Quantum variance for Hecke eigenforms |
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Authors: | Wenzhi Luo |
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Affiliation: | Department of Mathematics, 100 Mathematics Building, 231 West 18th avenue, Columbus, OH 43210-1174, USA; Courant Institute, 251 Mercer Street, New York, NY 10012, USA and Department of Mathematics, Princeton University, Princeton, NJ 08540, USA |
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Abstract: | 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. |
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