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

Received:
Accepted:
Published online:
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
@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},
     pages = {1133--1151},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {3},
     year = {2015},
     doi = {10.5802/aif.2953},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2953/}
}
TY  - JOUR
TI  - Cone theta functions and spherical polytopes with rational volumes
JO  - Annales de l'Institut Fourier
PY  - 2015
DA  - 2015///
SP  - 1133
EP  - 1151
VL  - 65
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2953/
UR  - https://doi.org/10.5802/aif.2953
DO  - 10.5802/aif.2953
LA  - en
ID  - AIF_2015__65_3_1133_0
ER  - 
%0 Journal Article
%T Cone theta functions and spherical polytopes with rational volumes
%J Annales de l'Institut Fourier
%D 2015
%P 1133-1151
%V 65
%N 3
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2953
%R 10.5802/aif.2953
%G en
%F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.2953/

[1] Bruinier, Jan Hendrik Nonvanishing modulo l of Fourier coefficients of half-integral weight modular forms, Duke Math. J., Tome 98 (1999) no. 3, pp. 595-611 | Article | MR: 1695803 | Zbl: 0966.11019

[2] Bruinier, Jan Hendrik; van der Geer, Gerard; Harder, Günter; Zagier, Don The 1-2-3 of modular forms, Universitext, Springer-Verlag, Berlin, 2008, x+266 pages (Lectures from the Summer School on Modular Forms and their Applications held in Nordfjordeid, June 2004, Edited by Kristian Ranestad) | Article | MR: 2385372 | Zbl: 1197.11047

[3] Deligne, P.; Rapoport, M. Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, p. 143-316. Lecture Notes in Math., Vol. 349 | MR: 330050 | Zbl: 0281.14010

[4] Desario, David; Robins, Sinai Generalized solid-angle theory for real polytopes, Q. J. Math., Tome 62 (2011) no. 4, pp. 1003-1015 | Article | MR: 2853227 | Zbl: 1236.52012

[5] Dupont, Johan L.; Sah, Chih-Han Three questions about simplices in spherical and hyperbolic 3-space, The Gelfand Mathematical Seminars, 1996–1999 (Gelfand Math. Sem.), Birkhäuser Boston, Boston, MA, 2000, pp. 49-76 | MR: 1731633 | Zbl: 1009.52027

[6] Grove, L. C.; Benson, C. T. Finite reflection groups, Graduate Texts in Mathematics, Tome 99, Springer-Verlag, New York, 1985, x+133 pages | Article | MR: 777684 | Zbl: 0579.20045

[7] Guo, Li; Paycha, Sylvie; Zhang, Bin Conical zeta values and their double subdivision relations (http://arxiv.org/abs/1301.3370) | MR: 3144233 | Zbl: 1294.11145

[8] Kohnen, Winfried On certain generalized modular forms, Funct. Approx. Comment. Math., Tome 43 (2010) no. part 1, pp. 23-29 | Article | MR: 2683571 | Zbl: 1275.11076

[9] McMullen, P. Non-linear angle-sum relations for polyhedral cones and polytopes, Math. Proc. Cambridge Philos. Soc., Tome 78 (1975) no. 2, pp. 247-261 | Article | MR: 394436 | Zbl: 0313.52005

[10] McMullen, P. Valuations and Euler-type relations on certain classes of convex polytopes, Proc. London Math. Soc. (3), Tome 35 (1977) no. 1, pp. 113-135 | Article | MR: 448239 | Zbl: 0353.52001

[11] Nahm, Werner Conformal field theory and torsion elements of the Bloch group, Frontiers in number theory, physics, and geometry. II, Springer, Berlin, 2007, pp. 67-132 | Article | MR: 2290759 | Zbl: 1193.81092

[12] Ogg, Andrew Modular forms and Dirichlet series, W. A. Benjamin, Inc., New York-Amsterdam, 1969, xvi+173 pp. (not consecutively paged) paperbound pages | MR: 256993 | Zbl: 0191.38101

[13] Perles, M. A.; Shephard, G. C. Angle sums of convex polytopes, Math. Scand., Tome 21 (1967), p. 199-218 (1969) | EuDML: 166020 | MR: 243425 | Zbl: 0172.23703

[14] Robins, Sinai An extension of the Gram relations, using cone theta functions (preprint)

[15] Schmoll, S. D. Eine Charakterisierung von Spitzenformen (2011) (Ph. D. Thesis)

[16] Schoeneberg, Bruno Elliptic modular functions: an introduction, Springer-Verlag, New York-Heidelberg, 1974, viii+233 pages (Translated from the German by J. R. Smart and E. A. Schwandt, Die Grundlehren der mathematischen Wissenschaften, Band 203) | MR: 412107 | Zbl: 0285.10016

[17] Shimura, Goro Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971, xiv+267 pages (Kanô Memorial Lectures, No. 1) | MR: 314766 | Zbl: 0221.10029

[18] Shimura, Goro On modular forms of half integral weight, Ann. of Math. (2), Tome 97 (1973), pp. 440-481 | Article | MR: 332663 | Zbl: 0266.10022

[19] Stanley, Richard P. Decompositions of rational convex polytopes, Ann. Discrete Math., Tome 6 (1980), pp. 333-342 (Combinatorial mathematics, optimal designs and their applications (Proc. Sympos. Combin. Math. and Optimal Design, Colorado State Univ., Fort Collins, Colo., 1978)) | Article | MR: 593545 | Zbl: 0812.52012

[20] Vlasenko, M.; Zwegers, S. Nahm’s conjecture: asymptotic computations and counterexamples (preprint) | MR: 2864462 | Zbl: 1256.81102

[21] Zagier, Don The dilogarithm function, Frontiers in number theory, physics, and geometry. II, Springer, Berlin, 2007, pp. 3-65 | Article | MR: 2290758 | Zbl: 1176.11026

Cited by Sources: