Cone theta functions and spherical polytopes with rational volumes
Annales de l'Institut Fourier, Volume 65 (2015) no. 3, p. 1133-1151

We study a class of polyhedral functions called cone theta functions, which are closely related to classical theta functions. Each polyhedral cone 𝕂 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.

Nous étudions une classe de fonctions polyédriques appelées fonctions theta de cône, qui sont étroitement liées à des fonctions theta classiques. Chaque cône polyédrique KR 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.

DOI : https://doi.org/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
@article{AIF_2015__65_3_1133_0,
     author = {Folsom, Amanda and Kohnen, Winfried and Robins, Sinai},
     title = {Cone theta functions and spherical polytopes with rational volumes},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {3},
     year = {2015},
     pages = {1133-1151},
     doi = {10.5802/aif.2953},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_3_1133_0}
}
Folsom, Amanda; Kohnen, Winfried; Robins, Sinai. Cone theta functions and spherical polytopes with rational volumes. Annales de l'Institut Fourier, Volume 65 (2015) no. 3, pp. 1133-1151. doi : 10.5802/aif.2953. http://www.numdam.org/item/AIF_2015__65_3_1133_0/

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