Background and motivation on the
Quantum Unique Ergodicity conjecture

by Zeev Rudnick

The original conjecture I made with Peter Sarnak concerns the quantum/classical correspondence for chaotic systems. Around 1991, when we were both at Stanford, we saw reports of some strange numerical simulations in chaotic billiards which seems to suggest that some highly excited quantum particles may concentrate on periodic trajectories for the corresponding classical particles. This phenomenon was called "scars" in the physical literature. We proposed a mathematical (rigorous) definition of these scars, nowadays called "strong scars". After looking for these scars in vain in a favorite toy model, namely the modular domain, we made the conjecture that in certain sufficiently chaotic systems, billiard particles on a negatively curved surface, these "scars" cannot occur. This is now called "Quantum Unique Ergodicity" (QUE).

In the special case of the modular domain (and other surfaces associated with congruence groups) a special instance of QUE (relating to Hecke-Maass eigenforms) is now known to be implied by the Generalized Riemann Hypothesis. In the case of COMPACT surfaces associated with quaternion algebras, Lindenstrauss proved QUE (again, only for Hecke eigenforms) but the original case of the modular domain remains open.

QUE is concerned with eigenfunctions of the Laplacian with large eigenvalue, and in the case of the modular domain these are Hecke-Maass forms. By analogy, one may consider holomorphic forms of large weight (again, which are Hecke eigenforms), and it is this case that Holowinsky and Soundararajan solve. Thus the motivation for their work is directly inspired by problems in quantum mechanics of chaotic systems, and in return one may hope that it will impact the case of Maass forms for the modular domain, which is closer to physical models. In fact before their work QUE in that case was known to follow from GRH, and as they say in their paper now "all" one needs is the Ramanujan conjectures.

The special case of modular forms

The problem of the distribution of zeros of modular forms has an amusing history. In the 1960's a number of authors investigated the zeros of the simplest kind of modular form, the holomorphic Eisenstein series

Ek(z)= 1/2 ∑(c,d)=1(cz+d)-k,

and for small weights explicitly located all zeros. In 1968, R.A. Rankin worked out some more cases and conjectured that all these zeros (lying in the fundamental domain) are on the unit circle. Shortly afterward, Swinnerton-Dyer solved Rankin's conjecture in a paper joint with F.K.C. Rankin (Fenny Rankin was Robert Rankin's daughter, at that time an undergraduate at Cambridge). One can surmise that Swinnerton-Dyer had seen how to do it for large weights and enlisted Rankin's daughter to help with some computations with medium-sized weights so as to tease her father...

Around 1998, guided by some developments in the theory of quantum chaos (due to Nonnenmacher and Voros), I realized that the holomorphic Eisenstein series behaved much differently than other modular forms of arithmetic interest, the cuspidal eigenforms. For these I realized that the zeros are not confined to a small set such as the unit circle, rather they should become dense (in fact uniformly distributed) in the fundamental domain as we increase the weight.

I knew this should follow from a version of Quantum Unique Ergodicity, which in turn should follow from the Generalized Riemann Hypothesis, so should be considered as a scientific fact. I circulated some notes on this but at the time some of the necessary technology (Watson's formula) wasn't written up anywhere and it was only a few years later that I published a formal proof that QUE implies this equidistribution of zeros. I found it a nice interaction between quantum mechanics and number theory.

Holowinsky and Soundararajan have now put the matter to rest. On a number of occasions I offered a cash prize (of 10 Euro) for an unconditional proof and the checks are already in the mail.