# zbMATH — the first resource for mathematics

On Galois representations for abelian varieties with complex and real multiplications. (English) Zbl 1056.11034
This paper studies the image of $$\ell$$-adic representations coming from the Tate module of an abelian variety. Let $$A$$ be a simple abelian variety of nondegerate CM-type defined over a number field $$F$$ with $$\text{End}(A)\otimes\mathbb{Q}= E\subset F$$, and let $$\phi:T_{E'}\to T_E$$ be the homomorphism of tori defined in [K. A. Ribet, Mém. Soc. Math. Fr., Nouv. Sér. 2, 75–94 (1980; Zbl 0452.14009)], where $$E'$$ denotes the reflex field of $$E$$. The authors determine the image of $$\overline{\rho_\ell}: G_F\to \text{GL}_{2g}(\mathbb{F}_\ell)$$ explicitly under the assumption that the kernel of the map $$\phi$$ is connected. When $$\text{End}(A)\otimes\mathbb{Q}$$ is a totally real field of degree $$e$$ with $$g/e$$ odd, they determine the commutator subgroup of $$\overline{\rho_\ell}(G_F)$$ under some conditions on $$\ell$$. As a consequence they show the validity of the Mumford-Tate conjecture for such $$A$$.

##### MSC:
 11G10 Abelian varieties of dimension $$> 1$$ 11G15 Complex multiplication and moduli of abelian varieties 11R34 Galois cohomology
Full Text:
##### References:
 [1] Banaszak, G.; Gajda, W.; Krasoń, P., Support problem for the intermediate Jacobians of l-adic representations, J. number theory, 100, 133-168, (2003) · Zbl 1088.11040 [2] Bogomolov, F.A., Sur l’algébricité des représentations l-adiques, C. R. acad. sci. Paris Sér. A-B, 290, A701-A703, (1980) [3] Bourbaki, N., Groupes et algèbres de Lie, (1975), Hermann Paris [4] Corralez-Rodrigáñez, C.; Schoof, R., Support problem and its elliptic analogue, J. number theory, 64, 276-290, (1997) · Zbl 0922.11086 [5] Chi, W., l-adic and λ-adic representations associated to abelian varieties defined over a number field, Amer. J. math., 114, 3, 315-353, (1992) · Zbl 0795.14024 [6] P. Deligne, Hodge Cycles on Abelian Varieties, Lecture Notes in Mathematics, Vol. 900, 1982, pp. 9-100. · Zbl 0537.14006 [7] par M. Demazure, dirigé; Grothendieck, A., Schémas en groupes III, Lecture notes in mathematics, Vols. 151, 152, 153, (1970), Springer Berlin [8] Faltings, G., Endlichkeitssätze für abelsche varietäten über zalhkörpern, Invent. math., 73, 349-366, (1983) · Zbl 0588.14026 [9] Gordon, B., A survey of the Hodge conjecture for abelian varieties, appendix B, (), 297-356 [10] Hazama, F., Hodge cycles on abelian varieties of Sn-type, J. algebraic geom., 9, 711-753, (2000) · Zbl 1004.14003 [11] Humphreys, J.E., Linear algebraic groups, (1975), Springer Berlin · Zbl 0507.20017 [12] Humphreys, J.E., Introduction to Lie algebras and representation theory, (1972), Springer-Verlag Berlin · Zbl 0254.17004 [13] Ichikawa, T., Algebraic groups associated with abelian varieties, Math. ann., 289, 133-142, (1991) · Zbl 0697.14031 [14] Kubota, T., On the field extension by complex multiplication, Trans. amer. math. soc., 118, 6, 113-122, (1965) · Zbl 0146.27902 [15] Larsen, M.; Pink, R., Abelian varieties, l-adic representations and l independence, Math. ann., 302, 561-579, (1995) · Zbl 0867.14019 [16] Larsen, M.; Pink, R., A connectedness criterion for l-adic representations, Israel J. math., 97, 1-10, (1997) · Zbl 0870.11037 [17] Lang, S., Complex multiplication, (1983), Springer Berlin · Zbl 0536.14029 [18] Mumford, D., Abelian varieties, (1988), Oxford University Press Oxford · Zbl 0199.24601 [19] Neukirch, J., Class field theory, (1986), Springer Berlin · Zbl 0587.12001 [20] Nori, M.V., On subgroups of $$GLn(Fp)$$, Invent. math., 88, 257-275, (1987) [21] Ono, T., Arithmetic of algebraic tori, Ann. math., 74, 1, 101-139, (1961) · Zbl 0119.27801 [22] Pink, R., l-adic algebraic monodromy groups, cocharacters, and the mumford – tate conjecture, J. reine angew. math., 495, 187-237, (1998) · Zbl 0920.14006 [23] Ribet, K.A., Galois action on division points of abelian varieties with real multiplications, Amer. J. math., 98, 3, 751-804, (1976) · Zbl 0348.14022 [24] Ribet, K.A., Dividing rational points of abelian varieties of CM type, Compositio math., 33, 69-74, (1976) · Zbl 0331.14020 [25] Ribet, K.A., Division points of abelian varieties with complex multiplication, Mem. soc. math. France 2e ser., 2, 75-94, (1980) · Zbl 0452.14009 [26] J.P. Serre, Résumés des cours au Collège de France, Ann. Collège France (1985-1986) 95-100. [27] J.P. Serre, Lettre à Daniel Bertrand du 8/6/1984, Oeuvres, Collected Papers IV, Springer, Berlin, 1985-1998, pp. 21-26. [28] J.P. Serre, Lettre à Marie-France Vignéras du 10/2/1986, Oeuvres, Collected Papers IV, Springer, Berlin, 1985-1998, pp. 38-55. [29] J.P. Serre, Lettres à Ken Ribet du 1/1/1981 et du 29/1/1981, Oeuvres, Collected Papers IV, Springer, Berlin, 1985-1998, pp. 1-20. [30] J.P. Serre, Représentations l-adiques, in: S. Iyanaga (Ed.), Algebraic Number Theory, Japan Society for the Promotion of Science, Kyoto University Press, 1977, pp. 177-193. [31] Serre, J.P.; Tate, J., Good reduction of abelian varieties, Ann. of math., 68, 492-517, (1968) · Zbl 0172.46101 [32] Tankeev, S.G., On algebraic cycles on surfaces and abelian varieties, Izv. acad. nauk SSSR. ser. mat., 45, 2, 398-434, (1981) · Zbl 0493.14014 [33] Tankeev, S.G., On the mumford – tate conjecture for abelian varieties, Algebraic geom. 4, J. math. sci., 81, 3, 2719-2737, (1996) · Zbl 0889.14022 [34] Voskresensky, V.E., Algebraiceskije tory, (1977), Izdatelstvo Nauka Moscow [35] White, S., Sporadic cycles on CM abelian varieties, Compositio math., 88, 123-142, (1993) · Zbl 0798.14025 [36] J.P. Wintenberger, Démonstration d’une conjecture de Lang dans des cas particuliers, preprint, October 3, 2000, pp. 1-25.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.