Quantitative quantum ergodicity and the nodal domains of Hecke-Maass cusp forms. (English) Zbl 1388.58021
Let \(\mathbb{X}=\Gamma\backslash\mathbb{H}\) be the modular surface, where \(\mathbb{H}\) is the Poincaré upper half-plane and \(\Gamma=SL_2(\mathbb{Z})\) is the modular group. Let \(\phi\) be an \(L^2\)-normalized Hecke-Maaß cusp form on \(\mathbb{X}\). It is an eigenform for the Laplace-Beltrami operator on \(\mathbb{X}\) and all the Hecke operators. Write \(\lambda_\phi=1/4+t_\phi^2\) for the eigenvalue of \(\phi\) for the Laplace-Beltrami operator on \(\mathbb{X}\).
The map \(x+iy\mapsto -x+iy\) induces an orientation reversing isometric involution \(\sigma\) on \(\mathbb{X}\). Since \(\sigma\) commutes with the Laplace–Beltrami operator and all the Hecke operators, the form \(\phi\) is an eigenfunction for \(\sigma\). Thus, \(\sigma\phi=\pm\phi\), and \(\phi\) is called even if the sign is positive.
Write the Fourier expansion of the form \(\phi\) at the cusp \(\infty\) as \[ \phi(z)=\sqrt{\cosh(\pi t_\phi)} \sum_{n\neq 0}\rho_\phi(n)\sqrt{y}K_{i{t_\phi}}(2\pi|n|y)e^{2\pi inx}, \] where \(z=x+iy\in\mathbb{H}\), \(\rho_\phi(n)\) are the Fourier coefficients of \(\phi\), and \(K_{it}(y)\) is the modified Bessel function of the second kind. For a fixed \(m\in\mathbb{Z}\), the shifted convolution sum of Fourier coefficients of \(\phi\) is defined as \[ \frac{1}{t_\phi}\sum_{n}\rho_\phi(n+m)\rho_\phi(n)\psi\left(\frac{\pi |n|}{t_\phi}\right), \] where \(\psi\) is a smooth compactly supported test function on \((0,\infty)\).
The main result of the paper estimates the variance of shifted convolution sums of Fourier coefficients of \(\phi\) over the range \(| t_{\phi} -T | < G\), where \(G\) is assumed to be a small power of \(T\). More precisely, let \(1/3<\theta <1\) and \(\epsilon >0\) be fixed. Assume a test function \(\psi\) is supported in the interval \((1/l,l)\) for some fixed \(l>1\). Let \(X\) be a parameter satisfying \(1\ll X\ll T\). Then, there is a constant \(A>0\), depending only on \(\theta\) and \(\epsilon\), such that \[ \begin{aligned} \sum_{|t_\phi -T| < T^\theta} \left| \sum_n \rho_\phi(n+m)\rho_\phi(n)\psi\left(\frac{\pi n}{X}\right) -\delta_{0,m}\frac{12X}{\pi^3}\int_0^\infty \psi(y)dy \right|^2\\ \ll_{\epsilon,\theta,l} \left( |m|^{3/2}+1 \right) XT^{1+\theta+\epsilon}\|\psi \|_{W^{A,\infty}}^2 \end{aligned} \] holds uniformly in \(|m| < X^{1/2}\) as \(T\) tends to \(\infty\), where \(\delta_{0,m}\) is the Kronecker symbol, i.e. \(\delta_{0,m}\) is one if \(m=0\) and zero otherwise, and \(\|\cdot\|_{W^{A,\infty}}\) is a Sobolev space norm defined as \[ \|\psi \|_{W^{A,\infty}}=\sum_{j=0}^A\sup_{x\in (0,\infty)}\left|\partial_x^j\psi(x)\right|. \] Such estimate for holomorphic Hecke eigenforms is obtained by W. Luo and P. Sarnak [Commun. Pure Appl. Math. 56, No. 7, 874–891 (2003; Zbl 1044.11022)].
The proof of the main result is based on writing the variance of shifted convolution sums over \(|t_\phi -T| < T^\theta\) as an exponential sum involving Kloosterman sums using N. V. Kuznetsov trace formula [Mat. Sb., N. Ser. 111(153), 334–383 (1980; Zbl 0427.10016)] for a convenient test function. Then, the rest of the proof is highly involved by technical estimation of the diagonal and off-diagonal terms in the formula.
