On genera containing non-split Eichler orders over function fields

Luis Arenas-Carmona; Claudio Bravo

doi:10.5802/jtnb.1221

Luis Arenas-Carmona ¹ ; Claudio Bravo ¹

¹ Universidad de Chile, Facultad de Ciencias, Casilla 653, Santiago, Chile

Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 3, pp. 647-677.

Résumé
Abstract

Le théorème de Grothendieck–Birkhoff établit que tout faisceaux vectoriel de dimension finie sur la droite projective $ℙ^{1}$ se scinde en somme de faisceaux vectoriels unidimensionnels (fibrés en droites). Il peut être reformulé en termes d’ordres comme l’énoncé que tous les $ℙ^{1}$ -ordres maximaux se scindent. Ceci est utile, car les ordres scindés jouent un rôle important dans le calcul des graphes quotients. Dans ce travail, on étudie dans quelle mesure ce résultat se généralise aux $ℙ^{1}$ -ordres d’Eichler, lorsque le corps de base $𝔽$ est fini. Pour être précis, on caractérise, d’une part, les genres des ordres d’Eichler contenant uniquement des ordres scindés et, d’autre part, les genres ne contenant qu’un nombre fini de classes d’isomorphie non scindées. La méthode développée ici nous permet également de calculer les graphes quotients pour certains sous-groupes de ${PGL}_{2} (𝔽 [t])$ d’intérêt arithmétique.

Grothendieck–Birkhoff Theorem states that every finite dimensional vector bundle over the projective line $ℙ^{1}$ splits as the sum of one dimensional vector bundles (line bundles). This can be rephrased, in terms of orders, as stating that all maximal $ℙ^{1}$ -orders in a matrix algebra split. This is useful, since split orders play an important role when computing quotient graphs. In this work we study the extent to which this result can be generalized to Eichler $ℙ^{1}$ -orders when the base field $𝔽$ is finite. To be precise, we characterize both the genera of Eichler orders containing only split orders and the genera containing only a finite number of non-split isomorphism classes. The method developed here also allows us to compute quotient graphs for some subgroups of ${PGL}_{2} (𝔽 [t])$ of arithmetical interest.

Reçu le : 2021-04-17
Révisé le : 2022-06-11
Accepté le : 2022-09-23
Publié le : 2023-01-27

DOI : 10.5802/jtnb.1221

Classification : 11R58, 14H60, 14G15, 20E08
Mots clés : Global function fields, eichler orders, quotient graphs, vector bundles

Affiliations des auteurs :

Luis Arenas-Carmona ¹ ; Claudio Bravo ¹

¹ Universidad de Chile, Facultad de Ciencias, Casilla 653, Santiago, Chile

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JTNB_2022__34_3_647_0,
     author = {Luis Arenas-Carmona and Claudio Bravo},
     title = {On genera containing non-split {Eichler} orders over function fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {647--677},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {34},
     number = {3},
     year = {2022},
     doi = {10.5802/jtnb.1221},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1221/}
}

TY  - JOUR
AU  - Luis Arenas-Carmona
AU  - Claudio Bravo
TI  - On genera containing non-split Eichler orders over function fields
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2022
SP  - 647
EP  - 677
VL  - 34
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1221/
DO  - 10.5802/jtnb.1221
LA  - en
ID  - JTNB_2022__34_3_647_0
ER  -

%0 Journal Article
%A Luis Arenas-Carmona
%A Claudio Bravo
%T On genera containing non-split Eichler orders over function fields
%J Journal de théorie des nombres de Bordeaux
%D 2022
%P 647-677
%V 34
%N 3
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1221/
%R 10.5802/jtnb.1221
%G en
%F JTNB_2022__34_3_647_0

Luis Arenas-Carmona; Claudio Bravo. On genera containing non-split Eichler orders over function fields. Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 3, pp. 647-677. doi : 10.5802/jtnb.1221. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1221/

Bibliographie
Cité par

[1] Manuel Arenas; Luis Arenas-Carmona; Jaime Contreras On optimal embeddings and trees, J. Number Theory, Volume 193 (2018), pp. 91-117 | DOI | Zbl

[2] Luis Arenas-Carmona Representation fields for commutative orders, Ann. Inst. Fourier, Volume 62 (2012) no. 2, pp. 807-819 | DOI | Numdam | Zbl

[3] Luis Arenas-Carmona Representation fields for cyclic orders, Acta Arith., Volume 156 (2012) no. 2, pp. 143-158 | DOI | Zbl

[4] Luis Arenas-Carmona Eichler orders, trees and representation fields, Int. J. Number Theory, Volume 9 (2013), pp. 1725-1741 | DOI | Zbl

[5] Luis Arenas-Carmona Computing quaternion quotient graphs via representations of orders, J. Algebra, Volume 402 (2014), pp. 258-279 | DOI | Zbl

[6] Luis Arenas-Carmona Roots of unity in definite quaternion orders, Acta Arith., Volume 170 (2015) no. 4, pp. 381-393 | DOI | Zbl

[7] Luis Arenas-Carmona Spinor class fields for generalized Eichler orders, J. Théor. Nombres Bordeaux, Volume 28 (2016) no. 3, pp. 679-698 | DOI | Numdam | Zbl

