Quantitative nodal domain count for Maass forms

Source abstract

We consider a real even Hecke-Maass cusp form $φ$ on the modular surface, with Laplace eigenvalue $1/4+T^2$. We prove that its restriction to any fixed compact segment $β$ of the vertical geodesic above $i$ has $\gg_{β,\varepsilon} T^{1/6-\varepsilon}$ sign changes, yielding the same lower bound for the number of distinct inert nodal domains meeting the segment. Building on ideas of Jang-Jung and Jung-Young, we study correlations between values at points on the vertical geodesic separated by distances of order $T^{-1}$. Using arithmetic quantum unique ergodicity and Dirichlet-polynomial estimates, we show that the correlation is negative for a suitable choice of separation. An elementary counting argument and the Iwaniec-Sarnak supremum bound then give the claimed lower bound.