On computing quaternion quotient graphs for function fields
Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 1, p. 73-99

Let Λ be a maximal 𝔽 q [T]-order in a division quaternion algebra over 𝔽 q (T) which is split at the place . The present article gives an algorithm to compute a fundamental domain for the action of the group of units Λ * on the Bruhat-Tits tree 𝒯 associated to PGL 2 (𝔽 q ((1/T))). This action is a function field analog of the action of a co-compact Fuchsian group on the upper half plane. The algorithm also yields an explicit presentation of the group Λ * in terms of generators and relations. Moreover we determine an upper bound for its running time using that Λ * 𝒯 is almost Ramanujan.

Soit Λ un 𝔽 q [T]-ordre maximal d’un corps de quaternions sur 𝔽 q (T) non-ramifié à la place . Cet article donne un algorithme pour calculer un domaine fondamental de l’action du groupe des unités Λ * sur l’arbre de Bruhat-Tits 𝒯 associé à PGL 2 (𝔽 q ((1/T))), l’action étant un analogue en corps de fonctions de l’action d’un groupe cocompact Fuchsian sur le demi-plan supérieur. L’algorithme donne également une présentation explicite du groupe Λ * par générateurs et relations. En outre nous trouvons une borne supérieure pour le temps de calcul en utilisant que le graphe quotient Λ * 𝒯 est presque de Ramanujan.

@article{JTNB_2012__24_1_73_0,
     author = {B\"ockle, Gebhard and Butenuth, Ralf},
     title = {On computing quaternion quotient graphs for function fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {24},
     number = {1},
     year = {2012},
     pages = {73-99},
     doi = {10.5802/jtnb.789},
     mrnumber = {2914902},
     zbl = {pre06075023},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2012__24_1_73_0}
}
Böckle, Gebhard; Butenuth, Ralf. On computing quaternion quotient graphs for function fields. Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 1, pp. 73-99. doi : 10.5802/jtnb.789. http://www.numdam.org/item/JTNB_2012__24_1_73_0/

[BCP] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language. J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. | MR 1484478 | Zbl 0898.68039

[Bu] R. Butenuth, Quaternionic Drinfeld modular forms. PhD thesis, in preparation.

[Cr] J. Cremona, The elliptic curve database for conductors to 130000. Algorithmic number theory (Berlin, 2006), Lecture Notes Comp. Sci. 4076, 11–29. Springer, Berlin, 2006. | MR 2282912

[De] L. Dembélé, Quaternionic Manin symbols, Brandt matrices and Hilbert modular forms. Math. Comp. 76 (2007), no. 258, 1039–1057. | MR 2291849

[GN] E.-U. Gekeler, U. Nonnengardt, Fundamental domains of some arithmetic groups over function fields. Int. J. Math. 6 (1995), 689–708. | MR 1351161 | Zbl 0858.11025

[GV] M. Greenberg, J. Voight, Computing systems of Hecke eigenvalues associated to Hilbert modular forms. Accepted in Math. Comp. | MR 2772112

[GY] P. Gunnells, D. Yasaki, Hecke operators and Hilbert modular forms. Algorithmic number theory (Berlin, 2008), Lecture Notes Comp. Sci. 5011, 387–401. Springer, Berlin, 2008. | MR 2467860

[He] F. Hess, Computing Riemann-Roch spaces in algebraic function fields and related topics J. Symbolic Computation 33 (2002), no. 4, 425–445. | MR 1890579

[JS] J. C. Jantzen, J. Schwermer, Algebra. Springer-Lehrbuch, 2006.

[KV] M. Kirschmer, J. Voight, Algorithmic enumeration of ideal classes for quaternion orders. SIAM J. Comput. 39 (2010), no. 5, 1714–1747. | MR 2592031

[Lu] A. Lubotzky, Discrete groups, expanding graphs and invariant measures. Birkhäuser, 1993. | MR 1308046

[LSV] A. Lubotzky, B. Samuels, U. Vishne, Ramanujan complexes of type A ˜ d * . Israel J. Math. 149 (2005), 267–299. | MR 2191217

[MS] V. K. Murty, J. Scherk, Effective versions of the Chebotarev density theorem for function fields. C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 6, 523–528. | MR 1298275 | Zbl 0822.11077

[Pa1] M. Papikian, Local diophantine properties of modular curves of 𝒟-elliptic sheaves. Accepted in J. reine angew. Math.

[Pa2] M. Papikian, On generators of arithmetic groups over function fields. Accepted in International Journal of Number Theory.

[Pau] S. Paulus, Lattice basis reduction in function fields. Proceedings of the Third Symposium on Algorithmic Number Theory, ANTS-III (1998), LNCS 1423, 567–575. | MR 1726102 | Zbl 0935.11045

[Ro] M. Rosen, Number theory in function fields. GTM 210. Springer, Berlin-New York, 2002. | MR 1876657

[Se1] J.-P. Serre, Trees. Springer, Berlin-New York, 1980. | MR 607504 | Zbl 0548.20018

[Se2] J.-P. Serre, A course in arithmetic. GTM 7. Springer, Berlin-New York, 1973. | MR 344216 | Zbl 0256.12001

[Te1] J.T. Teitelbaum, The Poisson Kernel For Drinfeld Modular Curves. J.A.M.S. 4 (1991), 491–511. | MR 1099281 | Zbl 0735.11025

[Te2] J.T. Teitelbaum, Modular symbols for A. Duke Math. J. 68 (1992), 271–295. | MR 1191561 | Zbl 0777.11021

[Sti] H. Stichtenoth, Algebraic Function Fields and Codes. GTM 254, Springer, Berlin-New York, (2009). | MR 2464941 | Zbl 0816.14011

[Ste] W. Stein, Modular forms database, (2004). http://modular.math.washington.edu/Tables.

[Vi] M.-F. Vignéras, Arithmétique des Algèbres de Quaternions. Lecture Notes in Math. 800. Springer, Berlin, 1980. | MR 580949 | Zbl 0422.12008

[Vo] J. Voight, Computing fundamental domains for Fuchsian groups. J. Théor. Nombres Bordeaux 21 (2009), 469–491. | Numdam | MR 2541438