Nodal domains of Maass forms, II

Abstract

In Part I we gave a polynomial growth lower-bound for the number of nodal domains of a Hecke-Maass cuspform in a compact part of the modular surface, assuming a Lindelöf hypothesis. That was a consequence of a topological argument and known subconvexity estimates, together with new sharp lower-bound restriction theorems for the Maass forms. This paper deals with the same question for general (compact or not) arithmetic surfaces which have a reflective symmetry. The topological argument is extended and representation theoretic methods are needed for the restriction theorems, together with results of Waldspurger. Various explicit examples are given and studied.

A uniform bound for geodesic periods of eigenfunctions on hyperbolic surfaces

“…@ n i / Ä 1 2 i obtained in [36] for the norm of the restriction. @ n i / c Y;l (see [14]). We note that one expects that the bound on the norm be essentially sharp since one can show that there exists a constant c Y;l > 0 such that n l .…”

Quantum unique ergodicity and the number of nodal domains of eigenfunctions

“…This result in the stronger form of a lower bound of ≫ ǫ λ 1 12 −ǫ for the number of nodal domains is obtained in [GRS14], however assuming the Generalized Lindelöf Hypothesis for a certain family of L-functions. Theorem 1.1 is a consequence of the below Theorem 1.5 which considers the number of nodal domains when we have Quantum Unique Ergodicity(QUE).…”

“…Let {φ j } j be the complete sequence of Hecke-Maass eigenforms on X, i.e., it is a joint eigenfunction of −∆ g and Hecke operators {T n } n≥1 . It is shown in [GRS14] that there exists an orientation-reversing isometric involution τ : X → X such that Fix(τ ) is separating and that τ commutes with all T n . From the multiplicity one theorem for Hecke eigenforms [AL70], the sequence of Hecke eigenvalues {λ φ (n)} n≥1 of T n (i.e., T n φ = λ φ (n)φ) determines φ uniquely.…”

“…(See [GRS13,Jun13,BR13,JZ13,Mag13,GRS14], where such an idea is used to prove a lower bound for the number of sign changes in various contexts.) In order to bound u n L 1 (β) from below using Hölder's inequality, the authors use the Quantum Ergodic Restriction (QER) theorem [TZ13,DZ13] for the lower bound of u n L 2 (β) and the point-wise Weyl law with an improved error term [Bér77] for the upper bound of u n L ∞ (β) .…”

Sup-Norm and Nodal Domains of Dihedral Maass Forms

“…where g is the genus of the surface Γ 0 (q)\H. (Ghosh-Reznikov-Sarnak [9] gave a proof of the above inequality for SL 2 (Z)\H, and in [10] they gave a proof for a compact surface of genus g ≥ 0. Since the proof is local, it is valid for non-compact surfaces in our case as mentioned in [9,10].…”