecke normalization of a modular form. Suppose $f$ has been normalized in such a way that

[TABLE]

Then the Fourier expansion of $f(z)$ is given by (see, for example, Chapter 13 of [7])

[TABLE]

where

[TABLE]

We can now state our main lemma, which refines Lemma 2 of [6], and allows us to estimate $L(1,\operatorname{sym}^{2}f)$ by a suitable Euler product. Below we use the notation $g\asymp h$ to denote $g\ll h$ and $h\ll g$.

Lemma 3**.**

For any Hecke eigenform $f$ of weight $k$ for the full modular group, we have

[TABLE]

Recall that $g\asymp h$ means $g\ll h$ and $h\ll g$.

Proof.

Let $1\leq\sigma\leq\frac{5}{4}$, and consider for some $c>0$ and $x\geq 1$, the integral

[TABLE]

which we shall evaluate in two ways. Here we shall take $x=k^{A}$ for a suitably large constant $A$. On one hand, we write

[TABLE]

where $\Lambda_{\operatorname{sym}^{2}f}(n)=0$ unless $n=p^{k}$ is a prime power, in which case

[TABLE]

so that $|\Lambda_{\operatorname{sym}^{2}f}(n)|\leq 3\Lambda(n)$ for all $n$. Using this in (2), and integrating term by term, using

[TABLE]

we obtain

[TABLE]

On the other hand, shift the line of integration in (2) to $\,\mathrm{Re}\,(s)=-3/2$. We encounter poles at $s=0$, and at $s=\rho-\sigma$ for non-trivial zeroes $\rho=\beta+i\gamma$ of $L(s,\operatorname{sym}^{2}f)$. Computing these residues, we see that (2) equals

[TABLE]

Differentiate the functional equation of $L(s,\operatorname{sym}^{2}f)$ logarithmically, and use Stirling’s formula. Thus with $s=-\frac{3}{2}+it$ we obtain that

[TABLE]

Therefore the integral in (4) may be bounded by $O((\log k)x^{-3/2})$, and we conclude that

[TABLE]

We now bound the sum over zeros in (5). Write $\rho=\beta+i\gamma$, and split into terms with $n\leq|\gamma|<n+1$, where $n=0$, $1$, $2$, $\ldots$. If $n\leq|\gamma|<n+1$, we may check using the exponential decay of the $\Gamma$-function that

[TABLE]

Therefore the contribution of zeros from this interval is

[TABLE]

Splitting the zeros further based on $1-(j+1)/\log k\leq\beta<1-j/\log k$ (and using the zero free region, so that $\sigma-\beta\gg(j+1)/\log k$) the above is

[TABLE]

Now using the log-free zero density estimate from Lemma 2, and recalling that $x=k^{A}$, the quantity above is

[TABLE]

provided $A\geq c+1$ is large enough. Now summing over $n$, we conclude that the sum over zeros in (5) is $\ll x^{1-\sigma}\log k$.

Use this bound in (5), and integrate that expression over $1\leq\sigma\leq 5/4$. It follows that

[TABLE]

since the contribution of prime powers above is easily seen to be $O(1)$, and since

[TABLE]

Exponentiating, we obtain

[TABLE]

since $x=k^{A}$, and $\sum_{k<p\leq k^{A}}1/p\ll 1$. This concludes our proof. ∎

3. Proof of Theorem 1

The Fourier expansion (1) and the Parseval formula give

[TABLE]

For $\xi\geq 0$, note that

[TABLE]

where the first bound follows because $\log(1+t)\leq t-t^{2}/4$ for $|t|\leq 1$ (with $t=(\xi-k)/k$), and the second bound from $\log(1+t)\leq t\log 2$ for $t\geq 1$.

The estimate (7) with $\xi=4\pi ny$ shows that the sum in (3) is concentrated around values of $n$ with $|4\pi ny-k|$ about size $\sqrt{k}$. To flesh this out, let us first show that the contribution to (3) from $n$ with $4\pi ny\geq 2k$ is negligible. Using the second bound in (7), such terms $n$ contribute (using that $L(1,\operatorname{sym}^{2}f)\gg(\log k)^{-1}$, which follows from Lemma 3 or [5])

[TABLE]

This contribution to (3) is clearly negligible.

It remains to handle the contribution from those $n$ with $4\pi ny\leq 2k$. Divide such $n$ into intervals of the form $j\sqrt{k}\leq|4\pi ny-k|<(j+1)\sqrt{k}$, where $0\leq j\ll\sqrt{k}$. We use the first bound in (7) with $\xi=4\pi ny$, and in the range $j\sqrt{k}\leq|4\pi ny-k|<(j+1)\sqrt{k}$ this gives

[TABLE]

provided $y\geq 1/k$ say. Thus the contribution from the terms $j\sqrt{k}\leq|4\pi ny-k|<(j+1)\sqrt{k}$ is

[TABLE]

At this stage, we appeal to a result of Shiu (see Theorem 1 of [18]) bounding averages of non-negative multiplicative functions in short intervals.

Lemma 4**.**

Let $g$ be a non-negative multiplicative function with (i) $g(p^{l})\leq A^{l}$ for some constant $A$, and (ii) $g(n)\ll_{\epsilon}n^{\epsilon}$ for any $\epsilon>0$. Then for any $\delta>0$, if $x^{\delta}\leq z\leq x$, we have

[TABLE]

Applying this lemma in (8), in the range $y\leq k^{1/2-\delta}$, we may bound that quantity by

[TABLE]

Since $\lambda_{f}(p)^{2}=\lambda_{f}(p^{2})+1$, the above bound when combined with Lemma 3 yields $\ll e^{-j^{2}/8}$, and summing this over all $j$ gives $\ll 1$. Thus we conclude that the quantity in (3) is bounded, completing the proof of our theorem.