International Congress of Mathematicians

Asymptotics of Polynomials and Eigenfunctions 7410(N(X)/(logX) p j) [28]. The asymptotics S 2 (X) ~ B(A)X have recently been obtainedby Luo-Sarnak [13] for Hecke eigenfunctions of the modular group, exploitingthe connections with L-functions. These asymptotics (though not the coefficient)are predicted by the random polynomial model. Other strong bounds in the arithmeticcase were obtained by Kurlberg-Rudnick for eigenfunctions of certain quantizedtorus automorphisms [10]. Bourgain-Lindenstrauss [3] and Wolpert [25] havedeveloped the 'non-scarring' result of [16] to give entropy estimates of possiblequantum limit measures in arithmetic cases.A natural problem is the converse:• Problem 6 What can be said of the dynamics if S P (X) = o(N(X)jiquantum ergodicity imply classical ergodicity?DoesIt is known that classical ergodicity is equivalent to this bound plus estimateson off-diagonal terms [21]. The existence of KAA1 quasimodes (due to Lazutkin[11], Colin de Verdiere [5], and Popov [15]) makes it very likely that KAA1 systemsare not quantum ergodic, nor are (M,g) which have stable elliptic orbits.A further problem which may be accessible is:• Problem 7 How are the nodal sets {(p v = 0} distributed in the limit v —t oo?In [14] (for elliptic curves) and [19] (general Kahler manifolds) it is proved thatthe complex zeros of quantum ergodic eigenfunctions become uniformly distributedrelative to the volume form. Can one prove an analogue for the real zeros?References[1] P. Bleher, B. Shiffman and S. Zelditch, Universality and scaling of correlationsbetween zeros on complex manifolds, Invent. Math. 142 (2000), 351^395.[2] P. Bleher, B. Shiffman and S. Zelditch, Correlations between zeros and supersymmetry,Commun. Alath. Phys. 224 (2001) 1, 255-269.[3] J. Bourgain and E. Lindenstrauss, Entropy of quantum limits (preprint, 2002).[4] Boutet de Alonvel, L.; Sjstrand, J. Sur la singularit des noyaux de Bergmanet de Szeg.Journes: quations aux Drives Partielles de Rennes (1975), 123^164.Astérisque, No. 34-35, Soc. Alath. France, Paris, 1976.[5] Colin de Verdire, Yves Quasi-modes sur les varits Riemanniennes. Invent.Alath. 43 (1977), no. 1, 15^52.[6] H. Donnelly, C. Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds.Invent. Alath. 93 (1988), no. 1, 161-183.[7] P. Gerard and E. Leichtnam, Ergodic properties of eigenfunctions for theDirichlet problem, Duke Alath. J. 71 (1993), 559^607.[8] A. Hassell and S. Zelditch, Quantum ergodicity and boundary values of eigenfunctions(preprint, 2002).[9] D. Jakobson, N. Nadirashvili, and J. Toth, Geometry of eigenfunctions (toappear in Russian Alath Surveys).[10] P. Kurlberg and Z. Rudnick, Value distribution for eigenfunctions of desymmetrizedquantum maps. Internat. Alath. Res. Notices 2001, no. 18, 985^1002.