Cone theta functions and spherical polytopes with rational volumes
[Cône fonctions thêta et polytopes sphériques avec des volumes rationnels]
Folsom, Amanda1 ; Kohnen, Winfried2 ; Robins, Sinai3
1 Yale University Mathematics Department P.O. Box 208283, New Haven Connecticut 06520-8283 (USA)
2 Ruprecht-Karls-Universität Heidelberg Mathematisches Institut Im Neuenheimer Feld 288 69120 Heidelberg (Germany)
3 Brown University Mathematics Department Box 1917, 151 Thayer Street Providence, RI 02912 (USA)
Annales de l'Institut Fourier, Tome 65 (2015) no. 3, pp. 1133-1151.

Nous étudions une classe de fonctions polyédriques appelées fonctions theta de cône, qui sont étroitement liées à des fonctions theta classiques. Chaque cône polyédrique K⊂R d a une fonction theta de cône associée, et nous montrons qu’elles codent des informations sur la rationalité du volume sphérique de K.

Nous montrons que si K est une chambre de Weyl pour tout groupe de Weyl fini, alors sa fonction theta de cône appartient à un anneau gradué de fonctions theta classiques et en ce sens est presque modulaire. Inversement, dans le cas où le volume sphérique est irrationnel, il est naturel de se demander si les fonctions theta de cône sont elles-mêmes modulaires, et nous prouvons qu’en général elles ne le sont pas.

We study a class of polyhedral functions called cone theta functions, which are closely related to classical theta functions. Each polyhedral cone 𝕂⊂ℝ d has an associated cone theta function, and we show that they encode information about the rationality of the spherical volume of K. We show that if K is a Weyl chamber for any finite Weyl group, then its cone theta function lies in a graded ring of classical theta functions and in this sense is “almost” modular. Conversely, in the case that the spherical volume is irrational, it is natural to ask whether the cone theta functions are themselves modular, and we prove that in general they are not.

DOI : 10.5802/aif.2953
Classification : 52C07, 52A55, 11F27, 14K25, 20M14
Keywords: Theta function, modular form, spherical volume, solid angle, rationality, cone, polytope, Weil chamber, root lattice
Mot clés : Fonction thêta, forme modulaire, volume sphérique, angle solide, rationalité, cône, polytope, chambre de Weil, réseau de racines
Folsom, Amanda 1 ; Kohnen, Winfried 2 ; Robins, Sinai 3

1 Yale University Mathematics Department P.O. Box 208283, New Haven Connecticut 06520-8283 (USA)
2 Ruprecht-Karls-Universität Heidelberg Mathematisches Institut Im Neuenheimer Feld 288 69120 Heidelberg (Germany)
3 Brown University Mathematics Department Box 1917, 151 Thayer Street Providence, RI 02912 (USA) 
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Folsom, Amanda; Kohnen, Winfried; Robins, Sinai. Cone theta functions and spherical polytopes with rational volumes. Annales de l'Institut Fourier, Tome 65 (2015) no. 3, pp. 1133-1151. doi : 10.5802/aif.2953. http://www.numdam.org/articles/10.5802/aif.2953/

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