Quantum variance for Hecke eigenforms

Wenzhi Luo, Peter Sarnak

Research output: Contribution to journalArticlepeer-review

35 Scopus citations


We calculate the quantum variance for the modular surface. This variance, introduced by S. Zelditch, describes the fluctuations of a quantum observable. The resulting quadratic form is then compared with the classical variance. The expectation that these two coincide only becomes true after inserting certain subtle arithmetic factors, specifically the central values of corresponding L-functions. It is the off-diagonal terms in the analysis that are responsible for the rich arithmetic structure arising from the diagonalization of the quantum variance.

Original languageEnglish (US)
Pages (from-to)769-799
Number of pages31
JournalAnnales Scientifiques de l'Ecole Normale Superieure
Issue number5
StatePublished - Sep 2004

All Science Journal Classification (ASJC) codes

  • General Mathematics


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