On the $\ell$-adic Galois representations attached to nonsimple abelian varieties

Lombardo, Davide

Annales de l'Institut Fourier, Tome 66 (2016) no. 3, pp. 1217-1245.

Nous étudions les représentations galoisiennes associées aux variétés abéliennes non simples définies sur des corps de type fini de caractéristique quelconque. Nous donnons des conditions suffisantes pour que ces représentations se décomposent en produit et nous les utilisons pour montrer des analogues arithmétiques de certains résultats antérieurs de Moonen et Zarhin concernant les variétés abéliennes complexes (de dimension au plus 5).

We study Galois representations attached to nonsimple abelian varieties over finitely generated fields of arbitrary characteristic. We give sufficient conditions for such representations to decompose as a product, and apply them to prove arithmetical analogues of results shown by Moonen and Zarhin in the context of complex abelian varieties (of dimension at most 5).

Classification : 11G10, 14K15, 11F80
Keywords: Tate classes, Hodge group, Galois representations, abelian varieties, Mumford-Tate conjecture
Mot clés : classes de Tate, groupe de Hodge, représentations de Galois, variétés abéliennes, conjecture de Mumford-Tate

[1] Banaszak, Grzegorz; Gajda, Wojciech; Krasoń, Piotr On the image of $\ell$-adic Galois representations for abelian varieties of type I and II, Doc. Math. (2006) no. Extra Vol., p. 35-75 (electronic)

[2] Banaszak, Grzegorz; Gajda, Wojciech; Krasoń, Piotr On the image of Galois $l$-adic representations for abelian varieties of type III, Tohoku Math. J. (2), Volume 62 (2010) no. 2, pp. 163-189 | DOI

[3] Bogomolov, Fedor Alekseivich Sur l’algébricité des représentations $l$-adiques, C. R. Acad. Sci. Paris Sér. A-B, Volume 290 (1980) no. 15, p. A701-A703

[4] Bourbaki, Nicolas Lie groups and Lie algebras. Chapters 7–9, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2005, xii+434 pages (Translated from the 1975 and 1982 French originals by Andrew Pressley)

[5] Chi, W. C. $l$-adic and $\lambda$-adic representations associated to abelian varieties defined over number fields, Amer. J. Math., Volume 114 (1992) no. 2, pp. 315-353 | DOI

[6] Deligne, Pierre; Milne, James S.; Ogus, Arthur; Shih, Kuang-yen Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, 900, Springer-Verlag, Berlin-New York, 1982, ii+414 pages

[7] Faltings, G. Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., Volume 73 (1983) no. 3, pp. 349-366 | DOI

[8] Faltings, G. Complements to Mordell, Rational points (Bonn, 1983/1984) (Aspects Math., E6), Vieweg, Braunschweig, 1984, pp. 203-227

[9] Hazama, Fumio Algebraic cycles on certain abelian varieties and powers of special surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 31 (1985) no. 3, pp. 487-520

[10] Hazama, Fumio Algebraic cycles on nonsimple abelian varieties, Duke Math. J., Volume 58 (1989) no. 1, pp. 31-37 | DOI

[11] Ichikawa, Takashi Algebraic groups associated with abelian varieties, Math. Ann., Volume 289 (1991) no. 1, pp. 133-142 | DOI

[12] Larsen, M.; Pink, R. Determining representations from invariant dimensions, Invent. Math., Volume 102 (1990) no. 2, pp. 377-398 | DOI

[13] Larsen, M.; Pink, R. On $l$-independence of algebraic monodromy groups in compatible systems of representations, Invent. Math., Volume 107 (1992) no. 3, pp. 603-636 | DOI

[14] Larsen, M.; Pink, R. Abelian varieties, $\ell$-adic representations, and $\ell$-independence, Math. Ann., Volume 302 (1995) no. 3, pp. 561-579 | DOI

[15] Moonen, B. J. J. Notes on Mumford-Tate Groups (http://www.math.ru.nl/~bmoonen/Lecturenotes/CEBnotesMT.pdf)

[16] Moonen, B. J. J.; Zarhin, Yu. G. Hodge classes and Tate classes on simple abelian fourfolds, Duke Math. J., Volume 77 (1995) no. 3, pp. 553-581 | DOI

