Abstract
We give an overview of some results and techniques related to the Mumford–Tate conjecture for motives over finitely generated fields of characteristic 0. In particular, we explain how working in families can lead to non-trivial results.
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References
Abdulali S.: Algebraic cycles in families of abelian varieties. Canad. J. Math. 46(6), 1121–1134 (1994)
Y. André, Une remarque à propos des cycles de Hodge de type CM. In: Sém. de Théorie des Nombres, Paris, 1989–90, pp. 1–7. Progr. Math. 102, Birkhäuser, Boston, 1992.
André Y.: Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part. Compositio Math. 82(1), 1–24 (1992)
André Y.: On the Shafarevich and Tate conjectures for hyperkähler varieties. Math. Ann. 305(2), 205–248 (1996)
André Y.: Pour une théorie inconditionnelle des motifs. Inst. Hautes Études Sci. Publ. Math. No. 83, 5–49 (1996)
Y. André, Une introduction aux motifs (motifs purs, motifs mixtes, périodes). Panoramas et Synthèses, 17. Société Mathématique de France, Paris, 2004.
André Y.: Déformation et spécialisation de cycles motivés. J. Inst. Math. Jussieu 5(4), 563–603 (2006)
Arapura D.: An abelian category of motivic sheaves. Adv. Math. 233, 135–195 (2013)
G. Banaszak, W. Gajda, P. Krasoń, On the image of \({\ell}\)-adic Galois representations for abelian varieties of type I and II. Doc. Math. 2006, Extra Vol., 35–75.
Banaszak G., Gajda W., Krasoń P.: On the image of Galois \({\ell}\)-adic representations for abelian varieties of type III. Tohoku Math. J. (2) 62(2), 163–189 (2010)
A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers. In: Analysis and topology on singular spaces, I (Luminy, 1981), 5–171. Astérisque, 100, Soc. Math. France, Paris, 1982.
Bhatt B., Scholze P.: The pro-étale topology for schemes. Astérisque No. 369, 99–201 (2015)
D. Blasius, A \({p}\)-adic property of Hodge classes on abelian varieties. In: Motives (Seattle, WA, 1991), 293–308. Proc. Sympos. Pure Math., 55, Part 2, Amer. Math. Soc., Providence, RI, 1994.
Bogomolov F.: Sur l’algébricité des représentations \({\ell}\)-adiques. C. R. Acad. Sci. Paris Sér. A–B 290(15), A701–A703 (1980)
M. Borovoi, The action of the Galois group on the rational cohomology classes of type \({(p, p)}\) of abelian varieties. (Russian) Mat. Sb. (N.S.) 94(136) (1974), 649–652, 656.
N. Bourbaki, Groupes et algèbres de Lie, Chap. 7 et 8 (Nouveau tirage). Éléments de mathématique, Masson, Paris, 1990.
J-L. Brylinski, “1-motifs” et formes automorphes (théorie arithmétique des domaines de Siegel). In: Conference on automorphic theory (Dijon, 1981), 43–106. Publ. Math. Univ. Paris VII, 15, Univ. Paris VII, Paris, 1983.
A. Cadoret,Motivated cycles under specialization. In: Geometric and differential Galois theories, 25–55. Sémin. Congr., 27, Soc. Math. France, Paris, 2013.
Cadoret A.: On \({\ell}\) –independency in families of motivic \({\ell}\) –adic representations. Manuscripta Math. 147(3–4), 381–398 (2015)
Cadoret A., Kret A.: Galois-generic points on Shimura varieties. Algebra Number Theory 10(9), 1893–1934 (2016)
A. Cadoret, B. Moonen, Integral and adelic aspects of the Mumford–Tate conjecture. Preprint, https://arxiv.org/abs/1508.06426.
Cadoret A., Tamagawa A.: A uniform open image theorem for \({\ell}\)-adic representations. I. Duke Math. J. 161(13), 2605–2634 (2012)
Cadoret A., Tamagawa A.: A uniform open image theorem for \({\ell}\)-adic representations. II. Duke Math. J. 162(12), 2301–2344 (2013)
Cattani E., Deligne P., Kaplan A.: On the locus of Hodge classes. J. Amer. Math. Soc. 8(2), 483–506 (1995)
F. Charles, C. Schnell, Notes on absolute Hodge classes. In: Hodge theory, 469–530, Math. Notes, 49, Princeton Univ. Press, Princeton, NJ, 2014.
Chi W.: \({\ell}\)-adic and \({\lambda}\)-adic representations associated to abelian varieties defined over number fields. Amer. J. Math. 114(2), 315–353 (1992)
J. Commelin, On \({\ell}\)-adic compatibility for abelian motives & the Mumford–Tate conjecture for products of K3 surfaces. PhD thesis, Radboud University Nijmegen, 2017.
