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Small scale distribution of zeros and mass of modular forms. (English) Zbl 1404.11041

In this article, the authors study the equidistribution of zeros and mass of holomorphic Hecke cusp forms \(f\) of weight \(k\) with respect to \(\text{SL}_2(\mathbb Z)\) at small scales. They consider the distribution of zeros of \(f\) in two domains \(B(z,r_k)\) and \(\mathcal{F}_Y\) separately. Here \(B(z,r_k)\) is a hyperbolic ball contained in the canonical fundamental domain \(\mathcal{F}\) of \(\text{SL}_2(\mathbb Z)\) centered at \(z\) and of radius \(r_k\) with \(r_k\rightarrow 0\) sufficiently slowly as \(k\longrightarrow\infty\) and \(\mathcal{F}_Y\) is a neighborhood of the cusp of \(\mathcal{F}\) such that \(\mathcal{F}_Y=\{z\in \mathcal{F}~:~\text{Im}(z)>Y\},~~Y\geq\sqrt{k\log k}\). They show the following results: Let \(N_f(D)\) be the number of zeros of \(f\) in a domain \(D\) and \(A(D)\) the area of \(D\) by hyperbolic measure. Let \(B(z_0,r)\subset \{z\in\mathcal{F}~:~\text{Im}(z)\leq B\}\) for a fixed \(B>0\) and \(r\geq(\log k)^{-\delta_0/2+\varepsilon}\), where \(\delta_0=31/8-\sqrt{15}\). Then \(\frac{N_f(B(z_0,r))}{N_f(\mathcal{F)}}=\frac{A(B(z_0,r))}{A(\mathcal{F})}+O_B(r(\log k)^{-\delta_0/2+\varepsilon})~(k\rightarrow\infty)\). Further, if assuming the generalized Lindelöf hypothesis, in the above, the factor \((\log k)^{-\delta_0/2+\varepsilon}\) can be replaced by the smaller factor \(k^{-1/8+\varepsilon}\) for \(r\geq k^{-1/8+\varepsilon}\). As for the distribution of zeros in \(\mathcal{F}_Y\), A. Ghosh and P. Sarnak [J. Eur. Math. Soc. (JEMS) 14, No. 2, 465–487 (2012; Zbl 1287.11054)] conjectured that almost all zeros \(\rho_f\) of \(f\) in \(\mathcal{F}_Y\) lie on two lines \(L_1:\text{Re}(z)=0,L_2:\text{Re}(z)=-1/2\) and gave a lower bound on the number of zeros on these lines. The authors show for almost all cusp forms \(f\) that \(\sharp\{\rho_f\in L_i\}\geq c(\varepsilon)\cdot \sharp\{\rho_f\in \mathcal{F}_Y\} ~(i=1,2)\), provided that \(\delta(\varepsilon)k>Y>\sqrt{k\log k}\) and \(k\rightarrow\infty\). The constants \(\delta(\varepsilon)\) and \(c(\varepsilon)\) are positive and depend only on \(\varepsilon\). Under the generalized Lindelöf hypothesis, they give a lower bound on the number of zeros on the line \(L_i\), which is significantly stronger than the previous results. Furthermore they show the following effective mass equidistribution for the mass of \(y^k|f(z)|\): Let \(\phi\) be a smooth function supported in \(\mathcal{F}\) with \(\sup_{z\in\mathcal{F}}\left|y\frac{\partial^a}{\partial x^a}\frac{\partial^b}{\partial y^b}\phi(z)\right|\ll_{a,b}M^{a+b}\), \((z=x+iy, a,b,M\geq 1)\). Then,  \(\left|\iint_{\mathcal{F}}y^k|f(z)|^2\phi(z)\frac{dxdy}{y^2}-\iint_{\mathcal{F}}\phi(z)\frac{dxdy}{y^2}\right|\ll_{\varepsilon}M^2(\log k)^{-4\delta_0+\varepsilon}, \) for all \(\varepsilon>0 \) fixed.

MSC:

11F11 Holomorphic modular forms of integral weight
11F30 Fourier coefficients of automorphic forms
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
58J51 Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity

Citations:

Zbl 1287.11054
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