Spectral theta series of operators with periodic bicharacteristic flow
Annales de l'Institut Fourier, Volume 57 (2007) no. 7, p. 2401-2427

The theta series ϑ(z)=exp(2πin 2 z) is a classical example of a modular form. In this article we argue that the trace ϑ P (z)=Trexp(2πiP 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’ Trexp(-iPt) whose singularities are well known to be located at the lengths of the periodic orbits of the bicharacteristic flow.

La série thêta ϑ(z)=exp(2πin 2 z) est un exemple classique de forme modulaire. Dans cet article, nous montrons que la trace ϑ P (z)=Trexp(2πiP 2 z), où P est un opérateur pseudo-différentiel elliptique auto-adjoint d’ordre 1 à flot bicaractéristique périodique, en est une généralisation naturelle. En particulier, nous établissons des égalités fonctionnelles approchées sous l’action du groupe modulaire. Ceci permet une analyse détaillée de l’asymptotique de ϑ P (z) au voisinage de l’axe réel, et prouve des lois du logarithme et des théorèmes limites pour la distribution de ses valeurs. Ces asymptotiques diffèrent de celles relatives à la trace de l’opérateur des ondes Trexp(-iPt), dont les singularités sont portées par les longueurs des bicaractéristiques périodiques.

DOI : https://doi.org/10.5802/aif.2338
Classification:  35P20,  11F72,  37D40
Keywords: Spectral theta series, Zoll manifolds, periodic geodesic flow, Shale-Weil representation, horocycle flow, logarithm laws
@article{AIF_2007__57_7_2401_0,
     author = {Marklof, Jens},
     title = {Spectral theta series of operators with periodic bicharacteristic flow},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {7},
     year = {2007},
     pages = {2401-2427},
     doi = {10.5802/aif.2338},
     mrnumber = {2394547},
     zbl = {1133.35075},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2007__57_7_2401_0}
}
Marklof, Jens. Spectral theta series of operators with periodic bicharacteristic flow. Annales de l'Institut Fourier, Volume 57 (2007) no. 7, pp. 2401-2427. doi : 10.5802/aif.2338. http://www.numdam.org/item/AIF_2007__57_7_2401_0/

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