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.

Classification:  52C07,  52A55,  11F27,  14K25,  20M14
Keywords: Theta function, modular form, spherical volume, solid angle, rationality, cone, polytope, Weil chamber, root lattice
     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 = {}
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.

[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, Springer-Verlag, Berlin, Universitext (2008), pp. x+266 (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, Birkhäuser Boston, Boston, MA (Gelfand Math. Sem.) (2000), pp. 49-76 | MR 1731633 | Zbl 1009.52027

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

[7] Guo, Li; Paycha, Sylvie; Zhang, Bin Conical zeta values and their double subdivision relations ( | 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), pp. xvi+173 pp. (not consecutively paged) paperbound | 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) | 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, Heidelberg (Germany) (2011) (Ph. D. Thesis)

[16] Schoeneberg, Bruno Elliptic modular functions: an introduction, Springer-Verlag, New York-Heidelberg (1974), pp. viii+233 (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), pp. xiv+267 (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