[17] Moonen, B. J. J.; Zarhin, Yu. G. Hodge classes on abelian varieties of low dimension, Math. Ann., Volume 315 (1999) no. 4, pp. 711-733 | DOI

[18] Moret-Bailly, L. Pinceaux de variétés abéliennes, Astérisque (1985) no. 129, 266 pages

[19] Murty, V. Kumar Exceptional Hodge classes on certain abelian varieties, Math. Ann., Volume 268 (1984) no. 2, pp. 197-206 | DOI

[20] Noot, R. Abelian varieties—Galois representation and properties of ordinary reduction, Compositio Math., Volume 97 (1995) no. 1-2, pp. 161-171 (Special issue in honour of Frans Oort)

[21] Pink, Richard $l$-adic algebraic monodromy groups, cocharacters, and the Mumford-Tate conjecture, J. Reine Angew. Math., Volume 495 (1998), pp. 187-237 | DOI

[22] Pink, Richard On Weil restriction of reductive groups and a theorem of Prasad, Math. Z., Volume 248 (2004) no. 3, pp. 449-457 | DOI

[23] Pohlmann, H. Algebraic cycles on abelian varieties of complex multiplication type, Ann. of Math. (2), Volume 88 (1968), pp. 161-180

[24] Ramón Marí, J. J. On the Hodge conjecture for products of certain surfaces, Collect. Math., Volume 59 (2008) no. 1, pp. 1-26 | DOI

[25] Ribet, Kenneth A. Galois action on division points of Abelian varieties with real multiplications, Amer. J. Math., Volume 98 (1976) no. 3, pp. 751-804

[26] Ribet, Kenneth A. Hodge classes on certain types of abelian varieties, Amer. J. Math., Volume 105 (1983) no. 2, pp. 523-538 | DOI

[27] Serre, Jean-Pierre Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math., Volume 15 (1972) no. 4, pp. 259-331

[28] Serre, Jean-Pierre Abelian $l$-adic representations and elliptic curves, Research Notes in Mathematics, 7, A K Peters, Ltd., Wellesley, MA, 1998, 199 pages (With the collaboration of Willem Kuyk and John Labute, Revised reprint of the 1968 original)

[29] Shimura, Goro On analytic families of polarized abelian varieties and automorphic functions, Ann. of Math. (2), Volume 78 (1963), pp. 149-192

[30] Shioda, Tetsuji Algebraic cycles on abelian varieties of Fermat type, Math. Ann., Volume 258 (1981/82) no. 1, pp. 65-80 | DOI

[31] Ullmo, Emmanuel; Yafaev, Andrei Mumford-Tate and generalised Shafarevich conjectures, Ann. Math. Qué., Volume 37 (2013) no. 2, pp. 255-284 | DOI

[32] Vasiu, A. Some cases of the Mumford-Tate conjecture and Shimura varieties, Indiana Univ. Math. J., Volume 57 (2008) no. 1, pp. 1-75 | DOI

[33] Zarhin, Yu. G. Endomorphisms of Abelian varieties over fields of finite characteristic, Izv. Akad. Nauk SSSR Ser. Mat., Volume 39 (1975) no. 2, p. 272-277, 471

[34] Zarhin, Yu. G. Abelian varieties in characteristic $p$, Mat. Zametki, Volume 19 (1976) no. 3, pp. 393-400

[35] Zarhin, Yu. G. Torsion of abelian varieties in finite characteristic, Math. Notes, Volume 22 (1978) no. 1, pp. 493-498

[36] Zarhin, Yu. G. Abelian varieties, $l$-adic representations and Lie algebras. Rank independence on $l$, Invent. Math., Volume 55 (1979) no. 2, pp. 165-176 | DOI

[37] Zarhin, Yu. G. Weights of simple Lie algebras in the cohomology of algebraic varieties, Izv. Akad. Nauk SSSR Ser. Mat., Volume 48 (1984) no. 2, pp. 264-304

[38] Zarhin, Yu. G. Abelian varieties without homotheties, Math. Res. Lett., Volume 14 (2007) no. 1, pp. 157-164 | DOI

[39] Zhao, B. On the Mumford-Tate conjecture of Abelian fourfolds (2013) (http://www.math.ucla.edu/~zhaobin/MTAV4.pdf)