Quantum Variance for Hecke Eigenforms ✩
نویسندگان
چکیده
– 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. 2004 Elsevier SAS RÉSUMÉ. – Nous calculons la variance quantique pour la surface modulaire. Cette variance, introduite par S. Zelditch, décrit les fluctuations d’une observable quantique. La forme quadratique ainsi obtenue est comparée avec la variance classique. On s’attend à ce que toutes les deux coïncident, mais cela ne se passe qu’après inclusion de certains facteurs arithmétiques subtils, précisément les valeurs centrales des fonctions L appropriées. Les termes non diagonaux apparaissant dans l’analyse de la diagonalisation de la variance quantique sont responsables de la riche structure arithmétique. 2004 Elsevier SAS
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