J. Commelin, On compatibility of the \({\ell}\)-adic realisations of an abelian motive. Preprint, https://arxiv.org/abs/1706.09444.
de Jong A.: Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic. Invent. Math. 134(2), 301–333 (1998)
P. Deligne, Équations différentielles à points singuliers réguliers. Lecture Notes in Mathematics, 163. Springer-Verlag, Berlin-New York, 1970.
Deligne P.: Théorie de Hodge. II. Inst. Hautes études Sci. Publ. Math. No. 40, 5–57 (1971)
Deligne P.: La conjecture de Weil pour les surfaces K3. Invent. Math. 15, 206–226 (1972)
P. Deligne, Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques. In: Automorphic forms, representations and L-functions. Proc. Sympos. Pure Math. 33, Amer. Math. Soc., Providence, R.I., 1979; Part 2, pp. 247–289.
P. Deligne, Valeurs de fonctions L et périodes d’intégrales. In: Automorphic forms, representations and L-functions. Proc. Sympos. Pure Math. 33, Amer. Math. Soc., Providence, R.I., 1979; Part 2, pp. 313–346.
P. Deligne, Hodge cycles on abelian varieties. (Notes by J. Milne.) In: P. Deligne, et al., Hodge cycles, motives, and Shimura varieties. Lecture Notes in Mathematics, 900. Springer-Verlag, Berlin-New York, 1982.
Deligne, P., La conjecture de Weil. II. Inst. Hautes études Sci. Publ. Math. No. 52 (1980), 137–252.
P. Deligne, J. Milne, Tannakian categories. In: P. Deligne, et al., Hodge cycles, motives, and Shimura varieties. Lecture Notes in Mathematics, 900. Springer-Verlag, Berlin-New York, 1982.
G. Faltings, Complements to Mordell. In: Rational points (G. Faltings, G. Wüstholz, eds.). Aspects of Math. E6, Friedr. Vieweg & Sohn, Braunschweig, 1984.
A. Grothendieck, Revêtements étales et groupe fondamental (SGA1). Séminaire de géométrie algébrique du Bois Marie 1960–61. Updated and annotated reprint of the 1971 original. Documents Math., 3. Société Mathématique de France, Paris, 2003.
A. Grothendieck, On the de Rham cohomology of algebraic varieties. Inst. Hautes Études Sci. Publ. Math. No. 29 (1966), 95–103.
Hazama F.: Algebraic cycles on certain abelian varieties and powers of special surfaces. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31(3), 487–520 (1985)
Hazama F.: Algebraic cycles on nonsimple abelian varieties. Duke Math. J. 58(1), 31–37 (1989)
Hindry M., Ratazzi N.: Torsion pour les variétés abéliennes de type I et II. Algebra Number Theory 10(9), 1845–1891 (2016)
A. Huber, S. Müller-Stach, Periods and Nori motives. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 65. Springer, Cham, 2017.
Hui C.: Specialization of monodromy group and \({\ell}\)-independence. C. R. Math. Acad. Sci. Paris 350(1–2), 5–7 (2012)
L. Illusie, Cohomologie de de Rham et cohomologie étale p-adique (d’après G. Faltings, J.-M. Fontaine et al.). Séminaire Bourbaki, Vol. 1989/90. Astérisque No. 189–190 (1990), Exp. No. 726, 325–374.
U. Jannsen, Mixed motives and algebraic K-theory. Lecture Notes in Mathematics, 1400. Springer-Verlag, Berlin, 1990.
Jannsen U.: Weights in arithmetic geometry. Jpn. J. Math. 5(1), 73–102 (2010)
P. Jossen, On the relation between Galois groups and Motivic Galois groups. Preprint, 2016, available at http://www.jossenpeter.ch/Maths.htm.
J. Lewis, A survey of the Hodge conjecture (Second edition). CRM Monograph Series, 10. American Mathematical Society, Providence, RI, 1999.
Larsen M., Pink R.: On \({\ell}\)-independence of algebraic monodromy groups in compatible systems of representations. Invent. Math. 107(3), 603–636 (1992)
M. Larsen, R. Pink, Abelian varieties, \({\ell}\)-adic representations, and \({\ell}\)-independence. Math. Ann. 302 (1995), no. 3, 561–579.
Laskar A.: \({\ell}\)-independence for a system of motivic representations. Manuscripta Math. 145(1-2), 125–142 (2014)
D. Lombardo, On the \({\ell}\)-adic Galois representations attached to nonsimple abelian varieties. Ann. Inst. Fourier (Grenoble) 66 (2016), no. 3, 1217–1245.
