Spectral theta series of operators with periodic bicharacteristic flow

Jens Marklof[1]

  • [1] University of Bristol School of Mathematics Bristol BS8 1TW (United Kingdom)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 7, page 2401-2427
  • ISSN: 0373-0956

Abstract

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The theta series ϑ ( z ) = ∑ exp ( 2 π i n 2 z ) is a classical example of a modular form. In this article we argue that the trace ϑ P ( z ) = Tr exp ( 2 π i P 2 z ) , where P is a self-adjoint elliptic pseudo-differential operator of order 1 with periodic bicharacteristic flow, may be viewed as a natural generalization. In particular, we establish approximate functional relations under the action of the modular group. This allows a detailed analysis of the asymptotics of ϑ P ( z ) near the real axis, and the proof of logarithm laws and limit theorems for its value distribution. These asymptotics are in fact distinctly different from those for the ‘wave trace’ Tr exp ( - i P t ) whose singularities are well known to be located at the lengths of the periodic orbits of the bicharacteristic flow.

How to cite

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Marklof, Jens. "Spectral theta series of operators with periodic bicharacteristic flow." Annales de l’institut Fourier 57.7 (2007): 2401-2427. <http://eudml.org/doc/10302>.

@article{Marklof2007,
abstract = {The theta series $\vartheta (z)=\sum \exp (2\pi \mathrm\{i\} n^2 z)$ is a classical example of a modular form. In this article we argue that the trace $\vartheta _P(z)=\operatorname\{Tr\}\,\exp (2\pi \mathrm\{i\} P^2 z)$, where $P$ is a self-adjoint elliptic pseudo-differential operator of order 1 with periodic bicharacteristic flow, may be viewed as a natural generalization. In particular, we establish approximate functional relations under the action of the modular group. This allows a detailed analysis of the asymptotics of $\vartheta _P(z)$ near the real axis, and the proof of logarithm laws and limit theorems for its value distribution. These asymptotics are in fact distinctly different from those for the ‘wave trace’ $\operatorname\{Tr\}\, \exp (-\mathrm\{i\} P t)$ whose singularities are well known to be located at the lengths of the periodic orbits of the bicharacteristic flow.},
affiliation = {University of Bristol School of Mathematics Bristol BS8 1TW (United Kingdom)},
author = {Marklof, Jens},
journal = {Annales de l’institut Fourier},
keywords = {Spectral theta series; Zoll manifolds; periodic geodesic flow; Shale-Weil representation; horocycle flow; logarithm laws; spectral theta series},
language = {eng},
number = {7},
pages = {2401-2427},
publisher = {Association des Annales de l’institut Fourier},
title = {Spectral theta series of operators with periodic bicharacteristic flow},
url = {http://eudml.org/doc/10302},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Marklof, Jens
TI - Spectral theta series of operators with periodic bicharacteristic flow
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 7
SP - 2401
EP - 2427
AB - The theta series $\vartheta (z)=\sum \exp (2\pi \mathrm{i} n^2 z)$ is a classical example of a modular form. In this article we argue that the trace $\vartheta _P(z)=\operatorname{Tr}\,\exp (2\pi \mathrm{i} P^2 z)$, where $P$ is a self-adjoint elliptic pseudo-differential operator of order 1 with periodic bicharacteristic flow, may be viewed as a natural generalization. In particular, we establish approximate functional relations under the action of the modular group. This allows a detailed analysis of the asymptotics of $\vartheta _P(z)$ near the real axis, and the proof of logarithm laws and limit theorems for its value distribution. These asymptotics are in fact distinctly different from those for the ‘wave trace’ $\operatorname{Tr}\, \exp (-\mathrm{i} P t)$ whose singularities are well known to be located at the lengths of the periodic orbits of the bicharacteristic flow.
LA - eng
KW - Spectral theta series; Zoll manifolds; periodic geodesic flow; Shale-Weil representation; horocycle flow; logarithm laws; spectral theta series
UR - http://eudml.org/doc/10302
ER -

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