On the image of the 𝑙-adic Abel-Jacobi map for a variety over the algebraic closure of a finite field

Schoen, Chad

doi:10.1090/S0894-0347-99-00303-3

On the image of the $l$-adic Abel-Jacobi map for a variety over the algebraic closure of a finite field
HTML articles powered by AMS MathViewer

by Chad Schoen

J. Amer. Math. Soc. 12 (1999), 795-838

DOI: https://doi.org/10.1090/S0894-0347-99-00303-3

Published electronically: April 23, 1999

PDF | Request permission

Abstract:

Let $Y$ be a smooth projective variety of dimension at most 4 defined over the algebraic closure of a finite field of characteristic $>2$. It is shown that the Tate conjecture implies the surjectivity of the $l$-adic Abel-Jacobi map, $\mathbf {a}^{r}_{Y,l}:CH^{r}_{hom}(Y)\to H^{2r-1}(Y,\mathbb Z_l (r))\otimes \mathbb Q_l /\mathbb Z_l$, for all $r$ and almost all $l$. For a special class of threefolds the surjectivity of $\mathbf {a}^{2}_{Y,l}$ is proved without assuming any conjectures.

References

J. L. Alperin, Local representation theory, Cambridge Studies in Advanced Mathematics, vol. 11, Cambridge University Press, Cambridge, 1986. Modular representations as an introduction to the local representation theory of finite groups. MR 860771, DOI 10.1017/CBO9780511623592
M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968), 277–291. MR 232018, DOI 10.1007/BF01389777
Arnaud Beauville, Le groupe de monodromie des familles universelles d’hypersurfaces et d’intersections complètes, Complex analysis and algebraic geometry (Göttingen, 1985) Lecture Notes in Math., vol. 1194, Springer, Berlin, 1986, pp. 8–18 (French). MR 855873, DOI 10.1007/BFb0076991
Spencer Bloch and Hélène Esnault, The coniveau filtration and non-divisibility for algebraic cycles, Math. Ann. 304 (1996), no. 2, 303–314. MR 1371769, DOI 10.1007/BF01446296
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956, DOI 10.1007/978-1-4684-9327-6
Joe Buhler, Chad Schoen, and Jaap Top, Cycles, $L$-functions and triple products of elliptic curves, J. Reine Angew. Math. 492 (1997), 93–133. MR 1488066, DOI 10.1515/crll.1997.492.93
J. W. S. Cassels, Rational quadratic forms, London Mathematical Society Monographs, vol. 13, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. MR 522835
Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258, DOI 10.1007/BF02684373
Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252 (French). MR 601520, DOI 10.1007/BF02684780
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Wolfgang Ebeling, An arithmetic characterisation of the symmetric monodromy groups of singularities, Invent. Math. 77 (1984), no. 1, 85–99. MR 751132, DOI 10.1007/BF01389136
Eberhard Freitag and Reinhardt Kiehl, Étale cohomology and the Weil conjecture, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 13, Springer-Verlag, Berlin, 1988. Translated from the German by Betty S. Waterhouse and William C. Waterhouse; With an historical introduction by J. A. Dieudonné. MR 926276, DOI 10.1007/978-3-662-02541-3
William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
William Fulton, Hurwitz schemes and irreducibility of moduli of algebraic curves, Ann. of Math. (2) 90 (1969), 542–575. MR 260752, DOI 10.2307/1970748
Mark L. Green, Griffiths’ infinitesimal invariant and the Abel-Jacobi map, J. Differential Geom. 29 (1989), no. 3, 545–555. MR 992330
Phillip A. Griffiths, On the periods of certain rational integrals. I, II, Ann. of Math. (2) 90 (1969), 460–495; 90 (1969), 496–541. MR 0260733, DOI 10.2307/1970746
Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tohoku Math. J. (2) 9 (1957), 119–221 (French). MR 102537, DOI 10.2748/tmj/1178244839
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773, DOI 10.1007/978-1-4684-9443-3
Uwe Jannsen, Mixed motives and algebraic $K$-theory, Lecture Notes in Mathematics, vol. 1400, Springer-Verlag, Berlin, 1990. With appendices by S. Bloch and C. Schoen. MR 1043451, DOI 10.1007/BFb0085080
Uwe Jannsen, Motives, numerical equivalence, and semi-simplicity, Invent. Math. 107 (1992), no. 3, 447–452. MR 1150598, DOI 10.1007/BF01231898