Moonen B.: On the Tate and Mumford-Tate conjectures in codimension 1 for varieties with \({h^{2,0} = 1}\). Duke Math. J. 166(4), 739–799 (2017)
B. Moonen, The Deligne–Mostow List and Special Families of Surfaces. Int. Math. Res. Not. Vol. 2017, 33pp. https://doi.org/10.1093/imrn/rnx055.
B. Moonen, A remark on the Tate conjecture. Preprint, https://arxiv.org/abs/1709.04489.
Moonen B., Zarhin Yu.: Hodge classes and Tate classes on simple abelian fourfolds. Duke Math. J. 77(3), 553–581 (1995)
Moonen B., Zarhin Yu.: Hodge classes on abelian varieties of low dimension. Math. Ann. 315(4), 711–733 (1999)
D. Mumford, Families of abelian varieties. In: Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., Boulder, Colo., 1965, pp. 347–351, Amer. Math. Soc., Providence, R.I., 1966.
Mumford D.: A note of Shimura’s paper "Discontinuous groups and abelian varieties". Math. Ann. 181, 345–351 (1969)
D. Mumford, Abelian varieties. Tata Institute of Fundamental Research Studies in Math. 5. Oxford University Press, Oxford, 1970.
Murty V.K.: Algebraic cycles on abelian varieties. Duke Math. J. 50(2), 487–504 (1983)
Murty V.K.: Exceptional Hodge classes on certain abelian varieties. Math. Ann. 268(2), 197–206 (1984)
S. Pepin Lehalleur, Subgroups of maximal rank of reductive groups. In: Autour des schémas en groupes. Vol. III, 147–172, Panor. Synthèses, 47, Soc. Math. France, Paris, 2015.
C. Peters, J. Steenbrink, Mixed Hodge structures. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 52. Springer-Verlag, Berlin, 2008.
Pink R.: \({\ell}\)-adic algebraic monodromy groups, cocharacters, and the Mumford-Tate conjecture. J. reine angew. Math. 495, 187–237 (1998)
Pohlmann H.: Algebraic cycles on abelian varieties of complex multiplication type. Ann. of Math. (2) 88, 161–180 (1968)
J. Ramón Marí, On the Hodge conjecture for products of certain surfaces. Collect. Math. 59 (2008), no. 1, 1–26.
Ribet K.: Hodge classes on certain types of abelian varieties. Amer. J. Math. 105(2), 523–538 (1983)
Schmid W.: Variation of Hodge structure: the singularities of the period mapping.Invent. Math. 22, 211–319 (1973)
Schoen C.: Hodge classes on self-products of a variety with an automorphism. Compositio Math. 65(1), 3–32 (1988)
Schoen C.: Varieties dominated by product varieties. Internat. J. Math. 7(4), 541–571 (1996)
C. Schoen, Addendum to [72]. Compositio Math. 114 (1998), no. 3, 329–336 (1998)
Serre J-P.: Sur les groupes de congruence des variétés abéliennes. Izv. Akad. Nauk SSSR Ser. Mat. 28, 3–20 (1964)
J-P. Serre, Résumé des cours de 1965–1966. In: OEuvres, Volume II, number 71. Springer-Verlag, Berlin, 1986.
J-P. Serre, Abelian \({\ell}\)-adic representations and elliptic curves. McGill University lecture notes. Benjamin, Inc., New York-Amsterdam, 1968.
Serre J-P.: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15(4), 259–331 (1972)
J-P. Serre, Représentations \({\ell}\)-adiques. In: Algebraic number theory (Kyoto Internat. Sympos., Res. Inst. Math. Sci., Univ. Kyoto, Kyoto, 1976), pp. 177–193. Japan Soc. Promotion Sci., Tokyo, 1977.
J-P. Serre, Groupes algébriques associés aux modules de Hodge-Tate. In: Journées de Géom. Algébrique de Rennes. Vol. III, pp. 155–188. Astérisque, 65, Soc. Math. France, Paris, 1979.
J-P. Serre, Lettres à Ken Ribet du 1/1/1981 et du 29/1/1981. In: OEuvres, Volume IV, number 133. Springer-Verlag, Berlin, 2000.
J-P. Serre, Lettre à John Tate du 2/1/1985. In: Correspondance Serre–Tate. Vol. II. Documents Mathématiques, 14. Soc. Math. France, Paris, 2015.
J-P. Serre, Résumé des cours de 1984–1985. In: OEuvres, Volume IV, number 135. Springer-Verlag, Berlin, 2000.
J-P. Serre, Propriétés conjecturales des groupes de Galois motiviques et des représentations \({\ell}\)-adiques. In: Motives (Seattle, WA, 1991), 377–400, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.