[8] Lesya Bodnarchuk; Igor Burban; Yuriy Drozd; Gert-Martin Greuel Vector bundles and torsion free sheaves on degenerations of elliptic curves, Global aspects of complex geometry, Springer, 2006, pp. 83-129 | DOI | Zbl

[9] François Bruhat; Jacques Tits Groupes réductifs sur un corps local, Publ. Math., Inst. Hautes Étud. Sci., Volume 41 (1972), pp. 5-251 | DOI | Numdam | Zbl

[10] J. Brzezinski Riemann–Roch Theorem for locally principal orders, Math. Ann., Volume 276 (1987), pp. 529-536 | DOI | Zbl

[11] Kai-Uwe Bux; Ralf Köhl; Stefan Witzel Higher Finiteness Properties of Reductive Arithmetic Groups in Positive Characteristic: the Rank Theorem, Ann. Math., Volume 177 (2013) no. 1, pp. 311-366 | Zbl

[12] Ted Chinburg; Eduardo Friedman An embedding theorem for quaternion algebras, J. Lond. Math. Soc., Volume 60 (1999) no. 1, pp. 33-44 | DOI | Zbl

[13] Hiroaki Hijikata Explicit formula of the traces of Hecke operators for $Γ_{0} (N)$ , J. Math. Soc. Japan, Volume 26 (1974), pp. 56-82

[14] Ralf Köhl; Bernhard Mühlherr; Koen Struyve Quotients of trees for arithmetic subgroups of ${PGL}_{2}$ over a rational function field, J. Group Theory, Volume 18 (2015) no. 1, pp. 61-74 | DOI | Zbl

[15] Benjamin Linowitz Selectivity in quaternion algebras, J. Number Theory, Volume 132 (2012) no. 7, pp. 1425-1437 | DOI | Zbl

[16] Benjamin Linowitz; Thomas R. Shemanske Embedding orders in central simple algebras, J. Théor. Nombres Bordeaux, Volume 24 (2012), pp. 405-424 | DOI | Zbl

[17] Benjamin Linowitz; John Voight Small isospectral and nonisometric orbifolds of dimension 2 and 3, Math. Z., Volume 281 (2015) no. 1-2, pp. 523-569 | DOI | Zbl

[18] Benedictus Margaux The structure of the group $G (k [t])$ : Variations on a theme of Soulé, Algebra Number Theory, Volume 3 (2009) no. 4, pp. 393-409 | DOI | Zbl

[19] Alexander W. Mason Serre’s generalization of Nagao’s theorem: an elementary approach, Trans. Am. Math. Soc., Volume 353 (2001) no. 2, pp. 749-767 | DOI | Zbl

[20] Alexander W. Mason The generalization of Nagao’s theorem to other subrings of the rational function field, Commun. Algebra, Volume 31 (2003) no. 11, pp. 5199-5242 | DOI | Zbl

[21] Alexander W. Mason; Andreas Schweizer The minimum index of a non-congruence subgroup of ${SL}_{2}$ over an arithmetic domain, Isr. J. Math., Volume 133 (2003), pp. 29-44 | DOI | Zbl

[22] Alexander W. Mason; Andreas Schweizer The minimum index of a non-congruence subgroup of ${SL}_{2}$ over an arithmetic domain II. The rank zero cases, J. Lond. Math. Soc., Volume 71 (2005) no. 1, pp. 53-68 | DOI | Zbl

[23] Alexander W. Mason; Andreas Schweizer The stabilizers in a Drinfeld modular group of the vertices of its Bruhat-Tits tree: an elementary approach, Int. J. Algebra Comput., Volume 23 (2013) no. 7, pp. 1653-1683 | DOI | Zbl

[24] Hirosi Nagao On $GL (2, K [x])$ , J. Inst. Polytechn., Osaka City Univ., Ser. A, Volume 10 (1959), pp. 117-121 | Zbl

[25] O. Timothy O’Meara Introduction to quadratic forms, Grundlehren der Mathematischen Wissenschaften, 117, Springer, 1963 | DOI

[26] Mihran Papikian Local Diophantine properties of modular curves of $𝒟$ -elliptic sheaves, J. Reine Angew. Math., Volume 664 (2012), pp. 115-140 | Zbl

[27] Jean-Pierre Serre Le Probleme des Groupes de Congruence Pour ${SL}_{2}$ , Ann. Math., Volume 92 (1970), pp. 489-527 | DOI | Zbl

[28] Jean-Pierre Serre Trees, Springer, 1980 | DOI | Numdam

[29] Thomas R. Shemanske Split orders and convex polytopes in buildings, J. Number Theory, Volume 130 (2010) no. 1, pp. 101-115 | DOI | Zbl

[30] Christophe Soulé Chevalley groups over polynomial rings, Homological group theory (Durham, 1977) (London Mathematical Society Lecture Note Series), Volume 36, Cambridge University Press, 1979, pp. 359-368 | DOI | Zbl

[31] Shuzo Takahashi The fundamental domain of the tree of $G L (2)$ over the function field of an elliptic curve, Duke Math. J., Volume 72 (1993) no. 1, pp. 85-97 | Zbl

[32] Marie-France Vignéras Variétés Riemanniennes isospectrales et non isométriques, Ann. Math., Volume 112 (1980), pp. 21-32 | DOI | Zbl

[33] André Weil Basic Number Theory, Grundlehren der Mathematischen Wissenschaften, 144, Springer, 1973 | DOI

Cité par Sources :