The main result has several important implications. Recall that the arithmetic Quantum Unique Ergodicity (QUE) theorem of E. Lindenstrauss [Ann. Math. (2) 163, No. 1, 165–219 (2006; Zbl 1104.22015)] and K. Soundararajan [Ann. Math. (2) 172, No. 2, 1529–1538 (2010; Zbl 1209.58019)] deals with the weak limit of the measure \(d\mu_\phi=|\phi(z)|^2dV\) as \(t_\phi\) tends to \(\infty\), where \(dV\) is the measure on \(\mathbb{X}\) induced from the hyperbolic measure \(y^{-2}dxdy\) on \(\mathbb{H}\). The Quantitative Quantum Ergodicity refers to the rate of convergence in QUE. This is all closely related to the limits of shifted convolution sums of Fourier coefficients as \(t_\phi\to\infty\). The first consequence of the main result in this paper is Quantitative Quantum Ergodicity in a short range \(|t_\phi -T| < T^{1/3}\). Let \(\epsilon >0\) be fixed, and let \(h\) be a smooth compactly supported test function with the support in \((1/L,L)\) for some \(L>1\). Then, there is a sufficiently small \(\kappa >0\) and a sufficiently large \(A>0\), both depending only on \(\epsilon\), such that \[ \sum_{|t_\phi-T| < T^{1/3}} \left| \int_\mathbb{X} P_{m,h}(z)|\phi(z)|^2dV - \frac{3}{\pi}\int_\mathbb{X} P_{m,h}(z)dV \right|^2 \ll_{\epsilon,L} T^{1/3+\epsilon}||h||_{W^{A,\infty}} \] holds uniformly in \(h\) and \(|m| < T^\kappa\), where \(P_{m,h}\) is the Poincaré series attached to the integer \(m\) and the test function \(h\).
The second consequence of the main result is concerned with \(L^2\)-norm of \(\phi\) restricted to compact geodesic segment on the imaginary axis \(\{iy\,:\,y>0\}\). Let \(\beta\) be a fixed compact geodesic segment of \(\{iy\,:\,y>0\}\). As an application of QUE, A. Ghosh et al. [Geom. Funct. Anal. 23, No. 5, 1515–1568 (2013; Zbl 1328.11044)] obtained a lower bound for the \(L^2\)-norm restriction \[ \int_\beta |\phi(z)|^2 ds \gg_\beta 1, \] where \(\phi\) is an even Hecke-Maaß cusp form, and \(\beta\) is sufficiently large. Quantitative Quantum Ergodicity result of this paper implies a sharp lower bound for any fixed geodesic \(\beta\) and almost all even Hecke-Maaß cusp forms. More precisely, let \(\epsilon >0\) be fixed. Then, for any fixed compact geodesic segment \(\beta\subset\{iy\,:\,y>0\}\), \[ \int_\beta |\phi(z)|^2 ds \gg_{\beta,\epsilon} 1 \] holds for all but \(O_{\beta,\epsilon}\left(T^{1/3+\epsilon}\right)\) forms in the set of even Hecke-Maaß cusp forms with \(|t_\phi-T| < T^{1/3}\), as \(T\) tends to \(\infty\).
The third consequence of the main result is an upper bound for the \(L^\infty\)-norm of Hecke-Maaß cusp forms. Using the Selberg trace formula and amplification method, H. Iwaniec and P. Sarnak [Ann. Math. (2) 141, No. 2, 301–320 (1995; Zbl 0833.11019)] obtained for any compact subset \(C\) of \(\mathbb{X}\) an upper bound \[ \sup_{z\in C} |\phi(z)|\ll_{C,\epsilon} t_\phi^{\frac{5}{12}+\epsilon}, \] where \(\phi\) is a Hecke-Maaß cusp form. The main result of this paper enables the author to find better amplifiers and thus improve the upper bound for almost all Hecke-Maaß cusp forms. Let \(C\) be a fixed compact subset of \(\mathbb{X}\), and let \(\epsilon >0\) be fixed. Then, all but \(O_\epsilon\left(T^{\frac{13}{12}+\epsilon}\right)\) Hecke-Maaß cusp forms \(\phi\) with \(|t_\phi-T| < T^{1/3}\) satisfy the upper bound \[ \sup_{z\in C}|\phi(z)|\ll_{C,\epsilon} t^{\frac{3}{8}+\epsilon} \] as \(T\) tends to \(\infty\).
