Nodal domains of Maass forms, II
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NODAL DOMAINS OF MAASS FORMS, II By AMIT GHOSH, ANDRE REZNIKOV, and PETER SARNAK 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. 1. Introduction. This paper is a continuation of our [GRS13], which will be referred to as I. The main result there gives a lower bound for the number of nodal domains for Maass forms on a compact part of the modular surface. We indicated in I that the techniques there can be modified and extended to deal with a general arithmetic surface. Our aim here is to carry this out. The treatment in I relies heavily on the Fourier expansions of the Maass forms in the cusp, and for a compact surface these are not available. Our primary focus here will be on the compact cases. For the most part we use the same notation and terminology as in I. Let X = Γ\H be a finite area hyperbolic surface. Let Δ be the Laplacian on functions on X, and ψ (or φ) will denote L2-eigenfunctions with eigenvalue λψ = 1 4 + t2 ψ > 0 (such ψ’s are also called Maass forms [Ma49]). We let σ : X → X be an isometric involution on X which is induced from a hyperbolic reflection on H. Next, we let Σ = Fix(σ) denote the points of X fixed by σ; it consists of a union (possibly disconnected) of piecewise geodesics in X (see Section 2). Our main interest is when Σ is compact unlike the case in I where Σ ran into the cusps of X. By φ we will always mean a σ-even eigenfunction of Δ (we could just as well deal with σ-odd as indicated in the Appendix of I). The nodal domains Ω of φ are inert or split according as σ(Ω)∩Ω is equal to Ω or is empty. Our main result concerns arithmetic congruence surfaces X ([Ta75, PR, Shi] and in Sections 5 and 6). These come with a commuting algebra of Hecke operators which can be simultaneously diagonalized together with Δ and respecting the σ even and odd functions on X. Manuscript received January 3, 2016; revised July 17, 2017. Research of the first author supported in part by IAS, the College of A&S and the Department of Mathematics of his home university, and the Simons Foundation for a Collaboration Grant; research of the second author supported in part by the Veblen Fund at IAS, the ERC grant 291612 and by the ISF grant 533/14; research of the second and third authors supported in part by a BSF grant; research of the third author supported by NSF grant 1302952. American Journal of Mathematics 139 (2017), 1395–1447. c  2017 by Johns Hopkins University Press. 1395 1396 A. GHOSH, A. REZNIKOV, AND P. SARNAK THEOREM 1.1. Let X be a congruence surface and φλ a Laplace-Hecke σeven eigenfunction as above. Assuming the Lindelöf Hypothesis for L-functions of GL2 automorphic forms, the number Nin(φλ) of inert nodal domains for φλ, satisfies Nin(φλ) ε λ 1 27 −ε φ , as λφ → ∞, for any ε > 0. The implied constant depends on ε and X only. Remark 1.2. (a) Various explicit examples of X and σ’s to which Theorem 1.1 applies are given in Section 6. The compact triangle surface in Figure 9 is a basic example. The (Hecke) eigenfunctions satisfying Neumann boundary conditions on the triangle obey Theorem 1.1. (b) As noted in I, this lower bound is no doubt far from the truth. From [TZ09] it follows that Nin(φλ)  λ 1 2 φ and this upper bound is probably sharp. As far as split nodal domains we do not know how to produce any of them in spite of the fact that they are probably the vast majority. The expectation is that Nsplit(φλ) ∼ cArea(X) 4π λφ, where...