On computing quaternion quotient graphs for function fields

Gebhard Böckle; Ralf Butenuth

doi:10.5802/jtnb.789

Gebhard Böckle; Ralf Butenuth ¹

¹ Interdisziplinäres Zentrum für wissenschaftliches Rechnen Im Neuenheimer Feld 368 69120 Heidelberg, Germany

Journal de Théorie des Nombres de Bordeaux, Volume 24 (2012) no. 1, pp. 73-99.

Abstract
Résumé

Let $Λ$ be a maximal $𝔽_{q} [T]$ -order in a division quaternion algebra over $𝔽_{q} (T)$ which is split at the place $\infty$ . 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 $\infty$ . 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.

Received: 2010-10-21
Published online: 2012-04-12

MR: 2914902 | Zbl: pre06075023

DOI: 10.5802/jtnb.789
Author's affiliations:

Gebhard Böckle ; Ralf Butenuth ¹

¹ Interdisziplinäres Zentrum für wissenschaftliches Rechnen Im Neuenheimer Feld 368 69120 Heidelberg, Germany

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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/

References
Cited by

[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 ${\tilde{A}}_{d}^{*}$ . 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 $𝒟$ -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

[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 $A$ . 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

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