J-P. Serre, Lectures on the Mordell-Weil theorem. Third edition. Aspects of Mathematics. Friedr. Vieweg & Sohn, Braunschweig, 1997.
Shioda T.: The Hodge conjecture for Fermat varieties. Math. Ann. 245(2), 175–184 (1979)
T. Shioda, What is known about the Hodge conjecture? In: Algebraic varieties and analytic varieties (Tokyo, 1981), 55–68. Adv. Stud. Pure Math. 1, North-Holland, Amsterdam-New York, 1983.
S. Tankeev, Algebraic cycles on abelian varieties. II. Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 2, 418–429.
S. Tankeev, Algebraic cycles on surfaces and abelian varieties. Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 2, 398–434, 463–464.
S. Tankeev, Cycles on simple abelian varieties of prime dimension. Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 1, 155–170, 192.
S. Tankeev, Cycles on simple abelian varieties of prime dimension over number fields. Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 6, 1214–1227, 1358; translation in Math. USSR-Izv. 31 (1988), no. 3, 527–540.
S. Tankeev, Surfaces of K3 type over number fields and the Mumford-Tate conjecture. Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 4, 846–861; translation in Math. USSRIzv. 37 (1991), no. 1, 191–208.
S. Tankeev, Surfaces of K3 type over number fields and the Mumford-Tate conjecture. II. Izv. Ross. Akad. Nauk Ser. Mat. 59 (1995), no. 3, 179–206; translation in Izv. Math. 59 (1995), no. 3, 619–646.
Tankeev S.: On the Mumford-Tate conjecture for abelian varieties. Algebraic geometry, 4. J. Math. Sci. 81(3), 2719–2737 (1996)
Tate J.: Endomorphisms of abelian varieties over finite fields. Invent. Math. 2, 134–144 (1966)
J. Tate, Conjectures on algebraic cycles in \({\ell}\)-adic cohomology. In: Motives (Seattle, WA, 1991), 71–83. Proc. Sympos. Pure Math., 55, Part 1, American Mathematical Society, Providence, RI, 1994.
R. Taylor, Galois representations. Ann. Fac. Sci. Toulouse Math. (6) 13 (2004), no. 1, 73–119.
Ullmo E., Yafaev A.: Mumford-Tate and generalised Shafarevich conjectures. Ann. Math. Qué. 37(2), 255–284 (2013)
van Geemen B.: Theta functions and cycles on some abelian fourfolds. Math. Z. 221(4), 617–631 (1996)
van Geemen B.: Half twists of Hodge structures of CM-type. J. Math. Soc. Japan 53(4), 813–833 (2001)
van Geemen B., Izadi E.: Half twists and the cohomology of hypersurfaces. Math. Z. 242(2), 279–301 (2002)
Vasiu A.: Some cases of the Mumford-Tate conjecture and Shimura varieties. Indiana Univ. Math. J. 57(1), 1–75 (2008)
Voisin C.: Some aspects of the Hodge conjecture. Jpn. J. Math. 2(2), 261–296 (2007)
Voisin C.: Hodge loci and absolute Hodge classes. Compos. Math. 143(4), 945–958 (2007)
C. Voisin, Hodge theory and complex algebraic geometry, vols. I and II. Cambridge Studies in Advanced Mathematics, vols. 76, 77. Cambridge University Press, Cambridge, 2002–03.
C. Voisin, The Hodge conjecture. Open problems in mathematics, 521–543, Springer, 2016.
A. Weil, Abelian varieties and the Hodge ring. Collected papers, Vol. III, 1977c., 421–429.
J-P. Wintenberger, Théorème de comparaison p-adique pour les schémas abéliens. I. Construction de l’accouplement de périodes. In: Périodes p-adiques (Bures-sur-Yvette, 1988). Astérisque No. 223 (1994), 349–397.
C. Yu, A note on the Mumford-Tate conjecture for CM abelian varieties. Taiwanese J. Math. 19 (2015), no. 4, 1073–1084.
Yu. Zarhin, Endomorphisms of Abelian varieties over fields of finite characteristic. Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 2, 272–277, 471.
Zarhin Yu.: Abelian varieties in characteristic p. Mat. Zametki 19(3), 393–400 (1976)
Yu. Zarhin, Weights of simple Lie algebras in the cohomology of algebraic varieties. Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 2, 264–304. (English translation: Math. USSR-Izv. 24 (1985), no. 2, 245282.)
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Lecture given at the Seminario Matematico e Fisico di Milano on April 6, 2016
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Moonen, B. Families of Motives and the Mumford–Tate Conjecture. Milan J. Math. 85, 257–307 (2017). https://doi.org/10.1007/s00032-017-0273-x
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DOI: https://doi.org/10.1007/s00032-017-0273-x