Some Relations between Spectral Geometry and Number Theory

In his paper Mc], whose object was to show that the spectrum of the Laplacian of a Riemannian surface S determines the surface up to nitely many possibilities , Henry McKean proved the following result, which he called the \Riemann hypothesis for Riemann surfaces": Theorem (McKean Mc]). If S is a hyperbolic Riemann surface, then the rst eigenvalue 1 (S) of the Laplacian on S satisses 1 (S) 14: Here, the term \Riemann surface" denotes a compact, oriented surface with a constant curvature metric. The term \hyperbolic" means that the constant is equal to ?1, so that S has genus > 1. The number 14 arises because it is the bottom 0 (H 2) of the L 2-spectrum of the Laplacian of the hyperbolic plane. The content of McKean's theorem was thus to relate the rst eigenvalue 1 (S) with the bottom of the spectrum of the universal cover e S = H 2. Actually, the term \proved" is used somewhat loosely here, because McKean's proof, to which we will return shortly, contained a fatal mistake. Indeed, it was observed shortly afterwards by Burt Randol that the result was indeed wrong, in the following strong sense: Theorem (B. Randol Ra]). Let S be a hyperbolic Riemann surface. Then there exist arbitrarily large i-fold coverings S i of S such that 1 (S i) ! 0 as i ! 1. The examples of Randol are fairly easy to describe: Let be a simple closed geodesic on S which does not divide S into two pieces. Then we may open S up by cutting it along , make i copies of S for some large i, and have them \link hands" to form a circle S i of surfaces: To see that 1 (S i) ! 0 as i ! 1, we may construct test funcxtions on S i in the following way: Assuming that i is even, divide S i into two sets A and B of copies of S| for instance, the left half of S i and the right half| so that A and B meet in two coipes of. Then let f i be +1 on A and ?1 on B, and change in some standard way from +1 to ?1 in neighborhoods of the two copies of .