Finally, the fourth consequence of the main result is concerned with the number of nodal domains intersecting a segment of the imaginary axis \(\{iy\,:\,y>0\}\). For a Hecke-Maaß cusp form \(\phi\), let \(Z_\phi\) denote its zero set. It is a finite union of real analytic curves. A nodal domain of \(\phi\) is, by definition, any connected component in \(\mathbb{X}\setminus Z_\phi\). Given any subset \(C\) of \(\mathbb{X}\), the number of nodal domains of \(\phi\) intersecting \(C\) is denoted by \(N^C(\phi)\). Assuming the Lindelöf Hypothesis for the \(L\)-function \(L(s,\phi)\), A. Ghosh et al. [Geom. Funct. Anal. 23, No. 5, 1515-1568 (2013; Zbl 1328.11044)] obtained a lower bound for the growth rate of \(N^\beta(\phi)\), where \(\beta\) is a fixed compact geodesic segment in \(\{iy\,:\,y>0\}\) which is sufficiently long. The bound is given by \[ N^{\beta}(\phi)\gg_{\beta,\epsilon} t_\phi^{\frac{1}{12}-\epsilon} \] as \(t_\phi\) tends to \(\infty\). Recently, S. U. Jang and the author [J. Am. Math. Soc. 31, No. 2, 303–318 (2018; Zbl 1385.58014)] showed that \(N^\beta(\phi)\) tends to \(\infty\) without any assumption. Here, the quantitative lower bound is obtained for almost all even Hecke-Maaß cusp forms. More precisely, let \(\beta\) be a fixed compact geodesic segment in \(\{iy\,:\,y>0\}\), and \(\epsilon >0\) fixed. Then, all but \(O\left(T^{\frac{4}{3}-\frac{\epsilon}{2}}\right)\) forms in the set of even Hecke-Maaß cusp forms with \(|t_\phi-T|<T^{1/3}\) satisfy \[ N^{\beta}(\phi)\gg_{\beta,\epsilon} t_\phi^{\frac{1}{8}-\epsilon} \] as \(T\) tends to \(\infty\).
The map \(x+iy\mapsto -x+iy\) induces an orientation reversing isometric involution \(\sigma\) on \(\mathbb{X}\). Since \(\sigma\) commutes with the Laplace–Beltrami operator and all the Hecke operators, the form \(\phi\) is an eigenfunction for \(\sigma\). Thus, \(\sigma\phi=\pm\phi\), and \(\phi\) is called even if the sign is positive.
Write the Fourier expansion of the form \(\phi\) at the cusp \(\infty\) as \[ \phi(z)=\sqrt{\cosh(\pi t_\phi)} \sum_{n\neq 0}\rho_\phi(n)\sqrt{y}K_{i{t_\phi}}(2\pi|n|y)e^{2\pi inx}, \] where \(z=x+iy\in\mathbb{H}\), \(\rho_\phi(n)\) are the Fourier coefficients of \(\phi\), and \(K_{it}(y)\) is the modified Bessel function of the second kind. For a fixed \(m\in\mathbb{Z}\), the shifted convolution sum of Fourier coefficients of \(\phi\) is defined as \[ \frac{1}{t_\phi}\sum_{n}\rho_\phi(n+m)\rho_\phi(n)\psi\left(\frac{\pi |n|}{t_\phi}\right), \] where \(\psi\) is a smooth compactly supported test function on \((0,\infty)\).
