Let be a maximal -order in a division quaternion algebra over 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 on the Bruhat-Tits tree associated to . 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 in terms of generators and relations. Moreover we determine an upper bound for its running time using that is almost Ramanujan.
Soit un -ordre maximal d’un corps de quaternions sur non-ramifié à la place . Cet article donne un algorithme pour calculer un domaine fondamental de l’action du groupe des unités sur l’arbre de Bruhat-Tits associé à , 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 par générateurs et relations. En outre nous trouvons une borne supérieure pour le temps de calcul en utilisant que le graphe quotient est presque de Ramanujan.
Published online:
DOI: 10.5802/jtnb.789
Author's affiliations:
@article{JTNB_2012__24_1_73_0, author = {Gebhard B\"ockle and Ralf Butenuth}, title = {On computing quaternion quotient graphs for function fields}, journal = {Journal de Th\'eorie des Nombres de Bordeaux}, pages = {73--99}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {24}, number = {1}, year = {2012}, doi = {10.5802/jtnb.789}, zbl = {pre06075023}, mrnumber = {2914902}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.789/} }
TY - JOUR TI - On computing quaternion quotient graphs for function fields JO - Journal de Théorie des Nombres de Bordeaux PY - 2012 DA - 2012/// SP - 73 EP - 99 VL - 24 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.789/ UR - https://zbmath.org/?q=an%3Apre06075023 UR - https://www.ams.org/mathscinet-getitem?mr=2914902 UR - https://doi.org/10.5802/jtnb.789 DO - 10.5802/jtnb.789 LA - en ID - JTNB_2012__24_1_73_0 ER -
Gebhard Böckle; Ralf Butenuth. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.789/
[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
[LSV] A. Lubotzky, B. Samuels, U. Vishne, Ramanujan complexes of type . Israel J. Math. 149 (2005), 267–299. | MR: 2191217
[Lu] A. Lubotzky, Discrete groups, expanding graphs and invariant measures. Birkhäuser, 1993. | MR: 1308046
[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 -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
[Ste] W. Stein, Modular forms database, (2004). http://modular.math.washington.edu/Tables.
[Sti] H. Stichtenoth, Algebraic Function Fields and Codes. GTM 254, Springer, Berlin-New York, (2009). | MR: 2464941 | Zbl: 0816.14011
[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 . Duke Math. J. 68 (1992), 271–295. | MR: 1191561 | Zbl: 0777.11021
[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
Cited by Sources: