Arithmetic, zeros, and nodal domains on the sphere

TL;DR

This paper establishes lower bounds and growth rates for the number of nodal domains of Hecke eigenfunctions on the sphere, under both conditional and unconditional assumptions, revealing new insights into their asymptotic behavior.

Contribution

It provides the first lower bounds for nodal domains of Hecke eigenfunctions on the sphere and compares growth rates under different hypotheses.

Findings

Number of nodal domains grows with eigenvalue under Lindelof hypothesis

Unconditionally, average nodal domains grow faster than under Lindelof

Lower bounds are established for nodal domain counts

Abstract

We obtain lower bounds for the number of nodal domains of Hecke eigenfunctions on the sphere. Assuming the generalized Lindelof hypothesis we prove that the number of nodal domains of any Hecke eigenfunction grows with the eigenvalue of the Laplacian. By a very different method, we show unconditionally that the average number of nodal domains of degree l Hecke eigenfunctions grows significantly faster than the uniform growth obtained under Lindelof.