The main result of the paper estimates the variance of shifted convolution sums of Fourier coefficients of \(\phi\) over the range \(| t_{\phi} -T | < G\), where \(G\) is assumed to be a small power of \(T\). More precisely, let \(1/3<\theta <1\) and \(\epsilon >0\) be fixed. Assume a test function \(\psi\) is supported in the interval \((1/l,l)\) for some fixed \(l>1\). Let \(X\) be a parameter satisfying \(1\ll X\ll T\). Then, there is a constant \(A>0\), depending only on \(\theta\) and \(\epsilon\), such that \[ \begin{aligned} \sum_{|t_\phi -T| < T^\theta} \left| \sum_n \rho_\phi(n+m)\rho_\phi(n)\psi\left(\frac{\pi n}{X}\right) -\delta_{0,m}\frac{12X}{\pi^3}\int_0^\infty \psi(y)dy \right|^2\\ \ll_{\epsilon,\theta,l} \left( |m|^{3/2}+1 \right) XT^{1+\theta+\epsilon}\|\psi \|_{W^{A,\infty}}^2 \end{aligned} \] holds uniformly in \(|m| < X^{1/2}\) as \(T\) tends to \(\infty\), where \(\delta_{0,m}\) is the Kronecker symbol, i.e. \(\delta_{0,m}\) is one if \(m=0\) and zero otherwise, and \(\|\cdot\|_{W^{A,\infty}}\) is a Sobolev space norm defined as \[ \|\psi \|_{W^{A,\infty}}=\sum_{j=0}^A\sup_{x\in (0,\infty)}\left|\partial_x^j\psi(x)\right|. \] Such estimate for holomorphic Hecke eigenforms is obtained by W. Luo and P. Sarnak [Commun. Pure Appl. Math. 56, No. 7, 874–891 (2003; Zbl 1044.11022)].
The proof of the main result is based on writing the variance of shifted convolution sums over \(|t_\phi -T| < T^\theta\) as an exponential sum involving Kloosterman sums using N. V. Kuznetsov trace formula [Mat. Sb., N. Ser. 111(153), 334–383 (1980; Zbl 0427.10016)] for a convenient test function. Then, the rest of the proof is highly involved by technical estimation of the diagonal and off-diagonal terms in the formula.
The main result has several important implications. Recall that the arithmetic Quantum Unique Ergodicity (QUE) theorem of E. Lindenstrauss [Ann. Math. (2) 163, No. 1, 165–219 (2006; Zbl 1104.22015)] and K. Soundararajan [Ann. Math. (2) 172, No. 2, 1529–1538 (2010; Zbl 1209.58019)] deals with the weak limit of the measure \(d\mu_\phi=|\phi(z)|^2dV\) as \(t_\phi\) tends to \(\infty\), where \(dV\) is the measure on \(\mathbb{X}\) induced from the hyperbolic measure \(y^{-2}dxdy\) on \(\mathbb{H}\). The Quantitative Quantum Ergodicity refers to the rate of convergence in QUE. This is all closely related to the limits of shifted convolution sums of Fourier coefficients as \(t_\phi\to\infty\). The first consequence of the main result in this paper is Quantitative Quantum Ergodicity in a short range \(|t_\phi -T| < T^{1/3}\). Let \(\epsilon >0\) be fixed, and let \(h\) be a smooth compactly supported test function with the support in \((1/L,L)\) for some \(L>1\). Then, there is a sufficiently small \(\kappa >0\) and a sufficiently large \(A>0\), both depending only on \(\epsilon\), such that \[ \sum_{|t_\phi-T| < T^{1/3}} \left| \int_\mathbb{X} P_{m,h}(z)|\phi(z)|^2dV - \frac{3}{\pi}\int_\mathbb{X} P_{m,h}(z)dV \right|^2 \ll_{\epsilon,L} T^{1/3+\epsilon}||h||_{W^{A,\infty}} \] holds uniformly in \(h\) and \(|m| < T^\kappa\), where \(P_{m,h}\) is the Poincaré series attached to the integer \(m\) and the test function \(h\).
The second consequence of the main result is concerned with \(L^2\)-norm of \(\phi\) restricted to compact geodesic segment on the imaginary axis \(\{iy\,:\,y>0\}\). Let \(\beta\) be a fixed compact geodesic segment of \(\{iy\,:\,y>0\}\). As an application of QUE, A. Ghosh et al. [Geom. Funct. Anal. 23, No. 5, 1515–1568 (2013; Zbl 1328.11044)] obtained a lower bound for the \(L^2\)-norm restriction \[ \int_\beta |\phi(z)|^2 ds \gg_\beta 1, \] where \(\phi\) is an even Hecke-Maaß cusp form, and \(\beta\) is sufficiently large. Quantitative Quantum Ergodicity result of this paper implies a sharp lower bound for any fixed geodesic \(\beta\) and almost all even Hecke-Maaß cusp forms. More precisely, let \(\epsilon >0\) be fixed. Then, for any fixed compact geodesic segment \(\beta\subset\{iy\,:\,y>0\}\), \[ \int_\beta |\phi(z)|^2 ds \gg_{\beta,\epsilon} 1 \] holds for all but \(O_{\beta,\epsilon}\left(T^{1/3+\epsilon}\right)\) forms in the set of even Hecke-Maaß cusp forms with \(|t_\phi-T| < T^{1/3}\), as \(T\) tends to \(\infty\).
