Geodesic
Hypergeometric Abelian Varieties
Canadian journal of mathematics, Tome 55 (2003) no. 5, pp. 897-932

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper, we construct abelian varieties associated to Gauss’ and Appell–Lauricella hypergeometric series. Abelian varieties of this kind and the algebraic curves we define to construct them were considered by several authors in settings ranging from monodromy groups (Deligne, Mostow), exceptional sets (Cohen, Wolfart, Wüstholz), modular embeddings (Cohen, Wolfart) to CM-type (Cohen, Shiga, Wolfart) and modularity (Darmon). Our contribution is to provide a complete, explicit and self-contained geometric construction.
DOI : 10.4153/CJM-2003-037-4
Mots-clés : 11, 14 
Archinard, Natália. Hypergeometric Abelian Varieties. Canadian journal of mathematics, Tome 55 (2003) no. 5, pp. 897-932. doi: 10.4153/CJM-2003-037-4
@article{10_4153_CJM_2003_037_4,
     author = {Archinard, Nat\'alia},
     title = {Hypergeometric {Abelian} {Varieties}},
     journal = {Canadian journal of mathematics},
     pages = {897--932},
     year = {2003},
     volume = {55},
     number = {5},
     doi = {10.4153/CJM-2003-037-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2003-037-4/}
}
TY  - JOUR
AU  - Archinard, Natália
TI  - Hypergeometric Abelian Varieties
JO  - Canadian journal of mathematics
PY  - 2003
SP  - 897
EP  - 932
VL  - 55
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2003-037-4/
DO  - 10.4153/CJM-2003-037-4
ID  - 10_4153_CJM_2003_037_4
ER  - 
%0 Journal Article
%A Archinard, Natália
%T Hypergeometric Abelian Varieties
%J Canadian journal of mathematics
%D 2003
%P 897-932
%V 55
%N 5
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-2003-037-4/
%R 10.4153/CJM-2003-037-4
%F 10_4153_CJM_2003_037_4

[1] [1] Appell, P., Sur les fonctions hypergéométriques de deux variables. Jour. de Math., 3ème sér. VIII(1882), 173–216. Google Scholar

[2] [2] Archinard, N., Abelian Varieties and Identies for Hypergeometric Series. PhD thesis, ETH Zurich, 2000. http://e-collection.ethbib.ethz.ch. Google Scholar

[3] [3] Archinard, N., Exceptional Sets of Hypergeometric Series. J. Number Theory 101(2003), 244–269. Google Scholar

[4] [4] Beukers, F. and Wolfart, J., Algebraic values of hypergeometric functions. In: New Advances in Transcendence Theory, (ed. Baker, A.), Cambrige University Press, 1986. Google Scholar

[5] [5] Brieskorn, E. and Knörrer, H., Ebene Algebraische Kurven. Birkhäuser, 1981. Google Scholar

[6] [6] Chevalley, C. and Weil, A., Über das Verhalten der Integrale 1. Gattung bei Automorphismen des Funktionenkörpers. Abh. HamburgerMath. Sem. 10(1934), 358–361. Google Scholar

[7] [7] Cohen, P. and Wolfart, J., Modular embeddings for some non-arithmetic Fuchsian groups. Acta Arith. 56(1990), 93–110. Google Scholar

[8] [8] Cohen, P. B. and Wolfart, J., Algebraic Appell-Lauricella functions. In: Special Differential Equations (ed. Yoshida, M.), Proceedings of the Taniguchi Workshop, 1991, 150–164, Google Scholar

[9] [9] Cohen, P. B. and Wolfart, J., Fonctions hypergéométriques en plusieurs variables et espaces des modules de variétés abéliennes. Ann. Sci. École Norm. Sup. (4) 26(1993), 665–690. Google Scholar

[10] [10] Cohen, P. B. and G.Wüstholz, Applications of the André-Oort Conjecture to transcendence. In: A Panorama in Number Theory, ed. Wüstholz, G., Cambrige University Press, 2001. Google Scholar

[11] [11] Darmon, H., Modularity of fibres in rigid local systems. Ann. of Math. 149(1999), 1079–1086. Google Scholar

[12] [12] Darmon, H., Rigid local systems, Hilbert modular forms, and Fermat's last theorem. Duke Math. J. 102(2000), 413–449. Google Scholar

[13] [13] Deligne, P. and Mostow, G. D., Monodromy of hypergeometric functions and non-lattice integral monodromy. Inst. Hautes Études Sci. Publ. Math. 63(1986), 5–89. Google Scholar

[14] [14] Edixhoven, S. and Yafaev, A., Subvarieties of Shimura varieties. Ann. of Math., to appear. Google Scholar

[15] [15] Lauricella, G., Sulle funzioni ipergeometriche a piu variabili. Rendiconti di Palermo VII(1893), 111–158. Google Scholar

[16] [16] Shafarevich, I. R., Basic Algebraic Geometry. Volumes 1 and 2, Springer-Verlag, 1994. Google Scholar

[17] [17] Shiga, H. and Wolfart, J., Criteria for complex multiplication and transcendence properties of automorphic functions. J. Reine Angew.Math. 463(1995), 1–25. Google Scholar

[18] [18] Terada, T., Problème de Riemann et fonctions automorphes provenant de fonctions hypergéométriques de plusieurs variables. J. Math. Kyoto Univ. 13-3(1973), 557–578. Google Scholar

[19] [19] van der Put, M. and Ulmer, F., Differential equations and finite groups. J. Algebra 226(2000), 920–966. Google Scholar

[20] [20] Gross, H. (with an appendix by Rohrlich, D.), On the periods of abelian integrals and a formula of Chowla and Selberg. Invent. Math. 45(1978), 193–211. Google Scholar

[21] [21] Wolfart, J., Fonctions hypergéometriques, arguments exceptionels et groupes de monodromie. Publ. Math. Univ. P. et M. Curie, Problèmes Diophantiens 79(1985–86), 1–24. Google Scholar

[22] [22] Wolfart, J., Werte hypergeometrischer Funktionen. Invent. Math. 92(1988), 187–216. Google Scholar

[23] [23] Wolfart, J. and Wüstholz, G., Der Ueberlagerungsradius gewisser algebraischer Kurven und die Werte der Betafunktion an rationalen Stellen. Math. Ann. 273(1985), 1–15. Google Scholar

[24] [24] Wüstholz, G., Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen. Ann. Math. 129(1989), 501–517. Google Scholar

Cité par Sources :