Spectral theta series of operators with periodic bicharacteristic flow

Marklof, Jens

doi:10.5802/aif.2338

Spectral theta series of operators with periodic bicharacteristic flow
[Série thêta spectrale d’opérateurs dont le flot bicaractéristique est périodique]

Marklof, Jens ¹

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

Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2401-2427.

Résumé
Abstract

La série thêta $ϑ (z) = \sum exp (2 π i n^{2} z)$ est un exemple classique de forme modulaire. Dans cet article, nous montrons que la trace $ϑ_{P} (z) = Tr exp (2 π i P^{2} z)$ , 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 $Tr exp (- i P t)$ , dont les singularités sont portées par les longueurs des bicaractéristiques périodiques.

The theta series $ϑ (z) = \sum exp (2 π i n^{2} z)$ 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.

MR Zbl

DOI : 10.5802/aif.2338

Classification : 35P20, 11F72, 37D40
Keywords: Spectral theta series, Zoll manifolds, periodic geodesic flow, Shale-Weil representation, horocycle flow, logarithm laws
Mot clés : série théta spectrale, variété de Zoll, flot géodésique périodique, représentation de Shale-Weil, flot horocyclique, lois du logarithme

Affiliations des auteurs :

Marklof, Jens ¹

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

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Marklof, Jens. Spectral theta series of operators with periodic bicharacteristic flow. Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2401-2427. doi : 10.5802/aif.2338. https://aif.centre-mersenne.org/articles/10.5802/aif.2338/

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