The third consequence of the main result is an upper bound for the \(L^\infty\)-norm of Hecke-Maaß cusp forms. Using the Selberg trace formula and amplification method, H. Iwaniec and P. Sarnak [Ann. Math. (2) 141, No. 2, 301–320 (1995; Zbl 0833.11019)] obtained for any compact subset \(C\) of \(\mathbb{X}\) an upper bound \[ \sup_{z\in C} |\phi(z)|\ll_{C,\epsilon} t_\phi^{\frac{5}{12}+\epsilon}, \] where \(\phi\) is a Hecke-Maaß cusp form. The main result of this paper enables the author to find better amplifiers and thus improve the upper bound for almost all Hecke-Maaß cusp forms. Let \(C\) be a fixed compact subset of \(\mathbb{X}\), and let \(\epsilon >0\) be fixed. Then, all but \(O_\epsilon\left(T^{\frac{13}{12}+\epsilon}\right)\) Hecke-Maaß cusp forms \(\phi\) with \(|t_\phi-T| < T^{1/3}\) satisfy the upper bound \[ \sup_{z\in C}|\phi(z)|\ll_{C,\epsilon} t^{\frac{3}{8}+\epsilon} \] as \(T\) tends to \(\infty\).
Finally, the fourth consequence of the main result is concerned with the number of nodal domains intersecting a segment of the imaginary axis \(\{iy\,:\,y>0\}\). For a Hecke-Maaß cusp form \(\phi\), let \(Z_\phi\) denote its zero set. It is a finite union of real analytic curves. A nodal domain of \(\phi\) is, by definition, any connected component in \(\mathbb{X}\setminus Z_\phi\). Given any subset \(C\) of \(\mathbb{X}\), the number of nodal domains of \(\phi\) intersecting \(C\) is denoted by \(N^C(\phi)\). Assuming the Lindelöf Hypothesis for the \(L\)-function \(L(s,\phi)\), A. Ghosh et al. [Geom. Funct. Anal. 23, No. 5, 1515-1568 (2013; Zbl 1328.11044)] obtained a lower bound for the growth rate of \(N^\beta(\phi)\), where \(\beta\) is a fixed compact geodesic segment in \(\{iy\,:\,y>0\}\) which is sufficiently long. The bound is given by \[ N^{\beta}(\phi)\gg_{\beta,\epsilon} t_\phi^{\frac{1}{12}-\epsilon} \] as \(t_\phi\) tends to \(\infty\). Recently, S. U. Jang and the author [J. Am. Math. Soc. 31, No. 2, 303–318 (2018; Zbl 1385.58014)] showed that \(N^\beta(\phi)\) tends to \(\infty\) without any assumption. Here, the quantitative lower bound is obtained for almost all even Hecke-Maaß cusp forms. More precisely, let \(\beta\) be a fixed compact geodesic segment in \(\{iy\,:\,y>0\}\), and \(\epsilon >0\) fixed. Then, all but \(O\left(T^{\frac{4}{3}-\frac{\epsilon}{2}}\right)\) forms in the set of even Hecke-Maaß cusp forms with \(|t_\phi-T|<T^{1/3}\) satisfy \[ N^{\beta}(\phi)\gg_{\beta,\epsilon} t_\phi^{\frac{1}{8}-\epsilon} \] as \(T\) tends to \(\infty\).
Reviewer: Neven Grbac (Rijeka)
MSC:
| 58J51 | Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity |
| 11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |
| 37A45 | Relations of ergodic theory with number theory and harmonic analysis (MSC2010) |
Keywords:
quantum unique ergodicity; quantitative quantum ergodicity; Hecke-Maaß cusp forms; modular surface; Fourier coefficients; shifted convolution sums; \(L^p\)-norm restrictions; nodal domainsCitations:
Zbl 1044.11022; Zbl 0427.10016; Zbl 1104.22015; Zbl 1209.58019; Zbl 1328.11044; Zbl 0833.11019; Zbl 1385.58014References:
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