Nicholas M. Katz and William Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73–77. MR 332791, DOI 10.1007/BF01405203
Steven L. Kleiman, The standard conjectures, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 3–20. MR 1265519, DOI 10.1090/pspum/055.1/1265519
S. L. Kleiman, Algebraic cycles and the Weil conjectures, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1968, pp. 359–386. MR 292838
Klaus Lamotke, The topology of complex projective varieties after S. Lefschetz, Topology 20 (1981), no. 1, 15–51. MR 592569, DOI 10.1016/0040-9383(81)90013-6
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
Minkowski, H., Zur Theorie der positiven quadratischen Formen, J. f. reine u. angewandte Mathe. 101, 196-202 (1887)
Madhav V. Nori, Algebraic cycles and Hodge-theoretic connectivity, Invent. Math. 111 (1993), no. 2, 349–373. MR 1198814, DOI 10.1007/BF01231292
A. P. Ogg, Elliptic curves and wild ramification, Amer. J. Math. 89 (1967), 1–21. MR 207694, DOI 10.2307/2373092
M. S. Raghunathan, Cohomology of arithmetic subgroups of algebraic groups. I, II, Ann. of Math. (2) 86 (1967), 409–424; 87 (1967), 279–304. MR 0227313, DOI 10.2307/1970585
M. S. Raghunathan, Cohomology of arithmetic subgroups of algebraic groups. I, II, Ann. of Math. (2) 86 (1967), 409–424; 87 (1967), 279–304. MR 0227313, DOI 10.2307/1970585
Wayne Raskind, Algebraic $K$-theory, étale cohomology and torsion algebraic cycles, Algebraic $K$-theory and algebraic number theory (Honolulu, HI, 1987) Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989, pp. 311–341. MR 991983, DOI 10.1090/conm/083/991983
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Chad Schoen, On the computation of the cycle class map for nullhomologous cycles over the algebraic closure of a finite field, Ann. Sci. École Norm. Sup. (4) 28 (1995), no. 1, 1–50. MR 1305423, DOI 10.24033/asens.1708
Chad Schoen, On certain modular representations in the cohomology of algebraic curves, J. Algebra 135 (1990), no. 1, 1–18. MR 1076075, DOI 10.1016/0021-8693(90)90147-G
Chad Schoen, Varieties dominated by product varieties, Internat. J. Math. 7 (1996), no. 4, 541–571. MR 1408839, DOI 10.1142/S0129167X9600030X
Chad Schoen, Some examples of torsion in the Griffiths group, Math. Ann. 293 (1992), no. 4, 651–679. MR 1176025, DOI 10.1007/BF01444739
Schoen, C., Complex varieties for which the Chow group mod $n$ is not finite, preprint (1996)
Schoen, C., An integral analog of the Tate Conjecture for one dimensional cycles on varieties over finite fields, Math. Ann. 311, p. 493-500 (1998)
Schoen, C. and Top, J., in preparation
Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380, DOI 10.1007/978-1-4684-9458-7
Jean-Pierre Serre, Groupes algébriques et corps de classes, Publications de l’Institut de Mathématique de l’Université de Nancago, VII, Hermann, Paris, 1959 (French). MR 0103191
Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237, DOI 10.1007/978-1-4757-5673-9
J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French. MR 0344216, DOI 10.1007/978-1-4684-9884-4
Jean-Pierre Serre, Construction de revêtements étales de la droite affine en caractéristique $p$, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 6, 341–346 (French, with English summary). MR 1071640
Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1); Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud. MR 0354651
Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Mathematics, Vol. 340, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II); Dirigé par P. Deligne et N. Katz. MR 0354657
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
C. Soulé, Groupes de Chow et $K$-théorie de variétés sur un corps fini, Math. Ann. 268 (1984), no. 3, 317–345 (French). MR 751733, DOI 10.1007/BF01457062
Tate, J., On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki 306 (1966)
R. O. Wells Jr., Differential analysis on complex manifolds, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0515872